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At what temperature are there Avogadro's number of translational states available for $\mathrm{O}_2$ confined to a volume of 1000. $\mathrm{cm}^3$ ?

The Avogadro number is the approximate number of nucleons (protons and neutrons) in one gram of ordinary matter. The Avogadro constant also relates the molar volume of a substance to the average volume nominally occupied by one of its particles, when both are expressed in the same units of volume. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. The name Avogadro's number was coined in 1909 by the physicist Jean Perrin, who defined it as the number of molecules in exactly 16 grams of oxygen. (The Avogadro number is closely related to the Loschmidt constant, and the two concepts are sometimes confused.) In older literature, the Avogadro number is denoted or , which is the number of particles that are contained in one mole, exactly . The Avogadro constant, commonly denoted or , is a ratio that relates the number of constituent particles (usually molecules, atoms, or ions) in a sample with the amount of substance in that sample. These definitions meant that the value of the Avogadro number depended on the experimentally determined value of the mass (in grams) of one atom of those elements, and therefore it was known only to a limited number of decimal digits. The value of 3R is about 25 joules per kelvin, and Dulong and Petit essentially found that this was the heat capacity of certain solid elements per mole of atoms they contained. Under the new definition, the mass of one mole of any substance (including hydrogen, carbon-12, and oxygen-16) is times the average mass of one of its constituent particles – a physical quantity whose precise value has to be determined experimentally for each substance. == History == === Origin of the concept === right|thumb|Jean Perrin in 1926 The Avogadro constant is named after the Italian scientist Amedeo Avogadro (1776–1856), who, in 1811, first proposed that the volume of a gas (at a given pressure and temperature) is proportional to the number of atoms or molecules regardless of the nature of the gas. The value of the Avogadro constant was chosen so that the mass of one mole of a chemical compound, expressed in grams, is approximately the number of nucleons in one constituent particle of the substance. Thus, the Avogadro constant is the proportionality factor that relates the molar mass of a substance to the average mass of one molecule. This value, the number density of particles in an ideal gas, is now called the Loschmidt constant in his honor, and is related to the Avogadro constant, , by : n_0 = \frac{p_0N_{\rm A}}{R\,T_0}, where is the pressure, is the gas constant, and is the absolute temperature. The numeric value of the Avogadro constant expressed in reciprocal moles, a dimensionless number, is called the Avogadro number. As a consequence of this definition, in the SI system the Avogadro constant had the dimension reciprocal of amount of substance rather than of a pure number, and had the approximate value . thumb|The transformation of one phase from another by the growth of nuclei forming randomly in the parent phase The Avrami equation describes how solids transform from one phase to another at constant temperature. The goal of this definition was to make the mass of a mole of a substance, in grams, be numerically equal to the mass of one molecule relative to the mass of the hydrogen atom; which, because of the law of definite proportions, was the natural unit of atomic mass, and was assumed to be 1/16 of the atomic mass of oxygen. === First measurements === right|thumb|Josef Loschmidt The value of Avogadro's number (not yet known by that name) was first obtained indirectly by Josef Loschmidt in 1865, by estimating the number of particles in a given volume of gas. In general, for uniform nucleation and growth, n = D + 1, where D is the dimensionality of space in which crystallization occurs. == Interpretation of Avrami constants == Originally, n was held to have an integer value between 1 and 4, which reflected the nature of the transformation in question. Perrin himself determined the Avogadro number by several different experimental methods. Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. Given the previous equations, this can be reduced to the more familiar form of the Avrami (JMAK) equation, which gives the fraction of transformed material after a hold time at a given temperature: : Y = 1 - \exp[-K\cdot t^n], where K = \pi\dot{N}\dot{G}^3/3, and n = 4. American Journal of Physics, 78 (4), 412-417 (https://doi.org/10.1119/1.3276053) and bound states in the continuum (red).

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The half-cell potential for the reaction $\mathrm{O}_2(g)+4 \mathrm{H}^{+}(a q)+4 \mathrm{e}^{-} \rightarrow 2 \mathrm{H}_2 \mathrm{O}(l)$ is $+1.03 \mathrm{~V}$ at $298.15 \mathrm{~K}$ when $a_{\mathrm{O}_2}=1.00$. Determine $a_{\mathrm{H}^{+}}$

The values below are standard apparent reduction potentials for electro- biochemical half-reactions measured at 25 °C, 1 atmosphere and a pH of 7 in aqueous solution. For a half cell equation, conventionally written as a reduction reaction (i.e., electrons accepted by an oxidant on the left side): : a \, A + b \, B + h \, \ce{H+} + z \, e^{-} \quad \ce{<=>} \quad c \, C + d \, D The half-cell standard reduction potential E^{\ominus}_\text{red} is given by : E^{\ominus}_\text{red} (\text{volt}) = -\frac{\Delta G^\ominus}{zF} where \Delta G^\ominus is the standard Gibbs free energy change, is the number of electrons involved, and is Faraday's constant. At pH = 7, \+ → P680 ~ +1.0 Half-reaction independent of pH as no is involved in the reaction ==See also== * Nernst equation * Electron bifurcation * Pourbaix diagram * Reduction potential ** Dependency of reduction potential on pH * Standard electrode potential * Standard reduction potential * Standard reduction potential (data page) * Standard state ==References== ==Bibliography== ;Electrochemistry * ;Bio-electrochemistry * * * * * * ;Microbiology * * Category:Biochemistry Standard reduction potentials for half-reactions important in biochemistry Category:Electrochemical potentials Category:Thermodynamics databases Category:Biochemistry databases For standard conditions in electrochemistry (T = 25 °C, P = 1 atm and all concentrations being fixed at 1 mol/L, or 1 M) the standard reduction potential of hydrogen E^{\ominus}_\text{red H+} is fixed at zero by convention as it serves of reference. Definitions must be clearly expressed and carefully controlled, especially if the sources of data are different and arise from different fields (e.g., picking and directly mixing data from classical electrochemistry textbooks (E^{\ominus}_\text{red} versus SHE, pH = 0) and microbiology textbooks (E^{\ominus'}_\text{red} at pH = 7) without paying attention to the conventions on which they are based). ==Example in biochemistry== For example, in a two electrons couple like : the reduction potential becomes ~ 30 mV (or more exactly, 59.16 mV/2 = 29.6 mV) more positive for every power of ten increase in the ratio of the oxidised to the reduced form. ==Some important apparent potentials used in biochemistry== Half-reaction Δ°' (V) E' Physiological conditions References and notes −0.58 Many carboxylic acid: aldehyde redox reactions have a potential near this value 2 + 2 → −0.41 Non-zero value for the hydrogen potential because at pH = 7, [H+] = 10−7 M and not 1 M as in the standard hydrogen electrode (SHE), and that: → NADPH −0.320 −0.370 The ratio of :NADPH is maintained at around 1:50. Solving the Nernst equation for the half-reaction of reduction of two protons into hydrogen gas gives: : : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 0 - \left(0.05916 \ \text{×} \ 7\right) = -0.414 \ V In biochemistry and in biological fluids, at pH = 7, it is thus important to note that the reduction potential of the protons () into hydrogen gas is no longer zero as with the standard hydrogen electrode (SHE) at 1 M (pH = 0) in classical electrochemistry, but that E_\text{red} = -0.414\mathrm V versus the standard hydrogen electrode (SHE). Te (aq) + 2 + 2 (s) + 4 1.02 2 . At pH = 7, when [] = 10−7 M, the reduction potential E_\text{red} of differs from zero because it depends on pH. Is it simply: * E_h = E_\text{red} calculated at pH 7 (with or without corrections for the activity coefficients), * E^{\ominus '}_\text{red}, a formal standard reduction potential including the activity coefficients but no pH calculations, or, is it, * E^{\ominus '}_\text{red apparent at pH 7}, an apparent formal standard reduction potential at pH 7 in given conditions and also depending on the ratio \frac{h} {z} = \frac{\text{(number of involved protons)}} {\text{(number of exchanged electrons)}}. The same also applies for the reduction potential of oxygen: : For , E^{\ominus}_\text{red} = 1.229 V, so, applying the Nernst equation for pH = 7 gives: : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 1.229 - \left(0.05916 \ \text{×} \ 7\right) = 0.815 \ V For obtaining the values of the reduction potential at pH = 7 for the redox reactions relevant for biological systems, the same kind of conversion exercise is done using the corresponding Nernst equation expressed as a function of pH. This is observed for the reduction of O2 into H2O, or OH−, and for reduction of H+ into H2. ==Formal standard reduction potential combined with the pH dependency== To obtain the reduction potential as a function of the measured concentrations of the redox- active species in solution, it is necessary to express the activities as a function of the concentrations. This equation predicts lower E_h at higher pH values. Fumarate + 2 + 2 → Succinate +0.03 +0.30 Formation of hydrogen peroxide from oxygen +0.82 In classical electrochemistry, E° for = +1.23 V with respect to the standard hydrogen electrode (SHE). The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . For the LJTS potential with r_\mathrm{end} = 2.5\,\sigma , the potential energy shift is approximately 1/60 of the dispersion energy at the potential well: V_\mathrm{LJ}(r_\mathrm{end} = 2.5\,\sigma) = -0.0163\,\varepsilon . At chemical equilibrium, the reaction quotient of the product activity (aRed) by the reagent activity (aOx) is equal to the equilibrium constant () of the half-reaction and in the absence of driving force () the potential () also becomes nul. This equation is the equation of a straight line for E_h as a function of pH with a slope of -0.05916\,\left(\frac{h}{z}\right) volt (pH has no units). The figure 8 shows the comparison of the vapor–liquid equilibrium of the 'full' Lennard-Jones potential and the 'Lennard-Jones truncated & shifted' potential. The activity coefficients \gamma_{red} and \gamma_{ox} are included in the formal potential E^{\ominus '}_\text{red}, and because they depend on experimental conditions such as temperature, ionic strength, and pH, E^{\ominus '}_\text{red} cannot be referred as an immuable standard potential but needs to be systematically determined for each specific set of experimental conditions. When the formal potential is measured under standard conditions (i.e. the activity of each dissolved species is 1 mol/L, T = 298.15 K = 25 °C = 77 °F, = 1 bar) it becomes de facto a standard potential. The properties of this ion are strongly related to the surface potential present on a corresponding solid. Without being able to correctly answering these questions, mixing data from different sources without appropriate conversion can lead to errors and confusion. ==Determination of the formal standard reduction potential when 1== The formal standard reduction potential E^{\ominus '}_\text{red} can be defined as the measured reduction potential E_\text{red} of the half-reaction at unity concentration ratio of the oxidized and reduced species (i.e., when 1) under given conditions.

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The partial molar volumes of water and ethanol in a solution with $x_{\mathrm{H}_2 \mathrm{O}}=0.45$ at $25^{\circ} \mathrm{C}$ are 17.0 and $57.5 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$, respectively. Calculate the volume change upon mixing sufficient ethanol with $3.75 \mathrm{~mol}$ of water to give this concentration. The densities of water and ethanol are 0.997 and $0.7893 \mathrm{~g} \mathrm{~cm}^{-3}$, respectively, at this temperature.

It can be calculated from this table that at 25 °C, 45 g of ethanol has volume 57.3 ml, 55 g of water has volume 55.2 ml; these sum to 112.5 ml. Mixing two solutions of alcohol of different strengths usually causes a change in volume. The volume of alcohol in the solution can then be estimated. Whereas to make a 50% v/v ethanol solution, 50 ml of ethanol and 50 ml of water could be mixed but the resulting volume of solution will measure less than 100 ml due to the change of volume on mixing, and will contain a higher concentration of ethanol. This source gives density data for ethanol:water mixes by %weight ethanol in 5% increments and against temperature including at 25 °C, used here. Mixing pure water with a solution less than 24% by mass causes a slight increase in total volume, whereas the mixing of two solutions above 24% causes a decrease in volume. The International Organization of Legal Metrology has tables of density of water–ethanol mixtures at different concentrations and temperatures. For example, to make 100 ml of 50% ABV ethanol solution, water would be added to 50 ml of ethanol to make up exactly 100 ml. Therefore, one can use the following equation to convert between ABV and ABW: \text{ABV} = \text{ABW} \times \frac{\text{density of beverage}}{\text{density of alcohol}} At relatively low ABV, the alcohol percentage by weight is about 4/5 of the ABV (e.g., 3.2% ABW is about 4% ABV). In the wine/water mixing problem, one starts with two barrels, one holding wine and the other an equal volume of water. The volume of the cup is irrelevant, as is any stirring of the mixtures. == Solution == Conservation of substance implies that the volume of wine in the barrel holding mostly water has to be equal to the volume of water in the barrel holding mostly wine. At 0% and 100% ABV is equal to ABW, but at values in between ABV is always higher, up to ~13% higher around 60% ABV. ==See also== *Apparent molar property *Excess molar quantity *Standard drink *Unit of alcohol *Volume fraction == Notes == == References == ==Bibliography== * * * * ==External links== * * Category:Alcohol measurement The following formulas can be used to calculate the volumes of solute (Vsolute) and solvent (Vsolvent) to be used: *Vsolute = Vtotal / F *Vsolvent = Vtotal \- Vsolute , where: *Vtotal is the desired total volume *F is the desired dilution factor number (the number in the position of F if expressed as "1:F dilution factor" or "xF dilution") However, some solutions and mixtures take up slightly less volume than their components. The density of sugar in water is greater than the density of alcohol in water. The phenomenon of volume changes due to mixing dissimilar solutions is called "partial molar volume". thumb|The Mollier enthalpy–entropy diagram for water and steam. However, because of the miscibility of alcohol and water, the conversion factor is not constant but rather depends upon the concentration of alcohol. The number of millilitres of pure ethanol is the mass of the ethanol divided by its density at , which is .Haynes, William M., ed. (2011). Alcohol by volume (abbreviated as ABV, abv, or alc/vol) is a standard measure of how much alcohol (ethanol) is contained in a given volume of an alcoholic beverage (expressed as a volume percent). In some countries, e.g. France, alcohol by volume is often referred to as degrees Gay-Lussac (after the French chemist Joseph Louis Gay-Lussac), although there is a slight difference since the Gay- Lussac convention uses the International Standard Atmosphere value for temperature, . ==Volume change== thumb|upright=1.48|Change in volume with increasing ABV. Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the amount concentrations and as long as the molar attenuation coefficients of the two components, and are known at both wavelengths. A cup of the wine/water mixture is then returned to the wine barrel, so that the volumes in the barrels are again equal.

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If the coefficient of static friction between the block and plane in the previous example is $\mu_s=0.4$, at what angle $\theta$ will the block start sliding if it is initially at rest?

The component of the force of gravity in the direction of the incline is given by: F_g = mg\sin{\theta} The normal force (perpendicular to the surface) is given by: N = mg\cos{\theta} Therefore, since the force of friction opposes the motion of the block, F_k =\mu_k \cdot mg\cos{\theta} To find the coefficient of kinetic friction on an inclined plane, one must find the moment where the force parallel to the plane is equal to the force perpendicular; this occurs when the block is moving at a constant velocity at some angle \theta \sum F = ma = 0 F_k = F_g or \mu_k mg\cos{\theta} = mg\sin{\theta} Here it is found that: \mu_k = \frac{mg\sin{\theta}}{mg\cos{\theta}} = \tan{\theta} where \theta is the angle at which the block begins moving at a constant velocity == References == Category:Classical mechanics The friction force between two surfaces after sliding begins is the product of the coefficient of kinetic friction and the normal force: F_{k} = \mu_\mathrm{k} F_{n}. However, the magnitude of the friction force itself depends on the normal force, and hence on the mass of the block. The maximum possible friction force between two surfaces before sliding begins is the product of the coefficient of static friction and the normal force: F_\text{max} = \mu_\mathrm{s} F_\text{n}. Thus, a force is required to move the back of the contact, and frictional heat is released at the front. thumb|Angle of friction, θ, when block just starts to slide. ===Angle of friction=== For certain applications, it is more useful to define static friction in terms of the maximum angle before which one of the items will begin sliding. Sliding commences only after this frictional force reaches the value F_f = \mu N. For surfaces at rest relative to each other, \mu = \mu_\mathrm{s}, where \mu_\mathrm{s} is the coefficient of static friction. The friction increases as the applied force increases until the block moves. Coefficients of friction range from near zero to greater than one. Prior to sliding, this friction force is F_f = -P_x, where P_x is the horizontal component of the external force. After the block moves, it experiences kinetic friction, which is less than the maximum static friction. It is defined as: \tan{\theta} = \mu_\mathrm{s} and thus: \theta = \arctan{\mu_\mathrm{s}} where \theta is the angle from horizontal and μs is the static coefficient of friction between the objects. If an object is on a level surface and subjected to an external force P tending to cause it to slide, then the normal force between the object and the surface is just N = mg + P_y, where mg is the block's weight and P_y is the downward component of the external force. Sliding friction is almost always less than that of static friction; this is why it is easier to move an object once it starts moving rather than to get the object to begin moving from a rest position. This is called the angle of friction or friction angle. When there is no sliding occurring, the friction force can have any value from zero up to F_\text{max}. For surfaces in relative motion \mu = \mu_\mathrm{k}, where \mu_\mathrm{k} is the coefficient of kinetic friction. In fact, the friction force always satisfies F_f\le \mu N, with equality reached only at a critical ramp angle (given by \tan^{-1}\mu) that is steep enough to initiate sliding. This formula can also be used to calculate μs from empirical measurements of the friction angle. ===Friction at the atomic level=== Determining the forces required to move atoms past each other is a challenge in designing nanomachines. In this case, rather than providing an estimate of the actual frictional force, the Coulomb approximation provides a threshold value for this force, above which motion would commence. The coefficient of friction is an empirical measurementit has to be measured experimentally, and cannot be found through calculations. The coefficient of static friction, typically denoted as μs, is usually higher than the coefficient of kinetic friction.

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Halley's comet, which passed around the sun early in 1986, moves in a highly elliptical orbit with an eccentricity of 0.967 and a period of 76 years. Calculate its minimum distances from the Sun.

The following is a list of comets with a very high eccentricity (generally 0.99 or higher) and a period of over 1,000 years that do not quite have a high enough velocity to escape the Solar System. On 23 March 2147 the comet will pass about from Earth with an uncertainty region of about ±2 million km. C/2001 OG108 (LONEOS) Closest Earth Approach on 2147-Mar-23 11:20 UT Date & time of closest approach Earth distance (AU) Sun distance (AU) Velocity wrt Earth (km/s) Velocity wrt Sun (km/s) Uncertainty region (3-sigma) Reference 2147-03-23 11:20 ± 13:38 40.3 35.3 ± 2 million km Horizons The comet has a rotational period of 2.38 ± 0.02 days (57.12 hr). The comet came to perihelion (closest approach to the Sun) on 15 March 2002. 170P/Christensen is a periodic comet in the Solar System. Perihelion distance at recent epochs Epoch Perihelion (AU) 2028 1.310 2022 1.306 2015 1.349 2008 1.355 Comet Borrelly or Borrelly's Comet (official designation: 19P/Borrelly) is a periodic comet, which was visited by the spacecraft Deep Space 1 in 2001. C/ (LONEOS) is a Halley-type comet with an orbital period of 48.51 years. The comet last came to perihelion (closest approach to the Sun) on February 1, 2022 and will next come to perihelion on December 11, 2028. 1308 Halleria, provisional designation , is a carbonaceous Charis asteroid from the outer regions of the asteroid belt, approximately 43 kilometers in diameter. Using data from Fernandez (2004–2005) JPL lists the comet with an albedo of 0.05 and a diameter of 13.6 ± 1.0 km. 164P/Christensen is a periodic comet in the Solar System. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 164P/Christensen – Seiichi Yoshida @ aerith.net * Elements and Ephemeris for 164P/Christensen – Minor Planet Center Category:Periodic comets 0164 164P 20041221 It came to perihelion in September 2014 at about apparent magnitude 18. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 170P on Seiichi Yoshida's comet list * Elements and Ephemeris for 170P/Christensen – Minor Planet Center Category:Periodic comets 0170 170P 20050617 The actual orbit of these comets significantly differs from the provided coordinates. Of the short-period comets with known diameters and perihelion inside the orbit of Earth, C/ is the second largest after Comet Swift–Tuttle. In 2003, the comet was estimated to have a mean absolute V magnitude (H) of 13.05 ± 0.10, with an albedo of 0.03, giving an effective radius of 8.9 ± 0.7 km. A Solar System barycentric orbit computed at an epoch when the object is located beyond all the planets is a more accurate measurement of its long-term orbit. ==List of near-parabolic comets== Comet designation Comet name /discoverer Semimajor axis (AU) Eccentricity Inclination (°) Perihelion distance (AU) Absolute magnitude (M1/M2) Perihelion date Period (3) (years) Ref C/1680 V1 Great Comet of 1680 444.4285 0.999986 60.6784 0.006222 1680/12/18 9370 C/1769 P1 Messier 163.4554 0.999249 40.7338 0.122755 1769/10/08 2090 C/1785 E1 Méchain 120.6893 0.99646 92.639 0.42724 1785/04/08 1325 C/1807 R1 Great comet of 1807 143.2012 0.995488 63.1762 0.646124 1807/09/19 1710 C/1811 F1 Great Comet of 1811 212.3922 0.995125 106.9342 1.035412 1811/09/12 3100 C/1822 N1 Pons 310.8303 0.996316 127.3429 1.145099 1822/10/24 5480 C/1823 Y1 Great Comet of 1823 170 0.9987 103.68 0.2252 1823/12/09 2300 C/1825 K1 Gambart 246.605 0.996395 123.3414 0.889011 1825/05/31 3870 C/1825 N1 Pons 271.5793 0.995431 146.4353 1.240846 1825/12/11 4480 C/1826 P1 Pons 340.063 0.997492 25.9496 0.852878 1826/10/09 6270 C/1840 B1 Galle 180.8076 0.99325 120.7807 1.220451 1840/03/13 2430 C/1844 N1 Mauvais 3520.1687 0.999757 131.4092 0.855401 1844/10/17 208900 C/1844 Y1 Great Comet of 1844 358.9355 0.999302 45.5651 0.250537 1844/12/14 6800 C/1846 B1 de Vico 194.9063 0.992403 47.4257 1.480703 1846/01/22 2720 C/1847 C1 Hind 473.2556 0.99991 48.6636 0.042593 1847/03/30 10300 C/1847 N1 Mauvais 1251.6357 0.998589 96.5817 1.766058 1847/08/09 44280 C/1849 G1 Schweizer 568.5696 0.998427 66.9587 0.89436 1849/06/08 13560 C/1850 J1 Petersen 771.8979 0.998599 68.1848 1.081429 1850/07/24 21450 C/1854 R1 Klinkerfues 118.2650 0.993246 40.9201 0.798762 1854/10/28 1290 C/1854 Y1 Winnecke-Dien 156.4219 0.991309 14.152 1.359463 1854/12/16 1960 C/1857 Q1 Klinkerfues 182.3447 0.996913 123.9614 0.562898 1857/10/01 2460 C/1858 L1 Donati 156.132 0.996295 116.9512 0.578469 1858/09/30 1950 C/1863 G1 Klinkerfues 1269.962 0.999159 112.6209 1.068038 1863/04/05 45260 C/1863 G2 Respighi 682.7155 0.999079 85.4961 0.628781 1863/04/21 17840 C/1863 V1 Tempel 630.1632 0.998879 78.0817 0.706413 1863/11/09 15820 C/1864 N1 Tempel 249.1888 0.996351 178.1269 0.90929 1864/08/16 3930 C/1864 O1 Donati-Toussaint 1450.486 0.999358 109.7124 0.931212 1864/10/11 55240 C/1871 G1 Winnecke 299.3138 0.997814 87.6034 0.6543 1871/06/11 5180 C/1871 V1 Tempel 161.2851 0.995714 98.2992 0.691268 1871/12/20 2050 C/1873 Q1 Borrelly 225.7138 0.996482 95.9662 0.794061 1873/09/11 3390 C/1873 Q2 Henry 1425.6037 0.99973 121.4625 0.384913 1873/10/02 53830 C/1874 H1 Coggia 572.6966 0.99882 66.3439 0.675782 1874/07/07 13710 C/1874 O1 Borrelly 840.5894 0.998831 41.8266 0.982649 1874/08/27 24370 C/1877 G1 Winnecke 730.7538 0.9987 121.1548 0.94998 1877/04/18 19750 C/1877 G2 Swift 485.8223 0.997923 77.1916 1.009053 1877/04/27 10700 C/1881 K1 Great Comet of 1881 178 0.99589 63.4253 0.734547 1881/06/16 2390 C/1881 W1 Swift 195.8064 0.990169 144.8016 1.924973 1881/11/20 2740 C/1882 F1 Wells 10127.1667 0.999994 73.7977 0.060763 1882/06/11 1019140 C/1887 J1 Barnard 356.7476 0.996093 17.5479 1.393813 1887/06/17 7640 C/1888 D1 Sawerthal 169.3582 0.995874 42.2482 0.698772 1888/03/17 2200 C/1888 P1 Brooks 9806.8043 0.999908 74.1904 0.902226 1888/07/31 971160 C/1888 U1 Barnard 179.494 0.991488 56.3425 1.527853 1888/09/13 2400 C/1889 G1 Barnard 12393.3846 0.999818 163.8517 2.255596 1889/06/11 1379700 C/1889 O1 Davidson 435.2118 0.997611 65.9916 1.039721 1889/07/19 9080 C/1890 O2 Denning 1550.0898 0.999187 98.9373 1.260223 1890/09/25 61030 C/1890 V1 Zona 495.8077 0.995872 154.307 2.046694 1890/08/07 11040 C/1892 E1 Swift 809.1663 0.998731 38.7002 1.026832 1892/04/07 23020 C/1893 N1 Rordame-Quénisset 1249.1648 0.99946 159.9804 0.674549 1893/07/07 44150 C/1893 U1 Brooks 231.2706 0.996489 129.8233 0.811991 1893/09/19 3520 C/1898 V1 Chase 4634.0588 0.999507 22.5046 2.2846 1898/09/20 315460 C/1902 R1 Perrine 12533.5625 0.999968 156.3548 0.4011 1902/11/24 1403170 C/1903 A1 Giacobini 1244.1242 0.99967 30.9416 0.4106 1903/03/16 43880 C/1906 B1 Brooks 1926.3913 0.999327 126.4425 1.296 7.0 1905/12/22 84552.21 C/1907 L2 Daniel 424.6874 0.998794 8.9577 0.512173 1907/09/04 8750 C/1909 L1 Borrelly-Daniel 160.9095 0.994762 52.0803 0.842844 1909/06/05 2040 C/1910 A1 Great January comet of 1910 25795 0.999995 138.7812 0.128975 1910/01/17 4142890 C/1910 P1 Metcalf 9596.1232 0.999797 121.0556 1.948013 1910/09/16 940030 C/1911 N1 Kiess 184 0.9963 148.42 0.68383 1911/06/30 2500 C/1911 O1 Brooks 163.1454 0.997005 16.4153589 0.489429 1911/10/28 2090 C/1911 S2 Quénisset 429.6907 0.998167 108.1 0.787623 1911/11/12 8910 C/1913 J1 Schaumasse 309.1747 0.995288 152.3673 1.456831 1913/05/15 5436 C/1913 R1 Metcalf 555.7869 0.99756 143.3547 1.35612 1913/09/14 13100 C/1914 S1 Campbell? 534.2999 0.998666 77.836 0.712756 1914/08/05 12350 C/1916 G1 Wolf 2834.363 0.999405 25.6592 1.686446 1917/06/17 150900 C/1920 X1 Skjellerup 193.9334 0.994081 22.0303 1.147892 1920/12/11 2700 C/1922 B1 Reid 125.0064 0.986968 32.4456 1.629083 1921/10/28 1400 C/1922 W1 Skjellerup 147.4206 0.993735 23.3659 0.92359 1923/01/04 1790 C/1924 F1 Reid 1252.2789 0.998598 72.3273 1.755695 1924/03/13 44320 C/1925 F2 Reid 334.418 0.995116 26.9797 1.633299 1925/07/29 6120 C/1926 B1 Blathwayt 176.6645 0.992384 128.2986 1.345477 1926/01/02 2350 C/1927 E1 Stearns 2023.0104 0.998179 87.6525 3.683902 1927/03/22 90990 C/1927 X1 Skjellerup-Maristany 1100.9813 0.99984 85.1126 0.176157 1927/12/18 36530 C/1929 Y1 Wilk 691.5998 0.999028 124.5103 0.672235 1930/01/22 18190 C/1930 D1 Peltier-Schwassmann- Wachmann 581.6554 0.998131 99.883 1.087114 1930/11/15 14030 C/1931 P1 Ryves Comet 111.1632 0.999326 169.2881 0.074924 1931/08/25 1180 C/1936 K1 Peltier 133.7226 0.991775 78.5447 1.099868 1936/06/08 1550 C/1937 N1 Finsler 57516 0.999985 146.4156 0.862744 1937/08/15 13793870 C/1939 B1 Kozik-Peltier 146.3133 0.995103 63.5238 0.716496 1939/02/06 1770 C/1939 H1 Jurlof-Achmarof- Hassel 346.8588 0.998477 138.1212 0.528266 1939/04/10 6460 C/1939 V1 Friend 336.6129 0.997192 92.952 0.945209 1939/11/05 6180 C/1941 B2 de Kock- Paraskevopoulos 879.7695 0.999102 168.2039 0.790033 1941/01/27 26100 C/1942 X1 Whipple-Fedtke-Tevzadze 173.4555 0.992196 19.7127 1.353647 1943/02/06 2280 C/1944 H1 Väisälä 370.9009 0.9935 17.2882 2.410856 1945/01/04 7140 C/1947 F1 Rondanina-Bester 217.5667 0.997427 39.3015 0.559799 1947/05/20 3210 C/1947 X1-A Southern Comet of 1947 243.4336 0.999548 138.5419 0.110032 1947/12/02 3800 C/1947 X1-B Southern Comet of 1947 296.558 0.999629 138.5332 0.110023 1947/12/02 5110 C/1948 L1 Honda-Bernasconi 1661.024 0.999875 23.1489 0.207628 1948/05/15 67700 C/1948 N1 Wirtanen 3884.5787 0.999352 130.2675 2.517207 1949/05/01 242110 C/1948 V1 Eclipse Comet of 1948 2083.4 0.999935 23.117 0.135421 1948/10/27 95100 C/1948 W1 Bester 509.1675 0.997499 87.6054 1.273428 1948/10/22 11490 C/1949 N1 Bappu-Bok-Newkirk 1517.8296 0.998644 105.7686 2.058177 1949/10/26 59130 C/1951 P1 Wilson-Harrington 2836.8851 0.999739 152.5337 0.740427 1952/01/12 151100 C/1952 M1 Peltier 4605.0766 0.999739 45.5521 1.201925 1952/06/15 312500 C/1952 Q1 Harrington 407.0848 0.99591 59.1154 1.664977 1953/01/05 8210 C/1953 G1 Mrkos-Honda 391.7716 0.997391 93.8573 1.022132 1953/05/26 7750 C/1955 N1 Bakharev-Macfarlane-Krienke 244.7197 0.994167 50.0329 1.42745 1955/07/11 3830 C/1957 P1 Mrkos 558.9496 0.999365 93.9411 0.354933 1957/08/01 13210 C/1958 D1 Burnham 23205.0702 0.999943 15.7879 1.322689 1958/04/16 3534880 C/1958 R1 Burnham-Slaughter 12150.7313 0.999866 61.2576 1.628198 1959/03/11 1339380 C/1959 X1 Mrkos 4974.0634921 0.999748 19.6339 1.253464 1959/11/13 350810 C/1960 Y1 Candy 105.1101 0.9899 150.9552 1.061612 1961/02/08 1080 C/1961 O1 Wilson 1057.8684 0.999962 24.2116 0.040199 1961/07/17 34410 C/1961 R1 Humason 204.5261 0.989569 153.278 2.133412 1962/12/10 2920 C/1963 F1 Alcock 792.3201 0.99806 86.2194 1.537101 1963/05/05 22300 C/1964 L1 Tomita-Gerber-Honda 123.03 0.995933 161.8323 0.500363 1964/06/30 1360 C/1964 P1 Everhart 361.1342 0.996513 67.9689 1.259275 1964/08/23 6860 C/1965 S1-B Ikeya-Seki 103.7067 0.999925 141.861 0.007778 1965/10/21 1060 C/1966 P1 Kilston 3821.7115 0.999376 40.2648 2.384748 1966/10/28 236260 C/1966 P2 Barbon 1111.0435 0.998183 28.7058 2.018766 1966/04/17 37033 C/1967 Y1 Ikeya-Seki 2000.6851 0.999152 129.3153 1.696581 1968/02/25 89490 C/1968 H1 Tago-Honda-Yamamoto 174 0.9961 102.1698 0.680378 9.8 1968/05/16 2300 C/1968 Y1 Thomas 705.6891 0.995301 45.2291 3.316033 1969/01/12 18750 C/1969 O1-A Kohoutek 1964.6686 0.999125 86.3128 1.719085 1970/03/21 87080 C/1969 T1 Tago-Sato-Kosaka 6400 0.999926 75.81773 0.4726395 6.5 1969/12/21 508060 C/1969 Y1 Bennett 141.21513 0.996193 90.0394 0.537606 1970/03/20 1680 C/1972 E1 Bradfield 494.778 0.998126 123.693 0.927214 1972/03/27 11010 C/1972 F1 Gehrels 1071.8224 0.996943 175.616 3.276561 1971/01/06 35090 C/1972 X1 Araya 54008.3111 0.99991 113.0902 4.860748 1972/12/18 12551360 C/1973 D1 Kohoutek 1082.2388 0.998723 121.5982 1.382019 1973/06/07 35600 C/1974 C1 Bradfield 1660.6964 0.999697 61.2842 0.503191 1974/03/18 67680 C/1974 F1 Lovas 7566.4724 0.999602 50.6485 3.011456 1975/08/22 658170 C/1975 T1 Mori-Sato-Fujikawa 632 0.997461 97.6077 1.603934 5.5 1975/12/25 15880 C/1975 V1-A Comet West 6780.2069 0.999971 43.0664 0.196626 1967/02/25 558300 C/1976 D1 Bradfield 136.9866 0.993811 46.834 0.84781 1976/02/24 1600 C/1976 J1 Harlan 5143.859 0.999695 38.8063 1.568877 1976/11/03 368920 C/1977 R1 Kohler 2170 0.999543 48.71188 0.9905761 7.3 1977/11/10 101000 C/1977 V1 Tsuchinshan 9817.545 0.999633 168.5495 3.603039 1977/06/24 972760 C/1978 T1 Seargent 220 0.99832 67.828 0.36988 1978/09/14 3300 C/1980 V1 Meier 285 0.99468 100.9864 1.51956 7.2 1980/12/09 4820 C/1980 Y1 Bradfield 944.8109 0.999725 138.585 0.259823 1980/12/29 29040 C/1980 Y2 Panther 1640 0.998991 82.64774 1.657269 6.1 1981/01/27 66500 C/1981 H1 Bus 2510.8713 0.999021 160.664 2.458143 1981/07/30 125816 C/1981 M1 Gonzalez 3857.1917 0.999395 107.1467 2.333601 1981/03/25 239560 C/1982 M1 Austin 1072 0.999396 84.4951 0.6478114 8.8 1982/08/24 35100 C/1983 J1 Sugano-Saigusa- Fujikawa 4779.898 0.999901 96.623 0.471 12.3 1983/05/01 330473.13 C/1983 N1 IRAS 4168.9638 0.99942 138.8364 2.417999 1983/05/02 269180 C/1984 N1 Austin 1891.4545 0.999846 164.1533 0.291284 1984/08/12 82260 C/1984 U1 Shoemaker 1145.723 0.995209 179.2123 5.489159 1984/09/03 38780 C/1984 V1 Levy-Rudenko 1160 0.99921 65.7146 0.917949 9.4 1984/12/14 39600 C/1984 W2 Hartley 9501.7435 0.999579 89.3273 4.000234 1985/09/28 926200 C/1985 R1 Hartley-Good 5800 0.999881 79.9294 0.694577 8.4 1985/12/09 450000 C/1986 N1 Churyumov- Solodovnikov 5669.7575 0.999534 114.9293 2.642107 1986/05/06 426920 C/1986 V1 Sorrells 18913.7912 0.999909 160.5801 1.721155 1987/03/09 2601160 C/1987 B1 Nishikawa-Takamizawa-Tago 207 0.9958 172.22989 0.869589 7.4 1987/03/17 2980 C/1987 P1 Bradfield 165.2 0.99474 34.08809 0.868956 6 1987/11/07 2123 C/1987 U3 McNaught 406 0.99792 97.5751 0.84393 6.9 1987/12/02 8200 C/1988 A1 Liller 244.9295 0.996565 73.3224 0.841333 1988/03/31 3830 C/1988 F1 Levy 537.6264 0.997816 62.8074 1.174176 1987/11/29 12470 C/1988 J1 Shoemaker-Holt 541.2286 0.99783 62.8066 1.174466 1988/02/14 12590 C/1989 A1 Yanaka 1410 0.99866 52.4092 1.89458 5.1 1988/10/31 53000 C/1989 A5 Shoemaker 547.6156 0.99518 96.5548 2.639507 1989/02/26 12810 C/1989 T1 Helin-Roman-Alu 112.097 0.990657 46.0369 1.047322 1989/12/15 1190 C/1990 N1 Tsuchiya-Kiuchi 233.2246 0.995316 143.7839 1.092424 1990/09/28 3560 C/1991 A2 Masaru Arai 151.0756 0.990507 70.9783 1.434161 1990/12/10 1860 C/1991 B1 Shoemaker-Levy 348.8963 0.993508 77.2881 2.265035 1991/12/31 6520 C/1991 Q1 McNaught-Russell 589.3992 0.994581 90.5062 3.19395 1992/05/03 14310 C/1991 R1 McNaught-Russell 11160.2875 0.999374 104.5086 6.98634 1990/11/12 1179000 C/1991 T2 Shoemaker-Levy 6000 0.999860 113.49709 0.8362597 7.7 1992/07/24 4650000 C/1992 F1 Tanaka-Machholz 312.7164 0.995966 79.2924 1.261498 1992/04/22 5530 C/1992 J1 Spacewatch 77102.7179 0.999961 124.3187 3.007006 1993/09/05 21409400 C/1992 U1 Shoemaker 3928.1053 0.999411 65.9859 2.313654 1993/03/25 246190 C/1993 Y1 McNaught- Russell 134.8 0.99356 51.5866 0.8676358 12.2 1994/03/31 1564 C/1994 E2 Shoemaker-Levy 431.4296 0.997314 131.2547 1.15882 1994/05/27 8960 C/1994 G1-A Takamizawa-Levy 1549.8632 0.999123 132.8728 1.35923 1994/05/22 61020 C/1994 J2 Takamizawa 545.4374 0.996429 135.9611 1.947757 1994/06/29 12740 C/1994 T1 Machholz 3820.7081 0.999517 101.7379 1.845402 1994/10/02 236170 C/1995 O1 Comet Hale–Bopp 185.86 0.9950817 89.430154 0.9141335 2.3 1997/04/01 2534 C/1995 Q1 Bradfield 220.6208 0.998022 147.3942 0.436388 1995/08/31 3280 C/1996 B1 Szczepanski 156.9 0.99076 51.9189 1.448788 7.1 1996/02/06 1965 C/1996 B2 Comet Hyakutake 2270 0.9998987 124.92266 0.2302293 7.3 1996/05/01 108000 C/1996 Q1 Tabur 800 0.9989 73.359 0.83984 11.0 1996/11/03 22000 C/1996 R1 Hergenrother-Spahr 132 0.9856 145.8144 1.89920 5.8 1996/08/28 1510 C/1996 R3 Lagerkvist 404.015 0.987 39.2 5.24 10.5 1995/07/24 8120.91 Spacewatch 3081 0.998884 72.71704 3.436463 4.9 1999/11/27 171000 C/1997 G2 Montani 529 0.99417 69.83548 3.084966 5.3 1998/04/16 12160 C/1997 J1 Mueller 255.5 0.990991 122.96833 2.302132 8.6 1997/05/03 4085 C/1997 L1 Zhu-Balam 2420 0.99797 72.9914 4.89956 6.5 1996/11/22 119000 C/1997 T1 Utsunomiya 920 0.998523 127.99262 1.3591096 8.0 1997/12/10 27910 C/1998 H1 Stonehouse 710 0.9979 104.693 1.48729 10.0 1998/04/14 19000 C/1998 K2 LINEAR 3210 0.999276 64.45667 2.323479 8.6 1998/09/01 182000 C/1998 K3 LINEAR 1700 0.9979 160.2056 3.5463 10.0 1998/03/07 70000 C/1998 M1 LINEAR 431 0.99277 20.38455 3.11812 5.4 1998/10/28 8950 C/1998 M2 LINEAR 1215 0.997758 60.18232 2.725333 8.5 1998/08/13 42400 C/1998 M4 LINEAR 1100 0.998 154.572 2.6001 9.5 1997/12/10 30000 C/1998 M5 LINEAR 438.3 0.996025 82.22889 1.7422899 8.0 1999/01/24 9176 C/1998 M6 Montani 5400 0.9989 91.540 5.9787 7.5 1998/10/06 400000 C/1998 P1 Williams 1700 0.999325 145.72831 1.146108 8.0 1998/10/17 70000 C/1998 Q1 LINEAR 358 0.99559 32.3058 1.57788 14.0 1998/06/29 6770 C/1998 T1 LINEAR 1657 0.999114 170.15995 1.467728 9.5 1999/06/25 67400 C/1998 U5 LINEAR 102.88 0.987981 131.76474 1.2364530 10.9 1998/12/21 1043.5 C/1999 A1 Tilbrook 177 0.99587 89.481 0.730741 12.0 1999/01/29 2350 C/1999 F1 Catalina (CSS) 6700 0.999136 92.03554 5.787022 4.6 2002/02/13 548000 C/1999 F2 Dalcanton 2640 0.99821 56.42742 4.71807 7.6 1998/08/23 135000 C/1999 H1 Lee 2775 0.9997449 149.35290 0.70810722 9.4 1999/07/11 146200 C/1999 J3 LINEAR 1600 0.99939 101.6561 0.976809 11.3 1999/09/20 64000 C/1999 K2 Ferris 155 0.9658 82.191 5.2903 7.0 1999/04/10 1920 C/1999 K3 LINEAR 235 0.9918 92.274 1.92878 12.0 1999/02/27 3600 C/1999 K6 LINEAR 346.8 0.993532 46.34384 2.246976 11.3 1999/07/24 6459 C/1999 K7 LINEAR 700 0.9966 135.159 2.3227 13.0 1999/02/24 18000 C/1999 L2 LINEAR 390 0.9951 43.942 1.90476 13.0 1999/08/04 7800 C/1999 N2 Lynn 298 0.99745 111.6559 0.7612844 10.3 1999/07/23 5150 C/1999 T1 McNaught-Hartley 8100 0.999856 79.97521 1.1716989 8.6 2000/12/13 740000 C/2000 B2 LINEAR 6000 0.9994 93.647 3.7762 10.3 1999/11/10 500000 LINEAR 1916 0.998353 49.21252 3.155967 7.4 2001/06/19 83900 C/2000 K2 LINEAR 522.0 0.995332 25.63358 2.437066 9.3 2000/10/11 11930 C/2000 Y2 Skiff 490 0.99435 12.0875 2.76871 11.4 2001/03/21 10850 C/2001 A1 LINEAR 266 0.99095 59.941 2.4064 12.7 2000/09/17 4330 C/2001 A2-A LINEAR 2500 0.99969 36.487 0.779054 13 2001/05/24 130000 C/2001 A2-B LINEAR 1119 0.999304 36.47582 0.7790172 7 2001/05/24 37400 C/2001 C1 LINEAR 38000 0.99987 68.96470 5.10432 6.5 2002/03/28 7000000 LINEAR-NEAT 1193.5 0.9976606 163.212126 2.7920832 7.4 2003/07/09 41230 C/2001 K3 Skiff 2870 0.99893 52.0265 3.06012 9.4 2001/04/22 153000 C/2001 K5 LINEAR 11410 0.999546 72.590342 5.184246 4.4 2002/10/11 1220000 C/2001 O2 NEAT 2200 0.9978 90.9262 4.8194 6.6 1999/10/17 103000 C/2001 Q1 NEAT 171.2 0.96593 66.9504 5.83397 7.7 2001/09/20 2241 C/2001 U6 LINEAR 1149 0.99617 107.25550 4.40642 6.5 2002/08/08 39000 C/2001 W1 LINEAR 2100 0.9989 118.645 2.39924 13.7 2001/12/24 100000 C/2001 X1 LINEAR 570 0.99700 115.6268 1.69793 11.3 2002/01/08 13500 C/2002 B2 LINEAR 1400 0.9972 152.8726 3.8422 10.1 2002/04/06 50000 C/2002 C2 LINEAR 9000 0.99964 104.88143 3.25375 9.9 2002/04/10 860000 C/2002 F1 Utsunomiya 950 0.999539 80.8770 0.4382989 10.5 2002/04/22 29300 C/2002 H2 LINEAR 276 0.99407 110.5011 1.63484 10.5 2002/03/23 4570 C/2002 J4 NEAT 29000 0.999874 46.52550 3.633722 8.4 2003/10/03 4900000 C/2002 K1 NEAT 9000 0.9997 89.723 3.23024 11.4 2002/06/16 900000 C/2002 K2 LINEAR 763 0.99314 130.8957 5.23506 8.2 2002/06/05 21100 C/2002 L9 NEAT 4460 0.99842 68.44211 7.03301 4.7 2004/04/05 297000 C/2002 O6 SWAN 350 0.99858 58.6240 0.494648 13.0 2002/09/09 6500 C/2002 P1 NEAT 414 0.98422 34.6061 6.5302 8.2 2001/11/23 8420 C/2002 Q3-A LINEAR 465.333 0.997194 96.87858 1.30583 16.4 2002/08/19 10038.16 C/2002 V1 NEAT 1011 0.9999018 81.70600 0.0992581 10.4 2003/02/18 32100 C/2002 V2 LINEAR 5010 0.99864 166.77622 6.81203 8.4 2003/03/13 355000 LINEAR 202.14 0.966377 70.51612 6.796713 7.1 2006/02/06 2874.1 C/2002 X1 LINEAR 1376 0.998192 164.08943 2.4867001 9.8 2003/07/12 51020 C/2002 X5 Kudo-Fujikawa 1210 0.999843 94.15226 0.189935 10.6 2003/01/29 42000 C/2002 Y1 Juels-Holvorcem 250.6 0.997152 103.78154 0.7138096 9.8 2003/04/13 3967 C/2003 G2 LINEAR 440 0.9965 96.167 1.55337 16.0 2003/04/29 9000 C/2003 H1 LINEAR 2653 0.999156 138.667242 2.2396301 8.7 2004/02/22 136700 C/2003 H3 NEAT 13200 0.999780 42.81171 2.901441 9.6 2003/04/24 1510000 C/2003 J1 NEAT 577 0.99112 98.3135 5.12542 8.8 2003/10/10 13900 C/2003 L2 LINEAR 154.40 0.981446 82.05107 2.864801 9.9 2004/01/19 1918.7 C/2003 T2 LINEAR 6400 0.99972 87.5315 1.786352 9.8 2003/11/14 520000 C/2003 T3 Tabur 5730 0.999742 50.44443 1.4810758 5.8 2004/04/29 434000 C/2003 V1 LINEAR 603 0.99704 28.67513 1.78314 9.9 2003/03/11 14800 C/2004 F2 LINEAR 151.6 0.99056 104.9600 1.43044 13.2 2003/12/26 1870 C/2004 F4 Bradfield 238 0.999294 63.16456 0.168266 11.3 2004/04/17 3680 C/2004 G1 LINEAR 328.47 0.996 114.486 1.201 14.4 2004/06/04 5953.27 C/2004 K1 Catalina (CSS) 1819 0.998131 153.747521 3.399147 7.9 2005/07/05 77600 C/2004 L1 LINEAR 858 0.997615 159.36082 2.04741344 12.6 2005/03/30 25100 C/2004 L2 LINEAR 790 0.995215 62.51864 3.778629 8.3 2005/11/15 22190 C/2004 P1 NEAT 8100 0.99925 28.8163 6.01377 10.1 2003/08/08 720000 C/2004 Q1 Tucker 186.78 0.989042 56.08768 2.0467255 9.8 2004/12/06 2552.8 C/2004 Q2 Comet Machholz 2403 0.9994986 38.588963 1.2050414 9.9 2005/01/24 117800 LINEAR 700 0.997227 21.61823 1.942359 13.9 2005/03/03 18540 C/2004 T3 Siding Spring 5600 0.99842 71.9642 8.8644 6.6 2003/04/15 420000 C/2004 U1 LINEAR 3610 0.999264 130.62532 2.659321 9.0 2004/12/08 217000 C/2004 X2 LINEAR 1450 0.99738 72.118 3.79308 10.2 2004/08/24 55000 LINEAR 13000 0.99987 52.47641 1.781202 17.3 2005/03/03 1500000 C/2005 G1 LINEAR 18800 0.99974 108.41395 4.960798 7.7 2006/02/27 2600000 C/2005 L3 McNaught 13390 0.999582 139.449248 5.593622 6.4 2008/01/16 1550000 C/2005 N1 Juels-Holvorcem 729 0.998457 51.18017 1.125447 11.3 2005/08/22 19700 C/2005 R4 LINEAR 2067 0.99749 164.01260 5.188473 7.7 2006/03/08 94000 C/2005 S4 McNaught 5690 0.998972 107.95897 5.850109 7.9 2007/07/18 430000 C/2005 X1 Beshore 690 0.9958 91.944 2.8623 11.1 2005/07/05 18000 C/2005 YW LINEAR 190.4 0.989534 40.54361 1.9930109 7.4 2006/12/07 2628 C/2006 A1 Pojmański 2370 0.999765 92.73611 0.5553959 10.5 2006/02/22 115000 C/2006 A2 Catalina (CSS) 3800 0.99862 148.3226 5.3160 9.8 2005/05/20 240000 C/2006 B1 McNaught 1340 0.99776 134.28193 2.997591 10.3 2005/11/19 49100 Catalina (CSS) 216.32 0.991900 144.26278 1.7521694 12.3 2006/07/03 3182 C/2006 K4 NEAT 1818 0.998246 111.33346 3.188618 8.8 2007/11/29 77500 C/2006 L1 Garradd 551 0.997345 143.24257 1.462070 8.6 2006/10/18 12930 C/2006 M1 LINEAR 153.67 0.976859 54.87693 3.556199 9.6 2007/02/13 1905.0 C/2006 O2 Garradd 420 0.99634 43.0287 1.55479 12.7 2006/10/05 8700 C/2006 Q1 McNaught 6890 0.9995986 59.050380 2.7637144 7.0 2008/07/03 571000 C/2006 U6 Spacewatch 1931 0.998706 84.87894 2.4983978 8.8 2008/06/05 84900 C/2006 V1 Catalina (CSS) 257.6 0.989618 31.11947 2.674906 9.0 2007/11/26 4136 C/2006 W3 Christensen 17990 0.9998262 127.074692 3.1262325 6.7 2009/07/06 2410000 Lemmon 583.8 0.998987 152.70463 0.5912444 17.4 2007/04/28 14110 LINEAR 252.0 0.992839 30.62941 1.804374 7.4 2007/07/21 4000 C/2007 B2 Skiff 737.2 0.995965 27.49527 2.9749171 8.1 2008/08/20 20020 C/2007 D1 LINEAR 171366.7 0.99995 41.50701 8.793 8.9 2007/06/18 C/2007 D3 LINEAR 652 0.99201 45.92022 5.20897 9.2 2007/05/27 16650 C/2007 E2 Lovejoy 1330 0.99918 95.8830 1.092939 10.9 2007/03/27 49000 C/2007 K1 Lemmon 436 0.97880 108.4325 9.23905 8.6 2007/05/07 9100 C/2007 K6 McNaught 224 0.9847 105.064 3.4330 10.6 2007/07/01 3350 C/2007 M1 McNaught 1564 0.99522 139.72142 7.47465 5.6 2008/08/11 61900 C/2007 M2 Catalina (CSS) 5360 0.999339 80.94565 3.541050 9.0 2008/12/08 392000 C/2007 M3 LINEAR 171.45 0.979768 161.76086 3.468759 9.9 2007/09/04 2245 C/2007 N3 Lulin 72000 0.9999833 178.373611 1.21225837 9.7 2009/01/10 19500000 C/2007 T1 McNaught 4040 0.999760 117.64244 0.9685028 11.1 2007/12/12 256000 Spacewatch 12200 0.999603 86.99476 4.842732 7.1 2010/04/26 1350000 C/2007 Y2 McNaught 1210 0.99652 98.50321 4.20896 9.2 2008/04/08 42100 C/2008 C1 Chen-Gao 101627.54 0.9999876 61.7845 1.262343 11.7 2008/04/16 32398532.38 C/2008 E3 Garradd 3740 0.99852 105.07653 5.53103 5.1 2008/08/02 229000 C/2008 G1 Gibbs 365 0.98908 72.856 3.9898 10.5 2009/01/11 6980 C/2008 J1 Boattini 166.07 0.989617 61.78002 1.7242934 8.8 2008/07/13 2140.1 C/2008 L3 Hill 330 0.9939 100.201 2.0113 10.6 2008/04/22 5900 C/2008 N1 Holmes 973.1 0.997140 115.52100 2.7835117 9.9 2009/09/25 30360 C/2008 Q1 Maticic 593.6 0.995015 118.62662 2.959143 9.8 2008/12/30 14460 C/2008 Q3 Garradd 8900 0.999799 140.70663 1.7982291 6.1 2009/06/23 840000 C/2009 F1 Larson 106 0.9827 171.3755 1.8307 15.1 2009/06/25 1090 C/2009 F2 McNaught 346.1 0.98303 59.36694 5.87503 4.9 2009/11/14 6440 C/2009 F6 Yi-SWAN 512.2 0.997512 85.76481 1.274159 9.7 2009/05/07 11590 C/2009 K2 Catalina (CSS) 1460 0.997776 66.82192 3.246173 11.8 2010/02/07 55800 C/2009 O2 Catalina (CSS) 278.3 0.997501 107.96052 0.6955493 12.3 2010/03/24 4643 C/2009 T1 McNaught 3680 0.99831 89.89396 6.22041 8.5 2009/10/08 223000 C/2009 T3 LINEAR 4300 0.999470 148.74183 2.281140 13.5 2010/01/12 282000 C/2009 U3 Hill 167.88 0.991575 51.26077 1.414424 12.6 2010/03/20 2175 C/2009 U5 Grauer 10600 0.99943 25.4726 6.09424 9.1 2010/06/22 1090000 C/2009 W2 Boattini 16000 0.99956 164.49053 6.90713 6.9 2010/05/01 1900000 C/2009 Y1 Catalina (CSS) 375.4 0.993285 107.31660 2.5204945 6.5 2011/01/28 7273 C/2010 A4 Siding Spring 292.4 0.990638 96.73015 2.737999 7.4 2010/10/08 5001 C/2010 B1 Cardinal 2932 0.998997 101.97777 2.9414900 10.0 2011/02/07 158700 C/2010 D3 WISE 11600 0.99963 76.39488 4.24754 10.0 2010/09/03 1250000 C/2010 E1 Garradd 110.4 0.9759 71.698 2.66219 11.8 2009/11/07 1160 WISE-Garradd 299.3 0.990500 107.62532 2.842764 8.5 2010/11/07 5177 C/2010 G1 Boattini 480 0.9975 78.3870 1.20455 13.1 2010/04/02 10000 C/2010 G3 WISE 2630 0.99814 108.26760 4.90765 8.9 2010/04/11 135000 C/2010 H1 Garradd 8400 0.99967 36.5317 2.74555 12.4 2010/06/18 800000 C/2010 J2 McNaught 6300 0.999460 125.85156 3.386994 10.4 2010/06/03 500000 C/2010 L3 Catalina (CSS) 12800 0.99923 102.63105 9.88290 4.7 2010/11/10 1400000 C/2011 A3 Gibbs 1167 0.997992 26.07435 2.344839 9.7 2011/12/16 39900 C/2011 C1 McNaught 344.1 0.997433 16.82561 0.8833784 12.7 2011/04/18 6380 C/2011 C3 Gibbs 320 0.99527 49.3760 1.51689 14.1 2011/04/07 5700 C/2011 F1 LINEAR 2780 0.999345 56.61904 1.818266 8.3 2013/01/07 146000 C/2011 N2 McNaught 10000 0.9997 33.675 2.5634 6.3 2011/10/18 C/2011 O1 LINEAR 1210 0.996785 76.49889 3.890653 7.2 2012/08/18 42100 C/2011 Q1 PANSTARRS 3300 0.9979 94.8620 6.78009 7.5 2011/06/29 190000 C/2012 A2 LINEAR 978.2 0.996384 125.868509 3.5374738 8.4 2012/11/05 30590 C/2012 C1 McNaught 1274 0.99620 96.27770 4.837975 5.4 2013/02/04 45500 MOSS 139913.5 0.999991 27.74418 1.296092 11.1 2012/09/28 52335655.79 C/2012 E1 Hill 3760 0.99801 122.54208 7.50290 5.7 2011/07/04 231000 C/2012 E3 PANSTARRS 221 0.9827 105.658 3.8274 9.9 2011/05/12 3280 C/2012 F6 Lemmon 487.1 0.9984987 82.60885 0.7312382 5.5 2013/03/24 10750 C/2012 K5 LINEAR 774.4 0.9985256 92.848032 1.14181083 10.5 2012/11/28 21550 C/2012 K6 McNaught 4130 0.999188 135.21497 3.353033 8.8 2013/05/21 265000 C/2012 L1 LINEAR 767.3 0.997051 87.21917 2.262410 11.9 2012/12/25 21250 C/2012 L2 LINEAR 563.4 0.997322 70.98049 1.5085342 9.5 2013/05/09 13370 C/2012 L3 LINEAR 331 0.99079 134.19664 3.04503 9.0 2012/06/12 6020 Palomar 6350 0.99897 25.37958 6.53605 8.8 2015/08/16 505000 C/2012 OP Siding Spring 1054 0.99658 114.82872 3.60707 11.2 2012/12/04 34200 C/2012 S4 PANSTARRS 252223.8 0.999983 126.54131 4.34873 9.2 2013/06/28 126673944.62 C/2012 T4 McNaught 110 0.983 24.092 1.953 12.7 2012/10/10 1200 C/2012 U1 PANSTARRS 12200 0.99957 56.33902 5.26390 8.3 2014/07/04 1350000 C/2012 V1 PANSTARRS 3800 0.99945 157.8399 2.0890 11.5 2013/07/21 230000 C/2012 V2 LINEAR 616.7 0.997641 67.18470 1.4547602 8.4 2013/08/16 15320 C/2012 X1 LINEAR 156.71 0.989803 44.36218 1.597956 5.7 2014/02/21 1962 C/2013 E2 Iwamoto 233.07 0.993936 21.85771 1.413322 10.6 2013/03/09 3558 C/2013 F2 Catalina (CSS) 8400 0.99926 61.74927 6.21785 7.1 2013/04/19 770000 C/2013 F3 McNaught 759 0.99703 85.4445 2.252612 12.3 2013/05/25 20900 C/2013 G5 Catalina (CSS) 2700 0.99965 40.617 0.92894 14.5 2013/09/01 140000 C/2013 G6 Lemmon 387.1 0.994708 124.08435 2.048499 6.8 2013/07/25 7620 C/2013 G7 McNaught 2190 0.99786 105.11012 4.677404 6.2 2014/03/18 102200 C/2013 G8 PANSTARRS 3340 0.99846 27.61506 5.14118 8.4 2013/11/14 193000 C/2013 H1 La Sagra 181.5 0.98542 27.0895 2.64696 6.2 2013/05/19 2445 C/2013 J3 McNaught 1950 0.99795 118.2255 3.98869 5.8 2013/02/22 86000 C/2013 J5 Boattini 10000 0.999 136.011 4.9049 10.0 2012/11/29 C/2013 O3 McNaught 819 0.99612 102.83974 3.18010 11.1 2013/09/09 23400 C/2013 P2 PANSTARRS 2590 0.998904 125.53216 2.834925 11.8 2014/02/17 132000 C/2013 R1 Lovejoy 515.4 0.9984250 64.04094 0.81182562 11.6 2013/12/22 11702 Spacewatch 450.8 0.98707 31.40046 5.83064 6.7 2014/08/17 9570 C/2013 U2 Holvorcem 891 0.99426 43.09366 5.116745 5.3 2014/10/25 26590 C/2013 V5 Oukaimeden 488.1 0.9987183 154.88544 0.6255811 10.8 2014/09/28 10784 C/2013 Y2 PANSTARRS 219.9 0.991275 29.41474 1.919086 9.7 2014/06/13 3262 C/2014 A5 PANSTARRS 152.3 0.96848 31.9046 4.79991 11.6 2014/08/14 1879 C/2014 C3 NEOWISE 108.4 0.98283 151.7843 1.86203 12.0 2014/01/16 1129 C/2014 E2 Jacques 688 0.999035 156.392752 0.6639172 10.4 2014/07/02 18060 C/2014 F1 Hill 3600 0.9990 108.2529 3.49638 10.4 2013/10/04 210000 C/2014 F2 Tenagra 148.20 0.97089 119.06119 4.314460 5.6 2015/01/02 1804 C/2014 G1 PANSTARRS 1000 0.9943 165.6403 5.4685 6.0 2013/11/06 30000 C/2014 H1 Christensen 141 0.9849 99.936 2.1389 14.8 2014/04/15 1700 C/2014 M2 Christensen 980 0.99293 32.4062 6.9085 7.9 2014/07/18 30500 C/2014 M3 Catalina (CSS) 138 0.9824 164.90964 2.43428 12.9 2014/06/21 1630 C/2014 N2 PANSTARRS 4700 0.99954 133.0132 2.184401 12.0 2014/10/08 330000 C/2014 N3 NEOWISE 5800 0.999331 61.63825 3.882231 4.7 2015/03/13 442000 PANSTARRS 20602.48 0.99969 81.3473 6.2444 7.6 2016/12/10 2957246.18 C/2014 Q1 PANSTARRS 1129 0.999721 43.10685 0.314570 9.8 2015/07/06 38000 C/2014 Q2 Lovejoy 579.4 0.9977728 80.301302 1.2903578 7.9 2015/01/30 13946 C/2014 Q6 PANSTARRS 6883 0.999386 49.7968 4.222 6.5 2015/01/06 PANSTARRS 260 0.9913 124.818 2.2233 13.3 2014/07/09 4100 C/2014 R1 Borisov 179.4 0.992501 9.93289 1.345431 9.8 2014/11/19 2403 C/2014 R3 PANSTARRS 14434.53 0.9995 90.84 7.2756 6.3 2016/08/08 1734251.91 C/2014 R4 Gibbs 3200 0.99943 42.4116 1.81797 8.7 2014/10/21 180000 C/2014 S2 PANSTARRS 169.71 0.987622 64.67037 2.100644 5.0 2015/12/09 2210.9 C/2014 U3 Kowalski 1100 0.9976 152.9921 2.5588 12.4 2014/09/03 40000 C/2014 W2 PANSTARRS 1610 0.998341 81.998347 2.6702156 7.9 2016/03/10 64570 C/2014 W8 PANSTARRS 174.518 0.9711 42.111 5.044 10.5 2015/09/08 2305.52 PANSTARRS 902 0.99666 149.7827 3.01028 6.8 2015/04/05 27100 C/2015 C2 SWAN 471 0.99849 94.5013 0.711372 14.9 2015/03/04 10200 PANSTARRS 1484 0.999295 6.25965 1.046217 7.9 2017/05/09 57200 C/2015 F3 SWAN 232 0.99640 73.3865 0.83444 14.2 2015/03/09 3530 C/2015 F4 Jacques 116.37 0.985873 48.70495 1.6439255 11.4 2015/08/10 1255.3 C/2015 J2 PANSTARRS 246.9 0.98250 17.28183 4.32039 10.1 2015/09/08 3880 C/2015 K1 MASTER 180.6 0.98584 29.3817 2.55749 9.1 2014/10/13 2426 C/2015 K2 PANSTARRS 260 0.9944 29.110 1.45527 20.7 2015/06/08 4200 C/2015 M1 PANSTARRS 390 0.9946 57.310 2.0916 15.9 2015/05/15 8000 C/2015 M3 PANSTARRS 133.0 0.97328 65.95107 3.55241 11.5 2015/08/26 1533 C/2015 O1 PANSTARRS 651202.3 0.999994 127.211 3.7296 7.2 2018/02/19 C/2015 R3 PANSTARRS 3400 0.9985 83.6135 4.9033 5.0 2014/02/11 190000 LINEAR 1500 0.99906 11.3925 1.41314 10.8 2016/08/27 58000 C/2015 V3 PANSTARRS 822 0.99485 86.2318 4.23569 6.3 2015/11/24 23600 C/2015 WZ PANSTARRS 193.16 0.992873 134.13494 1.3766377 10.5 2016/04/15 2685 C/2015 Y1 LINEAR 292.5 0.99141 71.2196 2.514080 6.7 2016/05/15 5000 C/2016 A5 PANSTARRS 1200 0.9976 40.319 2.9469 12.8 2015/06/28 43000 C/2016 A6 PANSTARRS 217.53 0.9889 120.92 2.4124 7.8 2015/11/05 3208.44 C/2016 B1 NEOWISE 453 0.99293 50.4644 3.20625 5.9 2016/12/04 9700 C/2016 E2 Kowalski 138.88 0.992 135.95 1.074 19.5 2016/02/06 1636.74 C/2016 J2 Denneau 700 0.998 130.343 1.5184 15.3 2016/04/11 C/2016 KA Catalina (CSS) 6000 0.9990 104.6293 5.4009 8.8 2016/02/01 400000 C/2016 M1 PANSTARRS 1760 0.99875 90.99839 2.21103 8.1 2018/08/10 74000 C/2016 N4 MASTER 5315.30 0.99940 72.5573 3.19912 11.1 2017/09/16 387525 C/2016 N6 PANSTARRS 1600 0.9984 105.8345 2.6699 5.0 2018/07/18 67000 C/2016 P4 PANSTARRS 330 0.9819 29.89 5.888 10.7 2016/10/16 5900 C/2016 Q2 PANSTARRS 5467.19 0.9987 109.409 7.087 8.3 2021/05/10 404254.11 C/2016 R2 PANSTARRS 780 0.9967 58.2134 2.6020 5.1 2018/05/09 22000 C/2016 T1 Matheny 126.10 0.9818 126.095 2.3000 12.1 2017/02/01 1415.98 C/2016 T2 Matheny 101.74 0.98125 81.311 1.9078 13.8 2016/12/29 1026.30 C/2016 T3 PANSTARRS 144 0.9816 22.6727 2.6496 8.1 2017/09/06 1730 PANSTARRS 194.0 0.99531 24.0354 0.910285 18.7 2017/03/07 2700 PANSTARRS 111625.443 0.99992 32.431 9.2164 11.2 2018/02/17 37295204.74 C/2017 D2 Barros 1369.820 0.9982 31.26579 2.48587 11.1 2017/07/14 51000 C/2017 D5 PANSTARRS 112.2883 0.9806 131.03858 2.1672 14.6 2017/01/08 1200 C/2017 E4 Lovejoy 477.669 0.9989 88.1867 0.49357 15.6 2017/04/23 10000 C/2017 E5 Lemmon 388.6996 0.9954 122.6377 1.7829 12.0 2016/06/10 7600 C/2017 G3 PANSTARRS 287.3256 0.99098 159.051 2.59048 14.2 2017/04/15 4900 C/2017 K6 Jacques 1054.174 0.99810 57.2511 2.00279 10.7 2018/01/03 34000 C/2017 M3 PANSTARRS 173.8170 0.9732 77.5073 4.6561 6.2 2017/04/28 2292 C/2017 O1 ASASSN 439.1911 0.99658 39.849 1.4987 10.4 2017/10/14 9200 C/2017 P2 PANSTARRS 1210 0.997967 50.08486 2.461777 9.2 2017/12/06 42100 C/2017 T2 PANSTARRS 5007 0.99968 57.231 1.6151 10.2 2020/05/05 354300 C/2017 T3 ATLAS 1344 0.99939 88.10362 0.82522 11.1 2018/07/19 49280 C/2017 U2 Fuls 8555.39 0.99921 95.4291 6.700 8.8 2017/08/28 C/2017 Y1 PANSTARRS 3791 0.99902 55.2287 3.719 9.3 2017/08/31 234400 C/2017 Y2 PANSTARRS 2502.89 0.99841 124.67 3.957 8.0 2020/08/19 C/2018 A3 ATLAS 487.788 0.99328 139.56 3.277 9.2 2019/01/12 10773 C/2018 E2 Barros 1769.556 0.99778 97.7428 3.92 6.4 2017/12/23 74439 Lemmon 640.788 0.99757 84.694 1.55663 18.2 2018/05/23 16221.08 C/2018 F1 Grauer 322.415 0.9907 46.0706 2.993 13.7 2018/12/14 5789.36 Lemmon 833.97 0.99565 136.66655 3.627 12.2 2019/09/10 24084.39 C/2018 L2 ATLAS 246.853 0.9931 67.4235 1.712 8.1 2018/12/02 3879 C/2018 N1 NEOWISE 693.83 0.9981 159.44 1.307 15.0 2018/08/01 C/2018 R3 Lemmon 1970.10 0.99934 69.7154 1.29 11.3 2019/06/07 87446.17 C/2018 R4 Fuls 311.353 0.99451 11.68371 1.7093 11.8 2018/03/03 5494 C/2018 V4 Africano 214.059 0.98506 69.0028 3.19901 15.7 2019/03/01 3131.89 C/2018 X2 Fitzsimmons 155.971 0.9864 23.06 2.125 6.4 2019/07/08 1947.93 C/2018 Y1 Iwamoto 109.736 0.988 160.4 1.287 12.3 2019/02/07 1149.57 C/2019 B1 Africano 151.97 0.9895 123.36 1.597 14.6 2019/03/19 1873.55 C/2019 D1 Flewelling 137.6571 0.989 34.098 1.5775 11.8 2019/05/11 1615.12 C/2019 H1 NEOWISE 230.7855 0.99201 104.579 1.8448 13.5 2019/04/27 3506.07 C/2019 J2 Palomar 610.07 0.99717 105.138 1.7269 11.6 2019/07/19 ATLAS 355.31 0.99424 148.2972 2.045 15.1 2019/05/31 C/2019 K4 Ye 2370.96 0.9990 105.31 2.2594 12.8 2019/06/16 115449.93 C/2019 K5 Young 150.99 0.9865 15.315 2.035 12.3 2019/06/22 1855.41 C/2019 K8 ATLAS 1440.4914 0.998 93.222 3.195 11.3 2019/07/21 C/2019 N1 ATLAS 13156.57 0.99987 82.424 1.7047 9.0 2020/12/01 C/2019 T3 ATLAS 11484.78 0.99948 121.86 5.9468 6.6 2021/03/02 C/2019 T4 ATLAS 1007.58 0.9958 53.62 4.245 5.6 2022/06/09 31983.74 C/2019 U6 Lemmon 435.36 0.9979 61.0049 0.914 13.3 2020/06/18 9084.03 C/2019 V1 Borisov 3033.17 0.99898 61.8636 3.0968 14.5 2020/07/16 C/2019 Y1 ATLAS 231.099 0.9964 73.347 0.8378 12.4 2020/03/15 3513.22 C/2019 Y4 ATLAS 331.14 0.9992 45.380 0.253 7.9 2020/05/31 6025.89 C/2019 Y4-B ATLAS 665.948 0.99962 45.454 0.2525 15.8 2020/05/31 17185 C/2020 A2 Iwamoto 1070.02 0.9991 120.75 0.978 15.0 2020/01/08 35002.27 C/2020 A3 ATLAS 6807.33 0.9991 146.7 5.767 7.7 2019/06/29 C/2020 B3 Rankin 1919.5 0.99826 20.703 3.3446 14.5 2019/10/19 84101.96 C/2020 F3 NEOWISE 377.32 0.9992 128.937 0.295 12.3 2020/07/03 7329.46 C/2020 F6 PANSTARRS 405.3 0.99134 174.58 3.511 13.2 2020/04/11 8159.58 C/2020 F8 SWAN 6642.61 0.99994 110.80 0.430 11.6 2020/05/27 C/2020 H2 Pruyne 183.596 0.9955 125.04 0.834 19.8 2020/04/27 2487.72 C/2020 H4 Leonard 140.477 0.9933 84.320 0.9383 16.5 2020/08/29 1665.00 C/2020 H5 Robinson 2497.44 0.9963 70.204 9.3500 4.5 2020/12/05 C/2020 H7 Lemmon 1476.6 0.997 135.92 4.42 11.1 2020/06/02 56742.20 C/2020 H8 PANSTARRS 594.908 0.99214 99.65 4.6744 10.4 2020/06/04 14510 C/2020 H11 PANSTARRS Lemmon 10470 0.99927 151.41 7.631 7.4 2020/09/15 1070000 C/2020 J1 SONEAR 9376.42 0.9996 142.305 3.356 7.2 2021/04/18 C/2020 K1 PANSTARRS 3141.03 0.99902 89.646 3.078 5.6 2023/05/09 C/2020 K2 PANSTARRS 8380.85 0.99894 91.0288 8.8762 6.1 2020/08/05 C/2020 K3 Leonard 210.450 0.9924 128.72 1.593 14.8 2020/05/30 3053.03 C/2020 K6 Rankin 2876.55 0.998 103.619 5.8844 8.1 2021/09/11 C/2020 K7 PANSTARRS 108.4 0.9411 32.059 6.3847 7.9 2019/10/30 1128.70 C/2020 M5 ATLAS 4936.2 0.9994 93.223 3.005 6.9 2021/08/19 346814.64 C/2020 N2 ATLAS 108.74 0.9835 161.034 1.746 15.6 2020/08/23 1134.00 C/2020 P3 ATLAS 4910.32 0.9986 61.89 6.812 6.7 2021/04/20 C/2020 R2 ATLAS 398.332 0.9882 53.22 4.693 7.1 2022/02/24 7950.15 C/2020 R6 Rankin 451.023 0.9931 82.83 3.129 7.4 2019/09/10 C/2020 R7 ATLAS 6397.95 0.99953 114.893 2.957 10.7 2022/09/16 C/2020 S3 Erasmus 187.987 0.99788 19.861 0.3985 13.0 2020/12/12 2577.50 C/2020 S4 PANSTARRS 4962.260 0.99932 20.5750 3.3673 7.4 2023/02/09 C/2020 S8 PANSTARRS 271.2715 0.99129 108.517 2.3639 8.1 2021/04/10 4468.01 C/2020 T2 Palomar 323.34 0.99364 27.873 2.055 8.8 2021/07/11 5814.24 C/2020 T5 Lemmon 930.79 0.99797 66.604 1.889 16.2 2020/10/09 28398.06 C/2020 U5 Lemmon 79736.84 0.99995 97.280 3.756 9.7 2022/04/27 C/2020 Y2 ATLAS 1217.888 0.99743 101.281 3.132 6.4 2022/06/17 42502.95 C/2020 Y3 ATLAS 151.18 0.98678 83.097 1.999 14.6 2020/12/03 1858.91 C/2021 A2 NEOWISE 257.26 0.9945 106.978 1.413 14.7 2021/01/22 4126.29 C/2021 A6 PANSTARRS 11846.50 0.99933 75.605 7.929 7.1 2021/05/05 C/2021 A7 NEOWISE 5448.01 0.99964 78.149 1.968 13.5 2021/07/15 C/2021 B2 PANSTARRS 336.015 0.99252 38.094 2.513 4.8 2021/07/15 6159.5 C/2021 C1 Rankin 8030.77 0.99957 143.04 3.481 8.8 2020/12/07 C/2021 C4 ATLAS 4645.31 0.99903 132.84 4.504 6.9 2021/01/17 C/2021 C5 ATLAS 14042.0 0.99977 50.787 3.241 12.0 2023/02/10 C/2021 G2 ATLAS 4011.4 0.99876 48.478 4.976 5.7 2024/09/10 C/2021 N3 PANSTARRS 158.23 0.9640 26.74 5.701 7.1 2020/08/17 1990.4 C/2021 P2 PANSTARRS 2211.11 0.9977 150.02 5.072 5.4 2023/01/21 C/2021 P4 ATLAS 305.44 0.9965 56.31 1.080 8.7 2022/07/30 5338.32 C/2021 Q6 PANSTARRS 10.932 0.9992 161.85 8.716 6.9 2024/03/21 C/2021 R2 PANSTARRS 2265.45 0.9968 134.46 7.312 7.7 2021/12/25 C/2021 R7 PANSTARRS 989.93 0.9943 158.85 5.640 7.5 2021/04/14 C/2021 S3 PANSTARRS 3317.47 0.9996 58.55 1.318 6.8 2024/02/14 C/2021 S4 Tsuchinshan (*CTC) 161.91 0.9583 17.478 6.694 7.0 2023/12/31 2035.2 C/2021 T1 Lemmon 1463.83 0.9979 140.35 3.058 5.7 2021/10/14 56007.2 C/2021 T4 Lemmon 43170 0.9999 160.757 1.482 6.9 2023/07/31 8970000 C/2021 U5 Catalina 216.86 0.9891 39.05 2.363 6.9 2022/01/26 3193.50 C/2021 V1 Rankin 679.40 0.9956 71.441 3.014 15.9 2022/04/30 17709.00 C/2022 A1 Sarneczky 411.49 0.9970 116.51 1.253 19.2 2022/01/31 8347 C/2022 A3 Lemmon - ATLAS 1006.28 0.9963 88.360 3.703 5.3 2023/09/28 31921.61 C/2022 B4 382.80 0.9964 20.043 1.380 21.7 2022/01/29 C/2022 D2 Kowalski 356.11 0.9956 22.655 1.555 14.1 2022/03/27 6720 C/2022 H1 Kowalski 1158.59 0.9934 49.870 7.693 6.3 2024/01/18 C/2022 L1 Catalina 528.47 0.9970 123.468 1.591 13.3 2022/09/28 12150 C/2022 L4 PANSTARRS 254.18 0.9881 141.224 3.015 16.6 2021/12/08 C/2022 P3 ZTF 243.88 0.9894 59.519 2.561 14.5 2022/07/27 3808 C/2022 R2 ATLAS 422.92 0.9985 52.895 0.633 16.6 2022/10/25 8697 C/2022 T1 Lemmon 4655 0.9993 22.543 3.444 5.1 2024/02/17 318000 C/2022 U1 Leonard 5401 0.9992 128.126 4.203 6.7 2025/03/25 C/2022 U4 Bok 3727 0.9992 52.038 2.898 9.7 2023/08/03 227000 C/2022 W2 ATLAS 332.03 0.9906 63.533 3.123 14.1 2023/03/08 6050 C/2022 W3 Leonard 280.85 0.9950 103.560 1.398 14.0 2023/06/22 4707 C/2023 A1 Leonard 232.82 0.9921 94.744 1.835 7.8 2023/03/18 3552 C/2023 A2 SWAN 987.3 0.9990 94.708 0.948 12.5 2023/01/20 31000 C/2023 B2 ATLAS 589.595 0.9970 40.771 1.743 8.7 2023/03/10 14316 C/2023 C2 ATLAS 4219 0.9994 48.319 2.368 7.1 2024/11/16 27400 C/2023 F1 PanSTARRS 221.9 0.9923 131.744 1.708 7.8 2023/06/08 3304 C/2023 H1 PanSTARRS 1611 0.9972 21.777 4.44 13.2 2024/11/28 C/2023 H2 Lemmon 246.6 0.9963 113.75 0.894 10.0 2023/10/29 3872 Comet designation Comet name /discoverer Semimajor axis (AU) Eccentricity Inclination (°) Perihelion distance (AU) Absolute magnitude (H/M1/M2) Perihelion date Period (3) (years) Ref == See also == * List of comets by type * List of Halley-type comets * List of hyperbolic comets * List of long-period comets * List of numbered comets * List of periodic comets near parabolic Its orbit has an eccentricity of 0.01 and an inclination of 6° with respect to the ecliptic. Often, these comets, due to their extreme semimajor axes and eccentricity, will have small orbital interactions with planets and minor planets, most often ending up with the comets fluctuating significantly in their orbital path. Deep Space 1 returned images of the comet's nucleus from 3400 kilometers away. Observations taken in January and February 2002 showed that the "asteroid" had developed a small amount of cometary activity as it approached perihelion. Damocloids have been studied as possible extinct cometary candidates due to the similarity of their orbital parameters with those of Halley-family comets. ==See also== * List of Halley-type comets == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris Category:Halley- type comets Category:Near-Earth comets Category:Damocloids 20010728 This comet probably represents the transition between typical Halley-family/long-period comets and extinct comets. The body's observation arc begins with its official discovery observation in March 1931. == Physical characteristics == Halleria is an assumed carbonaceous C-type asteroid, which agrees with the overall spectral type for members of the Charis family. === Rotation period === Between 2005 and 2011, three rotational lightcurves of Halleria were obtained from photometric observations by Donald Pray, René Roy, and Pierre Antonini ().

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Next, we treat projectile motion in two dimensions, first without considering air resistance. Let the muzzle velocity of the projectile be $v_0$ and the angle of elevation be $\theta$ (Figure 2-7). Calculate the projectile's range.

The Range is maximum when angle \theta = 45°, i.e. \sin 2\theta=1. ==See also== * Atlatl * Ballistics * Gunpowder * Bullet * Impact depth * Kinetic bombardment * Shell (projectile) * Projectile point * Projectile use by animals * Arrow * Dart * Missile * Sling ammunition * Spear * Torpedo * Range of a projectile * Space debris * Trajectory of a projectile ==Notes== ==References== * ==External links== * Open Source Physics computer model * Projectile Motion Applet * Another projectile Motion Applet Category:Ammunition Category:Ballistics Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. There are various calculations for projectiles at a specific angle \theta: 1\. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. Mathematically, it is given as t=U \sin\theta/g where g = acceleration due to gravity (app 9.81 m/s²), U = initial velocity (m/s) and \theta = angle made by the projectile with the horizontal axis. 2\. Range (R): The Range of a projectile is the horizontal distance covered (on the x-axis) by the projectile. The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. In ballistics, the elevation is the angle between the horizontal plane and the axial direction of the barrel of a gun, mortar or heavy artillery. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. The second solution is the useful one for determining the range of the projectile. A projectile is an object that is propelled by the application of an external force and then moves freely under the influence of gravity and air resistance. In physics, a projectile launched with specific initial conditions will have a range. Let tg be any time when the height of the projectile is equal to its initial value. : 0 = v t \sin \theta - \frac{1} {2} g t^2 By factoring: : t = 0 or : t = \frac{2 v \sin \theta} {g} but t = T = time of flight : T = \frac{2 v \sin \theta} {g} The first solution corresponds to when the projectile is first launched. A range table was a list of angles of elevation a particular artillery gun barrel needed to be set to, to strike a target at a particular distance with a projectile of a particular weight using a propellant cartridge of a particular weight. The laser rangefinder and computer-based FCS make guns highly accurate. ==Superelevation== When firing a missile such as a MANPADS at an aircraft target, superelevation is an additional angle of elevation above the angle sighted on which corrects for the effect of gravity on the missile. ==See also== *Altitude (astronomy) *Pitching moment ==References== * Gunnery Instructions, U.S. Navy (1913), Register No. 4090 * Gunnery And Explosives For Artillery Officers (1911) * Fire Control Fundamentals, NAVPERS 91900 (1953), Part C: The Projectile in Flight - Exterior Ballistics * FM 6-40, Tactics, Techniques, and Procedures for Field Artillery Manual Cannon Gunnery (23 April 1996), Chapter 3 - Ballistics; Marine Corps Warfighting Publication No. 3-1.6.19 * FM 23-91, Mortar Gunnery (1 March 2000), Chapter 2 Fundamentals of Mortar Gunnery * Fundamentals of Naval Weapons Systems: Chapter 19 (Weapons and Systems Engineering Department United States Naval Academy) * Naval Ordnance and Gunnery (Vol.1 - Naval Ordnance) NAVPERS 10797-A (1957) * Naval Ordnance and Gunnery (Vol.2 - Fire Control) NAVPERS 10798-A (1957) * Naval Ordnance and Gunnery Category:Ballistics Category:Angle Originally, elevation was a linear measure of how high the gunners had to physically lift the muzzle of a gun up from the gun carriage to compensate for projectile drop and hit targets at a certain distance. ==Until WWI== Though early 20th-century firearms were relatively easy to fire, artillery was not. With advances in the 21st century, it has become easy to determine how much elevation a gun needed to hit a target. thumb|A range of "diabolo" pellets with various nose profiles A pellet is a non-spherical projectile designed to be shot from an air gun, and an airgun that shoots such pellets is commonly known as a pellet gun. Thus, the solution is : t = \frac {v \sin \theta} {g} + \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} Solving for the range once again : d = \frac {v \cos \theta} {g} \left ( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2 g y_0} \right) To maximize the range at any height : \theta = \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} Checking the limit as y_0 approaches 0 : \lim_{y_0 \to 0} \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} = \frac {\pi} {4} ====Angle of impact==== The angle ψ at which the projectile lands is given by: : \tan \psi = \frac {-v_y(t_d)} {v_x(t_d)} = \frac {\sqrt { v^2 \sin^2 \theta + 2 g y_0 }} { v \cos \theta} For maximum range, this results in the following equation: : \tan^2 \psi = \frac { 2 g y_0 + v^2 } { v^2 } = C+1 Rewriting the original solution for θ, we get: : \tan^2 \theta = \frac { 1 - \cos^2 \theta } { \cos^2 \theta } = \frac { v^2 } { 2 g y_0 + v^2 } = \frac { 1 } { C + 1 } Multiplying with the equation for (tan ψ)^2 gives: : \tan^2 \psi \, \tan^2 \theta = \frac { 2 g y_0 + v^2 } { v^2 } \frac { v^2 } { 2 g y_0 + v^2 } = 1 Because of the trigonometric identity : \tan (\theta + \psi) = \frac { \tan \theta + \tan \psi } { 1 - \tan \theta \tan \psi } , this means that θ + ψ must be 90 degrees. == Actual projectile motion == In addition to air resistance, which slows a projectile and reduces its range, many other factors also have to be accounted for when actual projectile motion is considered. === Projectile characteristics === Generally speaking, a projectile with greater volume faces greater air resistance, reducing the range of the projectile.

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Calculate the time needed for a spacecraft to make a Hohmann transfer from Earth to Mars

* Ls 263 (Sol 505): Earth is closest to Mars (Sep 10, 1956). This is close to the modern value of 1/154 (many sources will cite somewhat different values, such as 1/193, because even a difference of only a couple of kilometers in the values of Mars' polar and equatorial radii gives a considerably different result). Solar time is a calculation of the passage of time based on the position of the Sun in the sky. Mars Year 1 is the first year of Martian timekeeping standard developed by Clancy et al. originally for the purposes of working with the cyclical temporal variations of meteorological phenomena of Mars, but later used for general timekeeping on Mars. They occur every 26, 79 and 100 years, and every 1,000 years or so there is an extra 53rd-year transit. ==Conjunctions== Transits of Earth from Mars usually occur in pairs, with one following the other after 79 years; rarely, there are three in the series. Start and End dates of Mars Years were determined for 1607-2141 by Piqueux et al. Earth and Mars dates can be converted in the Mars Climate Database, however, the Mars Years are only rational to apply to events that take place on Mars. The Observatory, 3 (1880), 471 * * SOLEX ==External links== * Transits of Earth on Mars – Fifteen millennium catalog: 5 000 BC – 10 000 AD * JPL HORIZONS System * Near miss of the Earth-moon system (2005-11-07) Earth from Mars Category:Earth Category:Mars However, Mars Year sols may be confused with rover mission times that are also expressed in sols. This short story was first published in the January 1971 issue of Playboy magazine.'Transit Of Earth' by Arthur C. Clarke read by himself, 16 October 2017. ==Dates of transits== Transits of Earth from Mars (grouped by series) November 10, 1595 May 5, 1621 May 8, 1700 November 9, 1800 November 12, 1879 May 8, 1905 May 11, 1984 November 10, 2084 November 15, 2163 May 10, 2189 May 13, 2268 November 13, 2368 May 10, 2394 November 17, 2447 May 13, 2473 May 16, 2552 November 15, 2652 May 13, 2678 ==Grazing and simultaneous transits== Sometimes Earth only grazes the Sun during a transit. A specific time within a day, always using UTC, is specified via a decimal fraction. ==References== ==External links== * Category:Types of year Category:Time in astronomy Year thumb|Transfer orbit from Earth to Mars. The last series ending was in 1211. ==View from Mars== No one has ever seen a transit of Earth from Mars, but the next transit will take place on November 10, 2084. Scientists generally use two sub-units of the Mars Year: * the Solar Longitude (Ls) system: 360 degrees per Mars Year that represent the position of Mars in its orbit around the Sun, or * the Sol system: 668 sols per Mars Year. In astronomy, a Julian year (symbol: a or aj) is a unit of measurement of time defined as exactly 365.25 days of SI seconds each.P. Kenneth Seidelmann, ed., The equivalent on Mars is termed Mars local true solar time (LTST). When the Sun has covered exactly 15 degrees (1/24 of a circle, both angles being measured in a plane perpendicular to Earth's axis), local apparent time is 13:00 exactly; after 15 more degrees it will be 14:00 exactly. * August 25, 2005: at 15:19:32 UTC, MRO was 100 million kilometers from Mars. Mean solar time is the hour angle of the mean Sun plus 12 hours. * January 29, 2006: at 06:59:24 UTC, MRO was 10 million kilometers from Mars. Date Duration in mean solar time February 11 24 hours March 26 24 hours − 18.1 seconds May 14 24 hours June 19 24 hours + 13.1 seconds July 25/26 24 hours September 16 24 hours − 21.3 seconds November 2/3 24 hours December 22 24 hours + 29.9 seconds These lengths will change slightly in a few years and significantly in thousands of years. ==Mean solar time== thumb|right|250px|The equation of time—above the axis a sundial will appear fast relative to a clock showing local mean time, and below the axis a sundial will appear slow. For example, in the next 1000 years, seven days will be dropped from the Gregorian calendar but not from 1000 Julian years, so J3000.0 will be . == Julian calendar distinguished == The Julian year, being a uniform measure of duration, should not be confused with the variable length historical years in the Julian calendar. Also, better measurements have been made by using artificial satellites that have been put into orbit around Mars, including Mariner 9, Viking 1, Viking 2, and Soviet orbiters, and the more recent orbiters that have been sent from the Earth to Mars. ==In science fiction== A science fiction short story published in 1971 by Arthur C. Clarke, called "Transit of Earth", depicts a doomed astronaut on Mars observing the transit in 1984.

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Calculate the maximum height change in the ocean tides caused by the Moon.

File:High tide sun moon same side beginning.png|Spring tide: Sun and Moon on the same side (0°) File:Low tide sun moon 90 degrees.png|Neap tide: Sun and Moon at 90° File:High tide sun moon opposite side.png|Spring tide: Sun and Moon at opposite sides (180°) File:Low tide sun moon 270 degrees.png|Neap tide: Sun and Moon at 270° File:High tide sun moon same side end.png|Spring tide: Sun and Moon at the same side (cycle restarts) === Lunar distance === The changing distance separating the Moon and Earth also affects tide heights. The two high tides a day alternate in maximum heights: lower high (just under three feet), higher high (just over three feet), and again lower high. To calculate the actual water depth, add the charted depth to the published tide height. The alternation in high tide heights becomes smaller, until they are the same (at the lunar equinox, the Moon is above the Equator), then redevelop but with the other polarity, waxing to a maximum difference and then waning again. === Current === The tides' influence on current or flow is much more difficult to analyze, and data is much more difficult to collect. The ocean bathymetry greatly influences the tide's exact time and height at a particular coastal point. Assuming (as a crude approximation) that the deceleration rate has been constant, this would imply that 70 million years ago, day length was on the order of 1% shorter with about 4 more days per year. === Bathymetry === The shape of the shoreline and the ocean floor changes the way that tides propagate, so there is no simple, general rule that predicts the time of high water from the Moon's position in the sky. At each place of interest, the tide heights are therefore measured for a period of time sufficiently long (usually more than a year in the case of a new port not previously studied) to enable the response at each significant tide-generating frequency to be distinguished by analysis, and to extract the tidal constants for a sufficient number of the strongest known components of the astronomical tidal forces to enable practical tide prediction. The daily inequality is not consistent and is generally small when the Moon is over the Equator. === Reference levels === The following reference tide levels can be defined, from the highest level to the lowest: * Highest astronomical tide (HAT) – The highest tide which can be predicted to occur. Furthermore, the coriolis and molecular mixing terms are omitted in the equations above since they are relatively small at the temporal and spatial scale of tides in shallow waters. The graph of Cook Strait's tides shows separately the high water and low water height and time, through November 2007; these are not measured values but instead are calculated from tidal parameters derived from years-old measurements. This causes a variation in the tidal force and theoretical amplitude of about ±18% for the Moon and ±5% for the Sun. In (The Reckoning of Time) of 725 Bede linked semidurnal tides and the phenomenon of varying tidal heights to the Moon and its phases. Using a simple harmonic fitting algorithm with a moving time window of 25 hours, the water level amplitude of different tidal constituents can be found. A tidal height is a scalar quantity and varies smoothly over a wide region. Bede then observes that the height of tides varies over the month. Tides are the rise and fall of sea levels caused by gravitational forces exerted by the Moon and Sun and by Earth's rotation. He goes on to emphasise that in two lunar months (59 days) the Moon circles the Earth 57 times and there are 114 tides. They are however only predictions, the actual time and height of the tide is affected by wind and atmospheric pressure. The difference between the height of a tide at perigean spring tide and the spring tide when the moon is at apogee depends on location but can be large as a foot higher. === Other constituents === These include solar gravitational effects, the obliquity (tilt) of the Earth's Equator and rotational axis, the inclination of the plane of the lunar orbit and the elliptical shape of the Earth's orbit of the Sun. Later the daily tides were explained more precisely by the interaction of the Moon's and the Sun's gravity. The two high waters on a given day are typically not the same height (the daily inequality); these are the higher high water and the lower high water in tide tables. 300px|thumb Tidal range is the difference in height between high tide and low tide.

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A particle of mass $m$ starts at rest on top of a smooth fixed hemisphere of radius $a$. Determine the angle at which the particle leaves the hemisphere.

thumb|upright=1.5|Spherical pendulum: angles and velocities. thumb|150px|right|Equatorial Inertial wave pulse caused patterns of fluid flow inside a steadily-rotating spherical chamber. thumb|upright=1.0|The inscribed angle θ is half of the central angle 2θ that subtends the same arc on the circle. Trajectory of the mass on the sphere can be obtained from the expression for the total energy :E=\underbrace{\left[\frac{1}{2}ml^2\dot\theta^2 + \frac{1}{2}ml^2\sin^2\theta \dot \phi^2\right]}_{T}+\underbrace{ \bigg[-mgl\cos\theta\bigg]}_{V} by noting that the horizontal component of angular momentum L_z = ml^2\sin^2\\!\theta \,\dot\phi is a constant of motion, independent of time. Therefore, angle AOV measures 180° − θ. Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of , where is fixed such that r = l. ==Lagrangian mechanics== Routinely, in order to write down the kinetic T=\tfrac{1}{2}mv^2 and potential V parts of the Lagrangian L=T-V in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. thumb|226px|An epispiral with equation r(θ)=2sec(2θ) The epispiral is a plane curve with polar equation :\ r=a \sec{n\theta}. The angle \theta lies between two circles of latitude, where :E>\frac{1}{2}\frac{L_z^2}{ml^2\sin^2\theta}-mgl\cos\theta. ==See also== *Foucault pendulum *Conical pendulum *Newton's three laws of motion *Pendulum *Pendulum (mathematics) *Routhian mechanics ==References== ==Further reading== * * * * * * * * Category:Pendulums That is a consequence of the rotational symmetry of the system around the vertical axis. == Trajectory == thumb|250x250px|Trajectory of a spherical pendulum. Arrows on this cross section show the direction and strength of flow in the equatorial plane as the sphere continues to rotate clockwise on its axis which shown at left . thumb|250px|right|The Western Hemisphere The Western Hemisphere is the half of the planet Earth that lies west of the Prime Meridian (which crosses Greenwich, London, England) and east of the 180th meridian. Angle BOA is a central angle; call it θ. Its portion lying east of the 180th meridian is the only part of the country lying in the Western Hemisphere. Therefore, : 2 \psi + 180^\circ - \theta = 180^\circ. Subtract : (180^\circ - \theta) from both sides, : 2 \psi = \theta, where θ is the central angle subtending arc AB and ψ is the inscribed angle subtending arc AB. ====Inscribed angles with the center of the circle in their interior==== thumb|Case: Center interior to angle Given a circle whose center is point O, choose three points V, C, and D on the circle. Angle DOC is a central angle, but so are angles DOE and EOC, and : \angle DOC = \angle DOE + \angle EOC. The angle θ does not change as its vertex is moved around on the circle. The last equation shows that angular momentum around the vertical axis, |\mathbf L_z| = l\sin\theta \times ml\sin\theta\,\dot\phi is conserved. Angle DOC is a central angle, but so are angles EOD and EOC, and : \angle DOC = \angle EOC - \angle EOD. In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Similarly, the Euler–Lagrange equation involving the azimuth \phi, : \frac{d}{dt}\frac{\partial}{\partial\dot\phi}L-\frac{\partial}{\partial\phi}L=0 gives : \frac{d}{dt} \left( ml^2\sin^2\theta \cdot \dot{\phi} \right) =0 . This is an elliptic orbit with semi-minor axis = 0 and eccentricity = 1. Combining these results with equation (4) yields : \theta_0 = 2 \psi_2 - 2 \psi_1 therefore, by equation (3), : \theta_0 = 2 \psi_0. thumb|400px|Animated gif of proof of the inscribed angle theorem.

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Consider the first stage of a Saturn $V$ rocket used for the Apollo moon program. The initial mass is $2.8 \times 10^6 \mathrm{~kg}$, and the mass of the first-stage fuel is $2.1 \times 10^6$ kg. Assume a mean thrust of $37 \times 10^6 \mathrm{~N}$. The exhaust velocity is $2600 \mathrm{~m} / \mathrm{s}$. Calculate the final speed of the first stage at burnout.

The S-I was the first stage of the Saturn I rocket used by NASA for the Apollo program. == Design == The S-I stage was powered by eight H-1 rocket engines burning RP-1 fuel with liquid oxygen (LOX) as oxidizer. Studied with the Saturn A-1 in 1959, the Saturn A-2 was deemed more powerful than the Saturn I rocket, consisting of a first stage, which actually flew on the Saturn IB, a second stage which contains four S-3 engines that flew on the Jupiter IRBM and a Centaur high-energy liquid-fueled third stage. == References == * Koelle, Heinz Hermann, Handbook of Astronautical Engineering, McGraw-Hill, New York, 1961. {{Infobox rocket |image = |imsize = |caption = |function = Launch vehicle for Project Horizon and Apollo |manufacturer = |country-origin = United States |height = (w/o payload) |diameter = |mass = gross (to LEO) |stages = |capacities = |family = Saturn |status = Study, not developed |sites = Kennedy Space Center |payloads = |stagedata = }} The Saturn C-2 was the second rocket in the Saturn C series studied from 1959 to 1962. Studied in 1959, the Saturn B-1, was a four-stage concept rocket similar to the Jupiter-C, and consisted of a Saturn IB first stage, a cluster of four Titan I first stages used for a second stage, a S-IV third stage and a Centaur high-energy liquid-fueled fourth stage. *Free return trajectory simulation, Robert A. Braeunig, August 2008 *Encyclopedia Astronautica Saturn C-2 C2 Category:Cancelled space launch vehicles It formed the second stage of the Saturn I and was powered by a cluster of six RL-10A-3 engines. The Army's original design used the S-III stage with two J-2 engines as the second stage; after the Saturn program was transferred to NASA, the second stage was replaced with an S-II second stage using four J-2 engines. The S-IV was the second stage of the Saturn I rocket used by NASA for early flights in the Apollo program. The Saturn C-8 was the largest member of the Saturn series of rockets to be designed. This saved up to 20% of structural weight. ==References== * * Category:Apollo program Category:Rocket stages The S-IV stage was a large LOX/LH2-fueled rocket stage used for the early test flights of the Saturn I rocket. The initial launch of the Saturn I consisted of an active S-I, an inactive S-IV and inactive S-V stage. Further development of the C-2 vehicle was cancelled on 23 June 1961. ==Launch vehicle design== The original Saturn C-2 design (1959-1960) was a four-stage launch vehicle, using an S-I first stage using eight Rocketdyne H-1 engines, later flown on the Saturn I. The design was for a four-stage launch vehicle that could launch 21,500 kg (47,300 lb) to low Earth orbit and send 6,800 kg (14,900 lb) to the Moon via Trans-Lunar Injection. The S-III stage would have been added atop the S-II, to convert the C-2 into the five-stage Saturn C-3. Later, a fifth J-2 engine was added to the S-II stage to be used on the Saturn C-5, which eventually was developed as the Saturn V launch vehicle. During a discussion on the Saturn program, several major problems were brought up: * The adequacy of the Saturn C-1 launch vehicle for the orbital qualification of the complete Apollo spacecraft was in question. Excellent account of the evolution, design, and development of the Saturn launch vehicles. Excellent account of the evolution, design, and development of the Saturn launch vehicles. Excellent account of the evolution, design, and development of the Saturn launch vehicles. While this S-V/Centaur stage would never fly on any Saturn rockets, it would be used on Atlas and Titan launch vehicles. The Saturn C-8 configuration was never taken further than the design process, as it was too large and costly. ==References== *Bilstein, Roger E, Stages to Saturn, US Government Printing Office, 1980. .

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How large should we take $n$ in order to guarantee that the Trapezoidal and Midpoint Rule approximations for $\int_1^2(1 / x) d x$ are accurate to within 0.0001 ?

The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. Therefore the total error is bounded by \text{error} = \int_a^b f(x)\,dx - \frac{b-a}{N} \left[ {f(a) + f(b) \over 2} + \sum_{k=1}^{N-1} f \left( a+k \frac{b-a}{N} \right) \right] = \frac{f(\xi)h^3N}{12}=\frac{f(\xi)(b-a)^3}{12N^2}. === Periodic and peak functions === The trapezoidal rule converges rapidly for periodic functions. As discussed below, it is also possible to place error bounds on the accuracy of the value of a definite integral estimated using a trapezoidal rule. thumb|Illustration of "chained trapezoidal rule" used on an irregularly-spaced partition of [a,b]. == History == A 2016 Science paper reports that the trapezoid rule was in use in Babylon before 50 BCE for integrating the velocity of Jupiter along the ecliptic. == Numerical implementation == === Non-uniform grid === When the grid spacing is non-uniform, one can use the formula \int_{a}^{b} f(x)\, dx \approx \sum_{k=1}^N \frac{f(x_{k-1}) + f(x_k)}{2} \Delta x_k , wherein \Delta x_k = x_{k} - x_{k-1} . === Uniform grid === For a domain discretized into N equally spaced panels, considerable simplification may occur. It follows that \int_{a}^{b} f(x) \, dx \approx (b-a) \cdot \tfrac{1}{2}(f(a)+f(b)). thumb|right|An animation that shows what the trapezoidal rule is and how the error in approximation decreases as the step size decreases The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The error in approximating an integral by Simpson's rule for n = 2 is -\frac{1}{90} h^5f^{(4)}(\xi) = -\frac{(b - a)^5}{2880} f^{(4)}(\xi), where \xi (the Greek letter xi) is some number between a and b. The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result: \text{E} = \int_a^b f(x)\,dx - \frac{b-a}{N} \left[ {f(a) + f(b) \over 2} + \sum_{k=1}^{N-1} f \left( a+k \frac{b-a}{N} \right) \right] There exists a number ξ between a and b, such that \text{E} = -\frac{(b-a)^3}{12N^2} f(\xi) It follows that if the integrand is concave up (and thus has a positive second derivative), then the error is negative and the trapezoidal rule overestimates the true value. Interestingly, in this case the trapezoidal rule often has sharper bounds than Simpson's rule for the same number of function evaluations. == Applicability and alternatives == The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. The evaluation of the full integral of a Gaussian function by trapezoidal rule with 1% accuracy can be made using just 4 points. Namely, composite Simpson's 1/3 rule requires 1.8 times more points to achieve the same accuracy as trapezoidal rule. It follows from the above formulas for the errors of the midpoint and trapezoidal rule that the leading error term vanishes if we take the weighted average \frac{2M + T}{3}. The trapezoidal rule states that the integral on the right- hand side can be approximated as \int_{t_n}^{t_{n+1}} f(t,y(t)) \,\mathrm{d}t \approx \tfrac12 h \Big( f(t_n,y(t_n)) + f(t_{n+1},y(t_{n+1})) \Big). Let \Delta x_k = \Delta x = \frac{b-a}{N} the approximation to the integral becomes \begin{align} \int_{a}^{b} f(x)\, dx &\approx \frac{\Delta x}{2} \sum_{k=1}^{N} \left( f(x_{k-1}) + f(x_{k}) \right) \\\\[1ex] &= \frac{\Delta x}{2} \Biggl( f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + \dotsb + 2f(x_{N-1}) + f(x_N) \Biggr) \\\\[1ex] &= \Delta x \left( \sum_{k=1}^{N-1} f(x_k) + \frac{f(x_N) + f(x_0) }{2} \right). \end{align} ==Error analysis== right|thumb|An animation showing how the trapezoidal rule approximation improves with more strips for an interval with a=2 and b=8. Several techniques can be used to analyze the error, including: #Fourier series #Residue calculus #Euler–Maclaurin summation formula #Polynomial interpolation It is argued that the speed of convergence of the trapezoidal rule reflects and can be used as a definition of classes of smoothness of the functions. === Proof === First suppose that h=\frac{b-a}{N} and a_k=a+(k-1)h. Note since it starts and ends at zero, this approximation yields zero area. alt=Two-piece approximation|thumb|Two-piece alt=Four-piece approximation|thumb|Four-piece alt=Eight-piece approximation|thumb|Eight-piece After trapezoid rule estimates are obtained, Richardson extrapolation is applied. Number of pieces Trapezoid estimates First iteration Second iteration Third iteration (4 MA − LA)/3* (16 MA − LA)/15 (64 MA − LA)/63 1 0 (4×16 − 0)/3 = 21.333... (16×34.667 − 21.333)/15 = 35.556... (64×42.489 − 35.556)/63 = 42.599... 2 16 (4×30 − 16)/3 = 34.666... (16×42 − 34.667)/15 = 42.489... 4 30 (4×39 − 30)/3 = 42 8 39 *MA stands for more accurate, LA stands for less accurate == Example == As an example, the Gaussian function is integrated from 0 to 1, i.e. the error function erf(1) ≈ 0.842700792949715. Simpson's rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval. The approximation becomes more accurate as the resolution of the partition increases (that is, for larger N, all \Delta x_k decrease). Using another approximation (for example, the trapezoidal rule with twice as many points), it is possible to take a suitable weighted average and eliminate another error term. Simpson's 1/3 rule is as follows: \begin{align} \int_a^b f(x)\, dx &\approx \frac{1}{3} h\left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]\\\ &= \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right], \end{align} where h = (b - a)/2 is the step size. (This derivation is essentially a less rigorous version of the quadratic interpolation derivation, where one saves significant calculation effort by guessing the correct functional form.) === Composite Simpson's 1/3 rule === If the interval of integration [a, b] is in some sense "small", then Simpson's rule with n = 2 subintervals will provide an adequate approximation to the exact integral. The result is then obtained by taking the mean of the two formulas. === Simpson's rules in the case of narrow peaks === In the task of estimation of full area of narrow peak-like functions, Simpson's rules are much less efficient than trapezoidal rule.

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Find the length of the cardioid $r=1+\sin \theta$.

The foot of the perpendicular from point O on the tangent is point (r\cos \varphi, r\sin \varphi) with the still unknown distance r to the origin O. Inserting the point into the equation of the tangent yields (r\cos\varphi - 2a)\cos\varphi + r\sin^2\varphi = 2a \quad \rightarrow \quad r = 2a(1 + \cos \varphi) which is the polar equation of a cardioid. For the cardioid r(\varphi) = 2a (1 - \cos\varphi) = 4a \sin^2\left(\tfrac{\varphi}{2}\right) one gets \rho(\varphi) = \cdots = \frac{\left[16a^2\sin^2\frac{\varphi}{2}\right]^\frac{3}{2}} {24a^2 \sin^2\frac{\varphi}{2}} = \frac{8}{3}a\sin\frac{\varphi}{2} \ . }} == Properties == thumb|Chords of a cardioid === Chords through the cusp === ; C1: Chords through the cusp of the cardioid have the same length 4a. Hence the cardioid has the polar representation r(\varphi) = 1 - \cos\varphi and its inverse curve r(\varphi) = \frac{1}{1 - \cos\varphi}, which is a parabola (s. parabola in polar coordinates) with the equation x = \tfrac{1}{2}\left(y^2 - 1\right) in Cartesian coordinates. Their intersection point is x(t) = 2(1 + \cos t)\cos t,\quad y(t) = 2(1 + \cos t)\sin t, which is a point of the cardioid with polar equation r = 2(1 + \cos t). thumb|Cardioid as caustic: light source Z, light ray \vec s, reflected ray \vec r thumb|Cardioid as caustic of a circle with light source (right) on the perimeter === Cardioid as caustic of a circle === The considerations made in the previous section give a proof that the caustic of a circle with light source on the perimeter of the circle is a cardioid. In order to keep the calculations simple, the proof is given for the cardioid with polar representation r = 2(1 \mathbin{\color{red}+} \cos\varphi) (§ Cardioids in different positions). ===== Equation of the tangent of the cardioid with polar representation r = 2(1 + \cos\varphi) ===== From the parametric representation \begin{align} x(\varphi) &= 2(1 + \cos\varphi) \cos \varphi, \\\ y(\varphi) &= 2(1 + \cos\varphi) \sin \varphi \end{align} one gets the normal vector \vec n = \left(\dot y , -\dot x\right)^\mathsf{T}. For the cardioids with the equations r=2a(1-\cos\varphi) \; and r = 2b(1 + \cos\varphi)\ respectively one gets: \frac{dy_a}{dx} = \frac{\cos(\varphi) - \cos(2\varphi)}{\sin(2\varphi) - \sin(\varphi)} and \frac{dy_b}{dx} = -\frac{\cos(\varphi) + \cos(2\varphi)}{\sin(2\varphi) + \sin(\varphi)}\ . These equations describe a cardioid a third as large, rotated 180 degrees and shifted along the x-axis by -\tfrac{4}{3} a. The catacaustic of a circle with respect to a point on the circumference is a cardioid. thumb|upright=1.25|r=\frac{\sin \theta}{\theta}, -20<\theta<20 thumb|upright=1.25|cochleoid (solid) and its polar inverse (dashed) In geometry, a cochleoid is a snail-shaped curve similar to a strophoid which can be represented by the polar equation :r=\frac{a \sin \theta}{\theta}, the Cartesian equation :(x^2+y^2)\arctan\frac{y}{x}=ay, or the parametric equations :x=\frac{a\sin t\cos t}{t}, \quad y=\frac{a\sin^2 t}{t}. For suitable formulas see polar coordinate system (arc length) and polar coordinate system (area) \; d\varphi = \cdots = 8a\int_0^\pi\sqrt{\tfrac{1}{2}(1 - \cos\varphi)}\; d\varphi = 8a\int_0^\pi\sin\left(\tfrac{\varphi}{2}\right) d\varphi = 16a. }} {r(\varphi)^2 + 2 \dot r(\varphi)^2 - r(\varphi) \ddot r(\varphi)} \ . thumb|A cardioid In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. For a cardioid one gets: : The evolute of a cardioid is another cardioid, one third as large, and facing the opposite direction (s. picture). === Proof === For the cardioid with parametric representation x(\varphi) = 2a (1 - \cos\varphi)\cos\varphi = 4a \sin^2\tfrac{\varphi}{2}\cos\varphi\, , y(\varphi) = 2a (1 - \cos\varphi)\sin\varphi = 4a \sin^2\tfrac{\varphi}{2}\sin\varphi the unit normal is \vec n(\varphi) = (-\sin\tfrac{3}{2}\varphi, \cos\tfrac{3}{2}\varphi) and the radius of curvature \rho(\varphi) = \tfrac{8}{3}a\sin\tfrac{\varphi}{2} \, . From here one gets the parametric representation above: \begin{array}{cclcccc} x(\varphi) &=& a\;(-\cos(2\varphi) + 2\cos\varphi - 1) &=& 2a(1 - \cos\varphi)\cdot\cos\varphi & & \\\ y(\varphi) &=& a\;(-\sin(2\varphi) + 2\sin\varphi) &=& 2a(1 - \cos\varphi)\cdot\sin\varphi &.& \end{array} (The trigonometric identities e^{i\varphi} = \cos\varphi + i\sin\varphi, \ (\cos\varphi)^2 + (\sin\varphi)^2 = 1, \cos(2\varphi) = (\cos\varphi)^2 - (\sin\varphi)^2, and \sin (2\varphi) = 2\sin\varphi\cos\varphi were used.) == Metric properties == For the cardioid as defined above the following formulas hold: * area A = 6\pi a^2, * arc length L = 16 a and * radius of curvature \rho(\varphi) = \tfrac{8}{3}a\sin\tfrac{\varphi}{2} \, . The reflected ray is part of the line with equation (see previous section) \cos\left(\tfrac{3}{2}\varphi\right) x + \sin \left(\tfrac{3}{2}\varphi\right) y = 4 \left(\cos\tfrac{1}{2}\varphi\right)^3 \, , which is tangent of the cardioid with polar equation r = 2(1 + \cos\varphi) from the previous section.}} For cardioids the following is true: : The orthogonal trajectories of the pencil of cardioids with equations r=2a(1-\cos\varphi)\ , \; a>0 \ , \ are the cardioids with equations r=2b(1+\cos\varphi)\ , \; b>0 \ . Hence any secant line of the circle, defined above, is a tangent of the cardioid, too: : The cardioid is the envelope of the chords of a circle. (Trigonometric formulae were used: \sin\tfrac{3}{2}\varphi = \sin\tfrac{\varphi}{2}\cos\varphi + \cos\tfrac{\varphi}{2}\sin\varphi\ ,\ \cos\tfrac{3}{2}\varphi = \cdots, \ \sin\varphi = 2\sin\tfrac{\varphi}{2}\cos\tfrac{\varphi}{2}, \ \cos\varphi= \cdots \ . ) == Orthogonal trajectories == 300px|thumb|Orthogonal cardioids An orthogonal trajectory of a pencil of curves is a curve which intersects any curve of the pencil orthogonally. Remark: If point O is not on the perimeter of the circle k, one gets a limaçon of Pascal. == The evolute of a cardioid == thumb| The evolute of a curve is the locus of centers of curvature. thumb|upright=1.0|The inscribed angle θ is half of the central angle 2θ that subtends the same arc on the circle. The proofs of these statement use in both cases the polar representation of the cardioid. # The envelope of these chords is a cardioid. thumb|Cremona's generation of a cardioid ==== Proof ==== The following consideration uses trigonometric formulae for \cos\alpha + \cos\beta, \sin\alpha + \sin\beta, 1 + \cos 2\alpha , \cos 2\alpha, and \sin 2\alpha. Hence a cardioid is a special pedal curve of a circle. ==== Proof ==== In a Cartesian coordinate system circle k may have midpoint (2a,0) and radius 2a.

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Estimate the volume of the solid that lies above the square $R=[0,2] \times[0,2]$ and below the elliptic paraboloid $z=16-x^2-2 y^2$. Divide $R$ into four equal squares and choose the sample point to be the upper right corner of each square $R_{i j}$.

The volume of a tricylinder is :V=8(2 - \sqrt{2}) r^3 and the surface area is :A=24(2 - \sqrt{2}) r^2. == More cylinders == With four cylinders, with axes connecting the vertices of a tetrahedron to the corresponding points on the other side of the solid, the volume is :V_4=12 \left( 2\sqrt{2} - \sqrt{6} \right) r^3 \, With six cylinders, with axes parallel to the diagonals of the faces of a cube, the volume is: :V_6=\frac{16}{3} \left( 3 + 2\sqrt{3} - 4\sqrt{2} \right) r^3 \, == See also == * Ungula == References == == External links == *A 3D model of Steinmetz solid in Google 3D Warehouse Category:Euclidean solid geometry For one half of a bicylinder a similar statement is true: * The relations of the volumes of the inscribed square pyramid (a=2r, h=r, V=\frac{4}{3}r^3), the half bicylinder (V=\frac{8}{3} r^3) and the surrounding squared cuboid ( a= 2r, h=r, V=4r^3) are 1 : 2 : 3. ==== Using Multivariable Calculus ==== Consider the equations of the cylinders: x^2+z^2=r^2 x^2+y^2=r^2 The volume will be given by: V = \iiint_V \mathrm{d}z\mathrm{d}y\mathrm{d}x With the limits of integration: -\sqrt{r^2-x^2} \leqslant z \leqslant \sqrt{r^2-x^2} -\sqrt{r^2-x^2} \leqslant y \leqslant \sqrt{r^2-x^2} -r \leqslant x \leqslant r Substituting, we have: V = \int_{-r}^{r}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}} \mathrm{d}z\mathrm{d}y\mathrm{d}x = 8r^3-\frac{8r^3}{3} = \frac{16r^3}{3} === Proof of the area formula === The surface area consists of two red and two blue cylindrical biangles. Hence the area of this development is thumb|cloister vault :B = \int_{0}^{\pi r} r\sin\left(\frac{\xi}{r}\right) \mathrm{d}\xi = 2r^2 and the total surface area is: :A=8\cdot B=16r^2. === Alternate proof of the volume formula === Deriving the volume of a bicylinder (white) can be done by packing it in a cube (red). This leads to :V = \int_{-r}^{r} (2x)^2 \mathrm{d}z = 4\cdot \int_{-r}^{r} x^2 \mathrm{d}z = 4\cdot \int_{-r}^{r} (r^2-z^2) \mathrm{d}z=\frac{16}{3} r^3. By the Pythagorean theorem, the radius of the cylinder is thumb|upright=1.2|Finding the measurements of the ring that is the horizontal cross-section. \sqrt{R^2 - \left(\frac{h}{2}\right)^2},\qquad\qquad(1) and the radius of the horizontal cross-section of the sphere at height y above the "equator" is \sqrt{R^2 - y^2}.\qquad\qquad(2) The cross-section of the band with the plane at height y is the region inside the larger circle of radius given by (2) and outside the smaller circle of radius given by (1). The volume and the surface area of the curved triangles can be determined by similar considerations as it is done for the bicylinder above. However, the same problem had been solved earlier, by Archimedes in the ancient Greek world, Zu Chongzhi in ancient China, and Piero della Francesca in the early Italian Renaissance. thumb|right|150px|Animated depiction of a bicylinder == Bicylinder == 300px|thumb|The generation of a bicylinder 180px|thumb|Calculating the volume of a bicylinder A bicylinder generated by two cylinders with radius r has the ;volume :V=\frac{16}{3} r^3 and the ;surface area :A=16 r^2. The key for the determination of volume and surface area is the observation that the tricylinder can be resampled by the cube with the vertices where 3 edges meet (s. diagram) and 6 curved pyramids (the triangles are parts of cylinder surfaces). Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. Their intercepts with the dashed lines show that when the volume increases 8 (2³) times, the surface area increases 4 (2²) times. In the end we find the volume is cubic units. ==See also== *Solid of revolution *Disc integration ==References== * *Frank Ayres, Elliott Mendelson. The volume of the band is : \int_{-h/2}^{h/2} (\text{area of cross-section at height }y) \, dy, and that does not depend on R. File:Sphere volume derivation using bicylinder.jpg|Zu Chongzhi's method (similar to Cavalieri's principle) for calculating a sphere's volume includes calculating the volume of a bicylinder. After integrating these two functions with the disk method we would subtract them to yield the desired volume. The cross-section's area is therefore the area of the larger circle minus the area of the smaller circle: \begin{align} & {}\quad \pi(\text{larger radius})^2 - \pi(\text{smaller radius})^2 \\\ & = \pi\left(\sqrt{R^2 - y^2}\right)^2 - \pi\left(\sqrt{R^2 - \left(\frac{h}{2}\right)^2\,{}}\,\right)^2 = \pi\left(\left(\frac{h}{2}\right)^2 - y^2\right). \end{align} The radius R does not appear in the last quantity. thumb|Steinmetz solid (intersection of two cylinders) In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r1, r2, r3 and the center of mass R. The volume of the 8 pyramids is: \textstyle 8 \times \frac{1}{3} r^2 \times r = \frac{8}{3} r^3 , and then we can calculate that the bicylinder volume is \textstyle (2 r)^3 - \frac{8}{3} r^3 = \frac{16}{3} r^3 == Tricylinder == 450px|thumb|Generating the surface of a tricylinder: At first two cylinders (red, blue) are cut. In this case the volume of the band is the volume of the whole sphere, which matches the formula given above. thumb|300px|Graphs of surface area, A against volume, V of the Platonic solids and a sphere, showing that the surface area decreases for rounder shapes, and the surface-area-to-volume ratio decreases with increasing volume. A problem on page 101 describes the shape formed by a sphere with a cylinder removed as a "napkin ring" and asks for a proof that the volume is the same as that of a sphere with diameter equal to the length of the hole.

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E

Find the average value of the function $f(x)=1+x^2$ on the interval $[-1,2]$.

In mathematics and its applications, the mean square is normally defined as the arithmetic mean of the squares of a set of numbers or of a random variable. A more general method for defining an average takes any function g(x1, x2, ..., xn) of a list of arguments that is continuous, strictly increasing in each argument, and symmetric (invariant under permutation of the arguments). The average y is then the value that, when replacing each member of the list, results in the same function value: . It may also be defined as the arithmetic mean of the squares of the deviations between a set of numbers and a reference value (e.g., may be a mean or an assumed mean of the data), in which case it may be known as mean square deviation. In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by: : \bar{f}=\frac{1}{b-a}\int_a^bf(x)\,dx. The function provides the arithmetic mean. For example, the average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. The type of calculations used in adjusting general average gave rise to the use of "average" to mean "arithmetic mean". In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). thumb|right|A graph of the function f(x) = e^{-x^2} and the area between it and the x-axis, (i.e. the entire real line) which is equal to \sqrt{\pi}. (If there are an even number of numbers, the mean of the middle two is taken.) That is, \int_{-\infty}^{\infty} e^{-x^2} \, dx = 2\int_{0}^{\infty} e^{-x^2}\,dx. For this reason, it is recommended to avoid using the word "average" when discussing measures of central tendency. ==General properties== If all numbers in a list are the same number, then their average is also equal to this number. If exactly one value is left, it is the median; if two values, the median is the arithmetic mean of these two. A typical estimate for the sample variance from a set of sample values x_i uses a divisor of the number of values minus one, n-1, rather than n as in a simple quadratic mean, and this is still called the "mean square" (e.g. in analysis of variance): :s^2=\textstyle\frac{1}{n-1}\sum(x_i-\bar{x})^2 The second moment of a random variable, E(X^{2}) is also called the mean square. There is also a harmonic average of functions and a quadratic average (or root mean square) of functions. ==See also== *Mean Category:Means Category:Calculus ==References== By analogy, a defining property of the average value \bar{f} of a function over the interval [a,b] is that : \int_a^b\bar{f}\,dx = \int_a^bf(x)\,dx In other words, \bar{f} is the constant value which when integrated over [a,b] equals the result of integrating f(x) over [a,b]. The square root of a mean square is known as the root mean square (RMS or rms), and can be used as an estimate of the standard deviation of a random variable. ==References== Category:Means In mathematics, the mean value problem was posed by Stephen Smale in 1981. Most types of average, however, satisfy permutation- insensitivity: all items count equally in determining their average value and their positions in the list are irrelevant; the average of (1, 2, 3, 4, 6) is the same as that of (3, 2, 6, 4, 1). ==Pythagorean means== The arithmetic mean, the geometric mean and the harmonic mean are known collectively as the Pythagorean means. ==Statistical location== The mode, the median, and the mid- range are often used in addition to the mean as estimates of central tendency in descriptive statistics. Depending on the context, an average might be another statistic such as the median, or mode.

0.3359

5.51

0.7812

210

2

E

Find the area of the region enclosed by the parabolas $y=x^2$ and $y=2 x-x^2$

thumb|right|200px|Two-dimensional plot (red curve) of the algebraic equation y = x^2 - x - 2. Adding the two equations together to get: : 8x = 16 which simplifies to : x = 2. The area under the curve decreases monotonically with increasing p. == Generalization == A natural generalization for the superparabola is to relax the constraint on the power of x. thumb|right|384px|In green, confocal parabolae opening upwards, 2y = \frac {x^2}{\sigma^2}-\sigma^2 In red, confocal parabolae opening downwards, 2y =-\frac{x^2}{\tau^2}+\tau^2 Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. Using the second equation: : 2x - y = 1 Subtracting 2x from each side of the equation: : \begin{align}2x - 2x - y & = 1 - 2x \\\ \- y & = 1 - 2x \end{align} and multiplying by −1: : y = 2x - 1. In mathematics, the definite integral :\int_a^b f(x)\, dx is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total. Using the method of exhaustion, it follows that the total area of the parabolic segment is given by :\text{Area}\;=\;T \,+\, \frac14T \,+\, \frac1{4^2}T \,+\, \frac1{4^3}T \,+\, \cdots. Archimedes provides the first attested solution to this problem by focusing specifically on the area bounded by a parabola and a chord. thumb|400x300px|Superparabola functions A superparabola is a geometric curve defined in the Cartesian coordinate system as a set of points with :\frac{y}{b} = \lbrack1-\left(\frac{x}{a}\right)^2\rbrack^p, where , , and are positive integers. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. Divide both sides by 2: \frac{2x}{2} = \frac{8}{2} 5\. This simplifies to: 2x = 8 4\. When the center of gravity of the triangle is known, the equilibrium of the lever yields the area of the parabola in terms of the area of the triangle which has the same base and equal height. The superparabola can vary in shape from a rectangular function , to a semi- ellipse (, to a parabola , to a pulse function . == Mathematical properties == thumb|400x300px| Without loss of generality we can consider the canonical form of the superparabola :f(x;p)=\left(1-x^2 \right)^p When , the function describes a continuous differentiable curve on the plane. Area function may refer to: *Inverse hyperbolic function *Antiderivative Here, however, we have the analytic solution for the area under the curve. The foci of all these parabolae are located at the origin. An interesting property is that any superparabola raised to a power n is just another superparabola; thus :\int_{-1}^{1}f^n (x) = \psi(n p) The centroid of the area under the curve is given by :C = \frac{\mathbf {i}}{A} \int_{-1}^{1} x\int_{0}^{f(x)} dydx + \frac{\mathbf {j}}{A}\ \int_{-1}^{1} \int_{0}^{f(x)}y dy dx :=\frac{\mathbf{j}}{2A}\int_{-1}^{1} f^2 (x) dx =\mathbf{j}\frac{\psi (2p)}{2\psi(p)} where the x-component is zero by virtue of symmetry. He then computes the sum of the resulting geometric series, and proves that this is the area of the parabolic segment. The indefinite and definite integrals are given by :\int f(x)dx=x \cdot_{2}F_{1} (-p, 1/q; 1+ 1/q ; x^2) :\text{Area}=\int _{-1}^{1}f(x)dx=\Psi (p,q) where \Psi is a universal function valid for all q and p>-1. Because of this a quadratic equation must contain the term ax^2, which is known as the quadratic term. The curve can be described parametrically on the complex plane as :z=\sin(u)+i\cos^{2p}(u);\quad-\tfrac{\pi}{2}\leq u\leq\tfrac{\pi}{2} Derivatives of the superparabola are given by :f'(x;p)=-2px(1-x^2)^{p-1} :\frac{\partial f}{\partial p} = (1-x^2)^p\ln(1-x^2) = f(x)\ln\lbrack f(x; 1)\rbrack The area under the curve is given by :\text{Area} = \int_{-1}^{1}\int_{0}^{f(x)}dydx = \int_{-1}^{1} (1-x^2)^p dx = \psi(p) where is a global function valid for all , :\psi( p)=\frac {\sqrt{\pi}\, \Gamma(p+1)}{\Gamma(p+\frac{3}{2})} The area under a portion of the curve requires the indefinite integral : \int (1-x^2)^p dx = x\,{_2}F{_1} (1/2, -p; 3/2; x^2) where _2F_1 is the Gaussian hypergeometric function.

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117

0.333333333333333

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C

The region $\mathscr{R}$ enclosed by the curves $y=x$ and $y=x^2$ is rotated about the $x$-axis. Find the volume of the resulting solid.

For example, the next figure shows the rotation along the -axis of the red "leaf" enclosed between the square- root and quadratic curves: thumb|Rotation about x-axis The volume of this solid is: :\pi\int_0^1\left(\left(\sqrt{x}\right)^2 - \left(x^2\right)^2 \right)\,dx\,. The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by :V = \pi \int_a^b \left| f(y)^2 - g(y)^2\right|\,dy\, . Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid multiplied by the figure's area (Pappus's second centroid theorem). In the end we find the volume is cubic units. ==See also== *Solid of revolution *Disc integration ==References== * *Frank Ayres, Elliott Mendelson. The areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given :A_x = \int_\alpha^\beta 2 \pi r\sin{\theta} \, \sqrt{ r^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta \, , :A_y = \int_\alpha^\beta 2 \pi r\cos{\theta} \, \sqrt{ r^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta \, , : ==See also== * Gabriel's Horn * Guldinus theorem * Pseudosphere * Surface of revolution * Ungula ==Notes== == References == * * () * Category:Integral calculus Category:Solids The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by :V = 2\pi \int_a^b x |f(x) - g(x)|\, dx\, . This works only if the axis of rotation is horizontal (example: or some other constant). ===Function of === If the function to be revolved is a function of , the following integral will obtain the volume of the solid of revolution: :\pi\int_c^d R(y)^2\,dy where is the distance between the function and the axis of rotation. The surface created by this revolution and which bounds the solid is the surface of revolution. This works only if the axis of rotation is vertical (example: or some other constant). ===Washer method=== To obtain a hollow solid of revolution (the “washer method”), the procedure would be to take the volume of the inner solid of revolution and subtract it from the volume of the outer solid of revolution. To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness , or a cylindrical shell of width ; and then find the limiting sum of these volumes as approaches 0, a value which may be found by evaluating a suitable integral. The volume of a tricylinder is :V=8(2 - \sqrt{2}) r^3 and the surface area is :A=24(2 - \sqrt{2}) r^2. == More cylinders == With four cylinders, with axes connecting the vertices of a tetrahedron to the corresponding points on the other side of the solid, the volume is :V_4=12 \left( 2\sqrt{2} - \sqrt{6} \right) r^3 \, With six cylinders, with axes parallel to the diagonals of the faces of a cube, the volume is: :V_6=\frac{16}{3} \left( 3 + 2\sqrt{3} - 4\sqrt{2} \right) r^3 \, == See also == * Ungula == References == == External links == *A 3D model of Steinmetz solid in Google 3D Warehouse Category:Euclidean solid geometry Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. If is the value of a horizontal axis, then the volume equals :\pi\int_a^b\left(\left(h-R_\mathrm{O}(x)\right)^2 - \left(h-R_\mathrm{I}(x)\right)^2\right)\,dx\,. The element is created by rotating a line segment (of length ) around some axis (located units away), so that a cylindrical volume of units is enclosed. ==Finding the volume== Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration. Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given by :A_x = \int_a^b 2 \pi y \, \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \, , :A_y = \int_a^b 2 \pi x \, \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \, . == Polar form == For a polar curve r=f(\theta) where \alpha\leq \theta\leq \beta, the volumes of the solids generated by revolving the curve around the x-axis or y-axis are :V_x = \int_\alpha^\beta \left(\pi r^2\sin^2{\theta} \cos{\theta}\, \frac{dr}{d\theta}-\pi r^3\sin^3{\theta}\right)d\theta\,, :V_y = \int_\alpha^\beta \left(\pi r^2\sin{\theta} \cos^2{\theta}\, \frac{dr}{d\theta}+\pi r^3\cos^3{\theta}\right)d\theta \, . One simply must solve each equation for before one inserts them into the integration formula. ==See also== *Solid of revolution *Shell integration ==References== * * *Frank Ayres, Elliott Mendelson. In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution) that lies on the same plane. Volume solid is the term which indicates the solid proportion of the paint on a volume basis. For one half of a bicylinder a similar statement is true: * The relations of the volumes of the inscribed square pyramid (a=2r, h=r, V=\frac{4}{3}r^3), the half bicylinder (V=\frac{8}{3} r^3) and the surrounding squared cuboid ( a= 2r, h=r, V=4r^3) are 1 : 2 : 3. ==== Using Multivariable Calculus ==== Consider the equations of the cylinders: x^2+z^2=r^2 x^2+y^2=r^2 The volume will be given by: V = \iiint_V \mathrm{d}z\mathrm{d}y\mathrm{d}x With the limits of integration: -\sqrt{r^2-x^2} \leqslant z \leqslant \sqrt{r^2-x^2} -\sqrt{r^2-x^2} \leqslant y \leqslant \sqrt{r^2-x^2} -r \leqslant x \leqslant r Substituting, we have: V = \int_{-r}^{r}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}} \mathrm{d}z\mathrm{d}y\mathrm{d}x = 8r^3-\frac{8r^3}{3} = \frac{16r^3}{3} === Proof of the area formula === The surface area consists of two red and two blue cylindrical biangles. For example, to rotate the region between and along the axis , one would integrate as follows: :\pi\int_0^3\left(\left(4-\left(-2x+x^2\right)\right)^2 - (4-x)^2\right)\,dx\,. After integrating these two functions with the disk method we would subtract them to yield the desired volume. This method may be derived with the same triple integral, this time with a different order of integration: :V = \iiint_D dV = \int_a^b \int_{g(r)}^{f(r)} \int_0^{2\pi} r\,d\theta\,dz\,dr = 2\pi \int_a^b\int_{g(r)}^{f(r)} r\,dz\,dr = 2\pi\int_a^b r(f(r) - g(r))\,dr. ==Parametric form== When a curve is defined by its parametric form in some interval , the volumes of the solids generated by revolving the curve around the -axis or the -axis are given by :V_x = \int_a^b \pi y^2 \, \frac{dx}{dt} \, dt \, , :V_y = \int_a^b \pi x^2 \, \frac{dy}{dt} \, dt \, .

+7.3

0.000216

0.41887902047

3.54

2.9

C

Use Simpson's Rule with $n=10$ to approximate $\int_1^2(1 / x) d x$.

The error in approximating an integral by Simpson's rule for n = 2 is -\frac{1}{90} h^5f^{(4)}(\xi) = -\frac{(b - a)^5}{2880} f^{(4)}(\xi), where \xi (the Greek letter xi) is some number between a and b. Simpson's 1/3 rule is as follows: \begin{align} \int_a^b f(x)\, dx &\approx \frac{1}{3} h\left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]\\\ &= \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right], \end{align} where h = (b - a)/2 is the step size. (This derivation is essentially a less rigorous version of the quadratic interpolation derivation, where one saves significant calculation effort by guessing the correct functional form.) === Composite Simpson's 1/3 rule === If the interval of integration [a, b] is in some sense "small", then Simpson's rule with n = 2 subintervals will provide an adequate approximation to the exact integral. This is called the trapezoidal rule \int_a^b f(x)\, dx \approx (b-a) \left(\frac{f(a) + f(b)}{2}\right). right|thumb|300px|Illustration of Simpson's rule. Simpson's rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval. The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads \int_a^b f(x) \, dx \approx \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]. Integration by Simpson's 1/3 rule can be represented as a weighted average with 2/3 of the value coming from integration by the trapezoidal rule with step h and 1/3 of the value coming from integration by the rectangle rule with step 2h. Simpson's 3/8 rule, also called Simpson's second rule, requires one more function evaluation inside the integration range and gives lower error bounds, but does not improve on order of the error. This leads to the adaptive Simpson's method. == Simpson's 3/8 rule == Simpson's 3/8 rule, also called Simpson's second rule, is another method for numerical integration proposed by Thomas Simpson. Adaptive Simpson's method uses an estimate of the error we get from calculating a definite integral using Simpson's rule. If the 1/3 rule is applied to n equal subdivisions of the integration range [a, b], one obtains the composite Simpson's 1/3 rule. The error committed by the composite Simpson's rule is -\frac{1}{180} h^4(b - a)f^{(4)}(\xi), where \xi is some number between a and b, and h = (b - a)/n is the "step length". thumb|right|Simpson's rule can be derived by approximating the integrand f (x) (in blue) by the quadratic interpolant P(x) (in red). thumb|right|An animation showing how Simpson's rule approximates the function with a parabola and the reduction in error with decreased step size thumb|right|An animation showing how Simpson's rule approximation improves with more strips. In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The formula above is obtained by combining the composite Simpson's 1/3 rule with the one consisting of using Simpson's 3/8 rule in the extreme subintervals and Simpson's 1/3 rule in the remaining subintervals. The 1/3 rule can be used for the remaining subintervals without changing the order of the error term (conversely, the 3/8 rule can be used with a composite 1/3 rule for odd-numbered subintervals). == Alternative extended Simpson's rule == This is another formulation of a composite Simpson's rule: instead of applying Simpson's rule to disjoint segments of the integral to be approximated, Simpson's rule is applied to overlapping segments, yielding \int_a^b f(x)\, dx \approx \frac{1}{48} h\left[17f(x_0) + 59f(x_1) + 43f(x_2) + 49f(x_3) + 48 \sum_{i= 4 }^{n - 4} f(x_i) + 49f(x_{n - 3}) + 43f(x_{n - 2}) + 59f(x_{n - 1}) + 17f(x_n)\right]. If the 3/8 rule is applied to n equal subdivisions of the integration range [a, b], one obtains the composite Simpson's 3/8 rule. Simpson's 3/8 rule is as follows: \begin{align} \int_a^b f(x)\, dx &\approx \frac{3}{8} h\left[f(a) + 3f\left(\frac{2a + b}{3}\right) + 3f\left(\frac{a + 2b}{3}\right) + f(b)\right]\\\ &= \frac{b - a}{8} \left[f(a) + 3f\left(\frac{2a + b}{3}\right) + 3f\left(\frac{a + 2b}{3}\right) + f(b)\right], \end{align} where h = (b - a)/3 is the step size. If the error exceeds a user-specified tolerance, the algorithm calls for subdividing the interval of integration in two and applying adaptive Simpson's method to each subinterval in a recursive manner. See: * Simpson's rule, a method of numerical integration * Simpson's rules (ship stability) * Simpson–Kramer method In case of odd number N of subintervals, the above formula are used up to the second to last interval, and the last interval is handled separately by adding the following to the result: \alpha f_N + \beta f_{N - 1} - \eta f_{N - 2}, where \begin{align} \alpha &= \frac{2h_{N - 1}^2 + 3h_{N - 1} h_{N - 2}}{6(h_{N - 2} + h_{N - 1})},\\\\[1ex] \beta &= \frac{h_{N - 1}^2 + 3h_{N - 1} h_{N - 2}}{6h_{N - 2}},\\\\[1ex] \eta &= \frac{h_{N - 1}^3}{6 h_{N - 2}(h_{N - 2} + h_{N - 1})}. \end{align} Example implementation in Python from collections.abc import Sequence def simpson_nonuniform(x: Sequence[float], f: Sequence[float]) -> float: """ Simpson rule for irregularly spaced data. :param x: Sampling points for the function values :param f: Function values at the sampling points :return: approximation for the integral See ``scipy.integrate.simpson`` and the underlying ``_basic_simpson`` for a more performant implementation utilizing numpy's broadcast. The result is then obtained by taking the mean of the two formulas. === Simpson's rules in the case of narrow peaks === In the task of estimation of full area of narrow peak-like functions, Simpson's rules are much less efficient than trapezoidal rule.

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635013559600.0

0.693150

3

D

Use the Midpoint Rule with $m=n=2$ to estimate the value of the integral $\iint_R\left(x-3 y^2\right) d A$, where $R=\{(x, y) \mid 0 \leqslant x \leqslant 2,1 \leqslant y \leqslant 2\}$.

For example, doing the previous calculation with order reversed gives the same result: : \begin{align} \int_{11}^{14} \int_{7}^{10} \, \left(x^2 + 4y\right) \, dy\, dx & = \int_{11}^{14} \Big[x^2 y + 2y^2 \Big]_{y=7}^{y=10} \, dx \\\ &= \int_{11}^{14} \, (3x^2 + 102) \, dx \\\ &= \Big[x^3 + 102x \Big]_{x=11}^{x=14} \\\ &= 1719. \end{align} === Double integral over a normal domain === thumb|160px|right|Example: double integral over the normal region D Consider the region (please see the graphic in the example): :D = \\{ (x,y) \in \mathbf{R}^2 \ : \ x \ge 0, y \le 1, y \ge x^2 \\} Calculate :\iint_D (x+y) \, dx \, dy. Let and :D = \left\\{ (x,y) \in \R^2 \ : \ 2 \le x \le 4 \ ; \ 3 > \le y \le 6 \right\\} in which case :\int_3^6 \int_2^4 \ 2 \ dx\, dy > =2\int_3^6 \int_2^4 \ 1 \ dx\, dy= 2\cdot\operatorname{area}(D) = 2 \cdot (2 > \cdot 3) = 12, since by definition we have: :\int_3^6 \int_2^4 \ 1 \ dx\, > dy=\operatorname{area}(D). ===Use of symmetry=== When the domain of integration is symmetric about the origin with respect to at least one of the variables of integration and the integrand is odd with respect to this variable, the integral is equal to zero, as the integrals over the two halves of the domain have the same absolute value but opposite signs. I = \left.\int_0^{3a}\rho^4 > d\rho = \frac{\rho^5}{5}\right\vert_0^{3a} = \frac{243}{5}a^5, II = > \int_0^\pi \sin^3\theta \, d\theta = -\int_0^\pi \sin^2\theta \, d(\cos > \theta) = \int_0^\pi (\cos^2\theta-1) \, d(\cos \theta) = > \left.\frac{\cos^3\theta}{3}\right|^\pi_0 - \left.\cos\theta\right|^\pi_0 = > \frac{4}{3}, III = \int_0^{2\pi} d \varphi = 2\pi. Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. The result of this integral, which is a function depending only on , is then integrated with respect to . :\begin{align} \int_{11}^{14} \left(x^2 + 4y\right) \, dx & = \left [\frac13 x^3 + 4yx \right]_{x=11}^{x=14} \\\ &= \frac13(14)^3 + 4y(14) - \frac13(11)^3 - 4y(11) \\\ &= 471 + 12y \end{align} We then integrate the result with respect to . :\begin{align} \int_7^{10} (471 + 12y) \ dy & = \Big[471y + 6y^2\Big]_{y=7}^{y=10} \\\ &= 471(10)+ 6(10)^2 - 471(7) - 6(7)^2 \\\ &= 1719 \end{align} In cases where the double integral of the absolute value of the function is finite, the order of integration is interchangeable, that is, integrating with respect to x first and integrating with respect to y first produce the same result. It is now possible to apply the formula: :\iint_D (x+y) \, dx \, dy = \int_0^1 dx \int_{x^2}^1 (x+y) \, dy = \int_0^1 dx \ \left[xy + \frac{y^2}{2} \right]^1_{x^2} (at first the second integral is calculated considering x as a constant). Then > we get :\begin{align} \int_0^{2\pi} d\varphi \int_0^{3a} \rho^3 d\rho > \int_{-\sqrt{9a^2 - \rho^2}}^{\sqrt{9 a^2 - \rho^2}}\, dz &= 2 \pi > \int_0^{3a} 2 \rho^3 \sqrt{9 a^2 - \rho^2} \, d\rho \\\ &= -2 \pi \int_{9 > a^2}^0 (9 a^2 - t) \sqrt{t}\, dt && t = 9 a^2 - \rho^2 \\\ &= 2 \pi > \int_0^{9 a^2} \left ( 9 a^2 \sqrt{t} - t \sqrt{t} \right ) \, dt \\\ &= 2 > \pi \left( \int_0^{9 a^2} 9 a^2 \sqrt{t} \, dt - \int_0^{9 a^2} t \sqrt{t} > \, dt\right) \\\ &= 2 \pi \left[9 a^2 \frac23 t^{ \frac32 } - \frac{2}{5} > t^{ \frac{5}{2}} \right]_0^{9 a^2} \\\ &= 2 \cdot 27 \pi a^5 \left ( 6 - > \frac{18}{5} \right ) \\\ &= \frac{648 \pi}{5} a^5. \end{align} Thanks to > the passage to cylindrical coordinates it was possible to reduce the triple > integral to an easier one-variable integral. In numerical analysis, Romberg's method is used to estimate the definite integral \int_a^b f(x) \, dx by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). See also the differential volume entry in nabla in cylindrical and spherical coordinates. ==Examples== === Double integral over a rectangle === Let us assume that we wish to integrate a multivariable function over a region : :A = \left \\{ (x,y) \in \mathbf{R}^2 \ : \ 11 \le x \le 14 \ ; \ 7 \le y \le 10 \right \\} \mbox{ and } f(x,y) = x^2 + 4y\, From this we formulate the iterated integral :\int_7^{10} \int_{11}^{14} (x^2 + 4y) \, dx\, dy The inner integral is performed first, integrating with respect to and taking as a constant, as it is not the variable of integration. right|thumb|Illustration of the midpoint method assuming that y_n equals the exact value y(t_n). The domain is the ball with center at the origin and radius , :D > = \left \\{ x^2 + y^2 + z^2 \le 9a^2 \right \\} and is the function to > integrate. Then, by Fubini's theorem: :\iint_D f(x,y)\, dx\, dy = \int_a^b dx \int_{ \alpha (x)}^{ \beta (x)} f(x,y)\, dy. ====-axis==== If is normal with respect to the -axis and is a continuous function; then and (both of which are defined on the interval ) are the two functions that determine . Collecting all parts, > \iiint_T \rho^4 \sin^3 \theta \, d\rho\, d\theta\, d\varphi = I\cdot II\cdot > III = \frac{243}{5}a^5\cdot \frac{4}{3}\cdot 2\pi = \frac{648}{5}\pi a^5. > Alternatively, this problem can be solved by using the passage to > cylindrical coordinates. Once the intervals are known, you have :\int_0^\pi \int_2^3 \rho^2 > \cos \varphi \, d \rho \, d \varphi = \int_0^\pi \cos \varphi \ d \varphi > \left[ \frac{\rho^3}{3} \right]_2^3 = \Big[ \sin \varphi \Big]_0^\pi \ > \left(9 - \frac{8}{3} \right) = 0. ====Cylindrical coordinates==== thumb|right|190px|Cylindrical coordinates. The explicit midpoint method is given by the formula the implicit midpoint method by for n=0, 1, 2, \dots Here, h is the step size -- a small positive number, t_n=t_0 + n h, and y_n is the computed approximate value of y(t_n). If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate. Using the linearity property, the > integral can be decomposed into three pieces: :\iint_T \left(2\sin x - 3y^3 > + 5\right) \, dx \, dy = \iint_T 2 \sin x \, dx \, dy - \iint_T 3y^3 \, dx > \, dy + \iint_T 5 \, dx \, dy The function is an odd function in the > variable and the disc is symmetric with respect to the -axis, so the value > of the first integral is 0. thumb|right|Integral as area between two curves. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r1, r2, r3 and the center of mass R. * Sphere: The volume of a sphere with radius can be calculated by integrating the constant function 1 over the sphere, using spherical coordinates. ::\begin{align} \text{Volume} &= \iiint_D f(x,y,z) \, dx\, dy\, dz \\\ &= \iiint_D 1 \, dV \\\ &= \iiint_S \rho^2 \sin \varphi \, d\rho\, d\theta\, d\varphi \\\ &= \int_0^{2\pi} \, d \theta \int_0^{ \pi } \sin \varphi\, d \varphi \int_0^R \rho^2\, d \rho \\\ &= 2 \pi \int_0^\pi \sin \varphi\, d \varphi \int_0^R \rho^2\, d \rho \\\ &= 2 \pi \int_0^\pi \sin \varphi \frac{R^3}{3 }\, d \varphi \\\ &= \frac23 \pi R^3 \Big[-\cos \varphi\Big]_0^\pi = \frac43 \pi R^3. \end{align} * Tetrahedron (triangular pyramid or 3-simplex): The volume of a tetrahedron with its apex at the origin and edges of length along the -, - and -axes can be calculated by integrating the constant function 1 over the tetrahedron. ::\begin{align} \text{Volume} &= \int_0^\ell dx \int_0^{\ell-x}\, dy \int_0^{\ell-x-y }\, dz \\\ &= \int_0^\ell dx \int_0^{\ell-x } (\ell - x - y)\, dy \\\ &= \int_0^\ell \left( l^2 - 2 \ell x + x^2 - \frac{(\ell-x)^2 }{2}\right)\, dx \\\ &= \ell^3 - \ell \ell^2 + \frac{\ell^3}{3 } - \left[\frac{\ell^2 x}{2} - \frac{ \ell x^2}{2} + \frac{x^3}{6 }\right]_0^ \ell \\\ &= \frac{\ell^3}{3} - \frac{\ell^3}{6} = \frac{ \ell^3}{6}\end{align} :This is in agreement with the formula for the volume of a pyramid ::\mathrm{Volume} = \frac13 \times \text{base area} \times \text{height} = \frac13 \times \frac{\ell^2}{2} \times \ell = \frac{ \ell^3}{6}. thumb|right|140px|Example of an improper domain. ==Multiple improper integral== In case of unbounded domains or functions not bounded near the boundary of the domain, we have to introduce the double improper integral or the triple improper integral. ==Multiple integrals and iterated integrals== Fubini's theorem states that if :\iint_{A\times B} \left|f(x,y)\right|\,d(x,y)<\infty, that is, if the integral is absolutely convergent, then the multiple integral will give the same result as either of the two iterated integrals: :\iint_{A\times B} f(x,y)\,d(x,y)=\int_A\left(\int_B f(x,y)\,dy\right)\,dx=\int_B\left(\int_A f(x,y)\,dx\right)\,dy. In the following example, the electric field produced by a distribution of charges given by the volume charge density is obtained by a triple integral of a vector function: :\vec E = \frac {1}{4 \pi \varepsilon_0} \iiint \frac {\vec r - \vec r'}{\left \| \vec r - \vec r' \right \|^3} \rho (\vec r')\, d^3 r'.

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The base radius and height of a right circular cone are measured as $10 \mathrm{~cm}$ and $25 \mathrm{~cm}$, respectively, with a possible error in measurement of as much as $0.1 \mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated volume of the cone.

For 0 \le K' \le 1 : y = R\left({2 \left({x \over L}\right) - K'\left({x \over L}\right)^2 \over 2 - K'}\right) can vary anywhere between and , but the most common values used for nose cone shapes are: Parabola Type Value Cone Half Three Quarter Full For the case of the full parabola () the shape is tangent to the body at its base, and the base is on the axis of the parabola. If the chosen ogive radius of a secant ogive is greater than the ogive radius of a tangent ogive with the same and , then the resulting secant ogive appears as a tangent ogive with a portion of the base truncated. :\rho > {R^2 + L^2 \over 2R} and \alpha = \arccos \left({\sqrt{L^2 + R^2} \over 2\rho}\right)-\arctan \left({R \over L}\right) Then the radius at any point as varies from to is: :y = \sqrt{\rho^2 - (\rho\cos(\alpha) - x)^2} - \rho\sin(\alpha) If the chosen is less than the tangent ogive and greater than half the length of the nose cone, then the result will be a secant ogive that bulges out to a maximum diameter that is greater than the base diameter. The radius of the circle that forms the ogive is called the ogive radius, , and it is related to the length and base radius of the nose cone as expressed by the formula: :\rho = {R^2 + L^2\over 2R} The radius at any point , as varies from to is: :y = \sqrt{\rho^2 - (L - x)^2}+R - \rho The nose cone length, , must be less than or equal to . The tangency point where the sphere meets the cone can be found from: : x_t = \frac{L^2}{R} \sqrt{ \frac{r_n^2}{R^2 + L^2} } : y_t = \frac{x_t R}{L} where is the radius of the spherical nose cap. The failure is governed by crack growth in concrete, which forms a typical cone shape having the anchor's axis as revolution axis. ==Mechanical models== ===ACI 349-85=== Under tension loading, the concrete cone failure surface has 45° inclination. For , Haack nose cones bulge to a maximum diameter greater than the base diameter. The sides of a conic profile are straight lines, so the diameter equation is simply: : y = {xR \over L} Cones are sometimes defined by their half angle, : : \phi = \arctan \left({R \over L}\right) and y = x \tan(\phi)\; ==== Spherically blunted conic ==== In practical applications such as re-entry vehicles, a conical nose is often blunted by capping it with a segment of a sphere. A conical measure is a type of laboratory glassware which consists of a conical cup with a notch on the top to allow for the easy pouring of liquids, and graduated markings on the side to allow easy and accurate measurement of volumes of liquid. Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium (such as a rocket or aircraft, missile, shell or bullet), an important problem is the determination of the nose cone geometrical shape for optimum performance. alt=Two-dimensional drawing of an elliptical nose cone with dimensions added to show how L is the total length of the nose cone, R is the radius at the base, and y is the radius at a point x distance from the tip.|thumb|300x300px|General parameters used for constructing nose cone profiles. For many applications, such a task requires the definition of a solid of revolution shape that experiences minimal resistance to rapid motion through such a fluid medium. == Nose cone shapes and equations == === General dimensions === In all of the following nose cone shape equations, is the overall length of the nose cone and is the radius of the base of the nose cone. is the radius at any point , as varies from , at the tip of the nose cone, to . The concrete cone failure load N_0 of a single anchor in uncracked concrete unaffected by edge influences or overlapping cones of neighboring anchors is given by: N_0 = f_{ct} {A_{N}} [N] Where: f_{ct} \- tensile strength of concrete A_{N} \- Cone's projected area === Concrete capacity design (CCD) approach for fastening to concrete=== Under tension loading, the concrete capacity of a single anchor is calculated assuming an inclination between the failure surface and surface of the concrete member of about 35°. The length/diameter relation is also often called the caliber of a nose cone. The center of the spherical nose cap, , can be found from: : x_o = x_t + \sqrt{ r_n^2 - y_t^2} And the apex point, can be found from: : x_a = x_o - r_n === Bi-conic === A bi-conic nose cone shape is simply a cone with length stacked on top of a frustum of a cone (commonly known as a conical transition section shape) with length , where the base of the upper cone is equal in radius to the top radius of the smaller frustum with base radius . The use of the conical measure usually dictates its construction material. For a circle with slightly smaller radius, the area is nearly the same, but the circle contains only 69 points, producing a larger error of approximately 9.54. thumb|A circle of radius 5 centered at the origin has area 25, approximately 78.54, but it contains 81 integer points, so the error in estimating its area by counting grid points is approximately 2.46. Conical measures are the most commonly used item of glassware used in the preparation of extemporaneous medicaments. While the equations describe the 'perfect' shape, practical nose cones are often blunted or truncated for manufacturing, aerodynamic, or thermodynamic reasons. === Conic === A very common nose-cone shape is a simple cone. :L=L_1+L_2 :For 0 \le x \le L_1 : y = {xR_1 \over L_1} :For L_1 \le x \le L : y = R_1 + {(x - L_1)(R_2-R_1)\over L_2} Half angles: :\phi_1 = \arctan \left({R_1 \over L_1}\right) and y = x \tan(\phi_1)\; :\phi_2 = \arctan \left({R_2 - R_1 \over L_2}\right) and y = R_1 + (x - L_1) \tan(\phi_2)\; === Tangent ogive === Next to a simple cone, the tangent ogive shape is the most familiar in hobby rocketry. The profile of this shape is formed by a segment of a circle such that the rocket body is tangent to the curve of the nose cone at its base, and the base is on the radius of the circle. They are not as precise as graduated cylinders for measuring liquids, but make up for this in terms of easy pouring and ability to mix solutions within the measure itself. ==History== During his experiments, Abū al-Rayhān al-Bīrūnī (973-1048) invented the conical measure,Marshall Clagett (1961).

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A force of $40 \mathrm{~N}$ is required to hold a spring that has been stretched from its natural length of $10 \mathrm{~cm}$ to a length of $15 \mathrm{~cm}$. How much work is done in stretching the spring from $15 \mathrm{~cm}$ to $18 \mathrm{~cm}$ ?

Let be the amount by which the free end of the spring was displaced from its "relaxed" position (when it is not being stretched). An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. If a spring has a rate of 100 then the spring would compress 1 inch (2.54 cm) with of load. == Types== Types of coil spring are: * Tension/extension coil springs, designed to resist stretching. The force an ideal spring would exert is exactly proportional to its extension or compression. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. The work of the net force is calculated as the product of its magnitude and the particle displacement. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. The force is applied through the ends of the spring. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. The change in length may be expressed as \Delta L = \varepsilon L = \frac{F L}{A E}\,. === Spring energy === The potential energy stored in a spring is given by U_\mathrm{el}(x) = \tfrac 1 2 kx^2 which comes from adding up the energy it takes to incrementally compress the spring. The work is the product of the distance times the spring force, which is also dependent on distance; hence the result. ===Work by a gas=== The work W done by a body of gas on its surroundings is: W = \int_a^b P \, dV where is pressure, is volume, and and are initial and final volumes. ==Work–energy principle== The principle of work and kinetic energy (also known as the work–energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. When a conventional spring, without stiffness variability features, is compressed or stretched from its resting position, it exerts an opposing force approximately proportional to its change in length (this approximation breaks down for larger deflections). This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. The work can be split into two terms \delta W = \delta W_\mathrm{s} + \delta W_\mathrm{b} where is the work done by surface forces while is the work done by body forces. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Explaining the Power of Springing Bodies, London, 1678. as: ("as the extension, so the force" or "the extension is proportional to the force"). Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle. A larger pitch in the larger-diameter coils and a smaller pitch in the smaller-diameter coils forces the spring to collapse or extend all the coils at the same rate when deformed. ===Simple harmonic motion=== Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like: :F = m a \quad \Rightarrow \quad -k x = m a. The manufacture normally specifies the spring rate. The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. We will use the word "torsion" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian.

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An automobile with a mass of $1000 \mathrm{~kg}$, including passengers, settles $1.0 \mathrm{~cm}$ closer to the road for every additional $100 \mathrm{~kg}$ of passengers. It is driven with a constant horizontal component of speed $20 \mathrm{~km} / \mathrm{h}$ over a washboard road with sinusoidal bumps. The amplitude and wavelength of the sine curve are $5.0 \mathrm{~cm}$ and $20 \mathrm{~cm}$, respectively. The distance between the front and back wheels is $2.4 \mathrm{~m}$. Find the amplitude of oscillation of the automobile, assuming it moves vertically as an undamped driven harmonic oscillator. Neglect the mass of the wheels and springs and assume that the wheels are always in contact with the road.

Thus, for g ≈ 9.8 ms−2, :L \approx \frac{894}{n^2} \; \text{m} | n 71 0.846 0.177 70 0.857 0.182 69 0.870 0.188 68 0.882 0.193 67 0.896 0.199 66 0.909 0.205 65 0.923 0.212 64 0.938 0.218 63 0.952 0.225 62 0.968 0.232 61 0.984 0.240 60 1.000 0.248 Parameters of the pendulum wave in the animation above ---|--- == See also == * Newton's cradle - a set of pendulums constrained to swing along the axis of the apparatus and collide with one another ==References== Category:Pendulums Category:Kinetic art Derivation of the frequency response Using the method of harmonic balance, an approximate solution to the Duffing equation is sought of the form: :x = a\, \cos(\omega t) + b\, \sin(\omega t) = z\, \cos(\omega t - \phi), with z^2=a^2+b^2 and \tan\phi = \frac{b}{a}. A vibration in a string is a wave. Constant envelope is achieved when a sinusoidal waveform reaches equilibrium in a specific system. In particular, the increase in train speeds from 140 to 180 km/h was accompanied by about tenfold increase in generated ground vibration level, which agrees with the theory. The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear spring and a linear damper. The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. Moreover, the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour. ==Parameters== The parameters in the above equation are: *\delta controls the amount of damping, *\alpha controls the linear stiffness, *\beta controls the amount of non-linearity in the restoring force; if \beta=0, the Duffing equation describes a damped and driven simple harmonic oscillator, *\gamma is the amplitude of the periodic driving force; if \gamma=0 the system is without a driving force, and *\omega is the angular frequency of the periodic driving force. String no.!!Thickness [in.] (d)!!Recommended tension [lbs.] (T)!!\rho [g/cm3] |- | 1 String no. Thickness [in.] (d) Recommended tension [lbs.] (T) \rho [g/cm3] 1 0.00899 13.1 7.726 (steel alloy) 2 0.0110 11.0 " 3 0.0160 14.7 " 4 0.0241 15.8 6.533 (nickel-wound steel alloy) 5 0.0322 15.8 " 6 0.0416 14.8 " Given the above specs, what would the computed vibrational frequencies (f) of the above strings' fundamental harmonics be if the strings were strung at the tensions recommended by the manufacturer? The horizontal tensions are not well approximated by T. == Frequency of the wave == Once the speed of propagation is known, the frequency of the sound produced by the string can be calculated. If the length of the string is L, the fundamental harmonic is the one produced by the vibration whose nodes are the two ends of the string, so L is half of the wavelength of the fundamental harmonic. String no.!!Computed frequency [Hz]!!Closest note in A440 12-TET tuning |- | 1 String no. Computed frequency [Hz] Closest note in A440 12-TET tuning 1 330 E4 (= 440 ÷ 25/12 ≈ 329.628 Hz) 2 247 B3 (= 440 ÷ 210/12 ≈ 246.942 Hz) 3 196 G3 (= 440 ÷ 214/12 ≈ 195.998 Hz) 4 147 D3 (= 440 ÷ 219/12 ≈ 146.832 Hz) 5 110 A2 (= 440 ÷ 224/12 = 110 Hz) 6 82.4 E2 (= 440 ÷ 229/12 ≈ 82.407 Hz) == See also == * Fretted instruments * Musical acoustics * Vibrations of a circular drum * Melde's experiment * 3rd bridge (harmonic resonance based on equal string divisions) * String resonance * Reflection phase change == References == * * ;Specific == External links == * "The Vibrating String" by Alain Goriely and Mark Robertson-Tessi, The Wolfram Demonstrations Project. For the parameters of the Duffing equation, the above algebraic equation gives the steady state oscillation amplitude z at a given excitation frequency. Hence one obtains Mersenne's laws: :f = \frac{v}{2L} = { 1 \over 2L } \sqrt{T \over \mu} where T is the tension (in Newtons), \mu is the linear density (that is, the mass per unit length), and L is the length of the vibrating part of the string. Ground vibration boom is a phenomenon of very large increase in ground vibrations generated by high-speed railway trains travelling at speeds higher than the velocity of Rayleigh surface waves in the supporting ground. == Technical background == thumb|Swedish high-speed train X 2000 approaching Ledsgard. Therefore: * the shorter the string, the higher the frequency of the fundamental * the higher the tension, the higher the frequency of the fundamental * the lighter the string, the higher the frequency of the fundamental Moreover, if we take the nth harmonic as having a wavelength given by \lambda_n = 2L/n, then we easily get an expression for the frequency of the nth harmonic: :f_n = \frac{nv}{2L} And for a string under a tension T with linear density \mu, then :f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} == Observing string vibrations == One can see the waveforms on a vibrating string if the frequency is low enough and the vibrating string is held in front of a CRT screen such as one of a television or a computer (not of an analog oscilloscope). As a result, : \begin{align} & -\omega^2\, a + \omega\,\delta\,b + \alpha\,a + \tfrac34\,\beta\,a^3 + \tfrac34\,\beta\,a\,b^2 = \gamma \qquad \text{and} \\\ & -\omega^2\, b - \omega\,\delta\,a + \tfrac34\,\beta\,b^3 + \alpha\,b + \tfrac34\,\beta\,a^2\,b = 0. \end{align} Squaring both equations and adding leads to the amplitude frequency response: :\left[\left(\omega^2-\alpha-\frac{3}{4}\beta z^2\right)^{2}+\left(\delta\omega\right)^2\right]\,z^2=\gamma^{2}, as stated above. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos. == Wave == The velocity of propagation of a wave in a string (v) is proportional to the square root of the force of tension of the string (T) and inversely proportional to the square root of the linear density (\mu) of the string: v = \sqrt{T \over \mu}. A pendulum wave is an elementary physics demonstration and kinetic art comprising a number of uncoupled simple pendulums with monotonically increasing lengths. In physics, the Toda oscillator is a special kind of nonlinear oscillator. File:Duffing frequency response.svg|alt=Frequency response as a function of for the Duffing equation, with and damping The dashed parts of the frequency response are unstable.[3]|Frequency response z/\gamma as a function of \omega/\sqrt{\alpha} for the Duffing equation, with \alpha=\gamma=1 and damping \delta=0.1. The equation is given by :\ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos (\omega t), where the (unknown) function x=x(t) is the displacement at time t, \dot{x} is the first derivative of x with respect to time, i.e. velocity, and \ddot{x} is the second time- derivative of x, i.e. acceleration.

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Find the shortest path between the $(x, y, z)$ points $(0,-1,0)$ and $(0,1,0)$ on the conical surface $z=1-\sqrt{x^2+y^2}$. What is the length of the path? Note: this is the shortest mountain path around a volcano.

In other words, the shortest path will be made by joining circular arcs of maximum curvature and straight lines. Given two points s and t, say on the surface of a convex polyhedron, the problem is to compute a shortest path that never leaves the surface and connects s with t. thumb|right|Example of a shortest path in a three-dimensional Euclidean space The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles. == Two dimensions == In two dimensions, the problem can be solved in polynomial time in a model of computation allowing addition and comparisons of real numbers, despite theoretical difficulties involving the numerical precision needed to perform such calculations. We also need to know what the actual shortest path is. There are simple geometric and analytical methods to compute the optimal path. Moving along each segment in this sequence for the appropriate length will form the shortest curve that joins a starting point A to a terminal point B with the desired tangents at each endpoint and that does not exceed the given curvature. Mount Short () is a mountain, 2,110 m, standing 1 mile (1.6 km) east of Sculpture Mountain, in the upper Rennick Glacier. There are many results on computing shortest paths which stays on a polyhedral surface. Wallis's Conical Edge with , In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt{a^2-b^2\cos^2u} where , and are constants. thumb|The shortest-path graph with t=2 In mathematics and geographic information science, a shortest-path graph is an undirected graph defined from a set of points in the Euclidean plane. Finding the shortest path in a graph using optimal substructure; a straight line indicates a single edge; a wavy line indicates a shortest path between the two vertices it connects (among other paths, not shown, sharing the same two vertices); the bold line is the overall shortest path from start to goal. If p is truly the shortest path, then it can be split into sub-paths p1 from u to w and p2 from w to v such that these, in turn, are indeed the shortest paths between the corresponding vertices (by the simple cut-and-paste argument described in Introduction to Algorithms). In particular, Harold H. Johnson presented necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length. This is termed as the weighted region problem in the literature. ==See also== * Shortest path problem, in a graph of edges and vertices * Any-angle path planning, in a grid space ==Notes== ==References== *. *. *. *. *. *. *. *. *. *. *. == External links == * Implementation of Euclidean Shortest Path algorithm in Digital Geometric Kernel software Category:Geometric algorithms Category:Computational geometry In 1957, Lester Eli Dubins (1920–2010) proved using tools from analysis that any such path will consist of maximum curvature and/or straight line segments. In the following pseudocode, `n` is the size of the board, `c(i, j)` is the cost function, and `min()` returns the minimum of a number of values: function minCost(i, j) if j < 1 or j > n return infinity else if i = 1 return c(i, j) else return min( minCost(i-1, j-1), minCost(i-1, j), minCost(i-1, j+1) ) + c(i, j) This function only computes the path cost, not the actual path. In geometry, the term Dubins path typically refers to the shortest curve that connects two points in the two-dimensional Euclidean plane (i.e. x-y plane) with a constraint on the curvature of the path and with prescribed initial and terminal tangents to the path, and an assumption that the vehicle traveling the path can only travel forward. The tangent direction of the path at initial and final points are constrained to lie within the specified intervals. Picking the square that holds the minimum value at each rank gives us the shortest path between rank `n` and rank `1`. The Dubins' path gives the shortest path joining two oriented points that is feasible for the wheeled-robot model. The edge set of the shortest-path graph varies based on a single parameter t ≥ 1. When the weight of an edge is defined as its Euclidean length raised to the power of the parameter t ≥ 1, the edge is present in the shortest-path graph if and only if it is the least weight path between its endpoints. == Properties of shortest-path graph == When the configuration parameter t goes to infinity, shortest-path graph become the minimum spanning tree of the point set.

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A simple pendulum of length $b$ and bob with mass $m$ is attached to a massless support moving vertically upward with constant acceleration $a$. Determine the period for small oscillations.

The weight of the bob itself has little effect on the period of the pendulum. The period depends on the length of the pendulum, and also to a slight degree on its weight distribution (the moment of inertia about its own center of mass) and the amplitude (width) of the pendulum's swing. Note that under the small-angle approximation, the period is independent of the amplitude ; this is the property of isochronism that Galileo discovered. ===Rule of thumb for pendulum length=== T_0 = 2\pi\sqrt{\frac \ell g} gives \ell = \frac{g}{\pi^2}\frac{T_0^2} 4. In this case the pendulum's period depends on its moment of inertia around the pivot point. First start by defining the torque on the pendulum bob using the force due to gravity. \boldsymbol{ \tau } = \mathbf{l} \times \mathbf{F}_\mathrm{g} , where is the length vector of the pendulum and is the force due to gravity. The expression for is of the same form as the conventional simple pendulum and gives a period of T = 2 \pi \sqrt{\frac{I} {mgL}} And a frequency of f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{mgL}{I}} If the initial angle is taken into consideration (for large amplitudes), then the expression for \alpha becomes: \alpha = \ddot{\theta} = -\frac{mgL \sin\theta}{I} and gives a period of: T = 4 \operatorname{K}\left(\sin^2\frac{\theta_0}{2}\right) \sqrt{\frac{I}{mgL}} where is the maximum angle of oscillation (with respect to the vertical) and is the complete elliptic integral of the first kind. == Physical interpretation of the imaginary period == The Jacobian elliptic function that expresses the position of a pendulum as a function of time is a doubly periodic function with a real period and an imaginary period. A common weight for the bob of a one second pendulum, widely used in grandfather clocks and many others, is around 2 kilograms. == See also == * Plumb-bob ==References== Category:Pendulums Deviation of the "true" period of a pendulum from the small-angle approximation of the period. The equations of motion for two identical simple pendulums coupled by a spring connecting the bobs can be obtained using Lagrangian Mechanics. Therefore, :r = L \sin \theta \, Substituting this value for r yields a formula whose only varying parameter is the suspension angle θ: For small angles θ, cos(θ) ≈ 1; in which case :t \approx 2 \pi \sqrt { \frac {L} {g} } so that for small angles the period t of a conical pendulum is equal to the period of an ordinary pendulum of the same length. A bob is the mass on the end of a pendulum found most commonly, but not exclusively, in pendulum clocks. == Reason for use == Although a pendulum can theoretically be any shape, any rigid object swinging on a pivot, clock pendulums are usually made of a weight or bob attached to the bottom end of a rod, with the top attached to a pivot so it can swing. This yields an alternative and faster-converging formula for the period: T = \frac{2\pi}{M\left(1, \cos\frac{\theta_0} 2 \right)} \sqrt\frac\ell g. Expressing the solutions in terms of \theta_1 and \theta_2 alone: \begin{align} \theta_1&=\frac{1}{2}A\cos(\omega_1t+\alpha)+\frac{1}{2}B\cos(\omega_2t+\beta) \\\ \theta_2&=\frac{1}{2}A\cos(\omega_1t+\alpha)-\frac{1}{2}B\cos(\omega_2t+\beta) \end{align} If the bobs are not given an initial push, then the condition \dot\theta_1(0)=\dot\theta_2(0)=0 requires \alpha=\beta=0, which gives (after some rearranging): \begin{align} A&=\theta_1(0)+\theta_2(0)\\\ B&=\theta_1(0)-\theta_2(0) \end{align} ==See also== *Harmonograph *Conical pendulum *Cycloidal pendulum *Double pendulum *Inverted pendulum *Kapitza's pendulum *Rayleigh–Lorentz pendulum *Elastic pendulum *Mathieu function *Pendulum equations (software) ==References== ==Further reading== * * * ==External links== *Mathworld article on Mathieu Function Category:Differential equations Category:Dynamical systems Category:Horology Category:Mathematical physics Mathematics Since there is no acceleration in the vertical direction, the vertical component of the tension in the string is equal and opposite to the weight of the bob: :T \cos \theta = mg \, These two equations can be solved for T/m and equated, thereby eliminating T and m and yielding the centripetal acceleration: :{g\tan\theta} = \frac {v^2} {r} A little rearrangement gives: :\frac{g} {\cos\theta} = \frac {v^2} {r\sin \theta} Since the speed of the pendulum bob is constant, it can be expressed as the circumference 2πr divided by the time t required for one revolution of the bob: : v = \frac {2\pi r}{t} Substituting the right side of this equation for v in the previous equation, we find: : \frac {g} {\cos \theta} = \frac {( \frac {2 \pi r} {t} )^2} {r \sin \theta} = \frac {(2 \pi)^2 r} {t^2 \sin \theta} Using the trigonometric identity tan(θ) = sin(θ) / cos(θ) and solving for t, the time required for the bob to travel one revolution is :t = 2 \pi \sqrt {\frac {r} {g \tan \theta}} In a practical experiment, r varies and is not as easy to measure as the constant string length L. r can be eliminated from the equation by noting that r, h, and L form a right triangle, with θ being the angle between the leg h and the hypotenuse L (see diagram). Since this activity has been encouraged by many instructors, a simple approximate formula for the pendulum period valid for all possible amplitudes, to which experimental data could be compared, was sought. For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth ( = ) at initial angle 10 degrees is 4\sqrt{\frac{1\text{ m}}{g}}\ K\left(\sin\frac{10^\circ} {2} \right)\approx 2.0102\text{ s}. If it is assumed that the pendulum is released with zero angular velocity, the solution becomes The motion is simple harmonic motion where is the amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). File:Pendulum_0deg.gif|Initial angle of 0°, a stable equilibrium File:Pendulum_45deg.gif|Initial angle of 45° File:Pendulum_90deg.gif|Initial angle of 90° File:Pendulum_135deg.gif|Initial angle of 135° File:Pendulum_170deg.gif|Initial angle of 170° File:Pendulum_180deg.gif|Initial angle of 180°, unstable equilibrium == Damped, Driven Pendulum == The above discussion focuses on a pendulum bob only acted upon by the force of gravity. For now just consider the magnitude of the torque on the pendulum. |\boldsymbol{\tau}| = -mg\ell\sin\theta, where is the mass of the pendulum, is the acceleration due to gravity, is the length of the pendulum, and is the angle between the length vector and the force due to gravity. The bob has mass m and is suspended by a string of length L. thumb|The second pendulum, with a period of two seconds so each swing takes one second A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.Seconds pendulum == Pendulum == A pendulum is a weight suspended from a pivot so that it can swing freely. For a point mass on a weightless string of length L swinging with an infinitesimally small amplitude, without resistance, the length of the string of a seconds pendulum is equal to L = g/π2 where g is the acceleration due to gravity, with units of length per second squared, and L is the length of the string in the same units.

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In the blizzard of ' 88 , a rancher was forced to drop hay bales from an airplane to feed her cattle. The plane flew horizontally at $160 \mathrm{~km} / \mathrm{hr}$ and dropped the bales from a height of $80 \mathrm{~m}$ above the flat range. She wanted the bales of hay to land $30 \mathrm{~m}$ behind the cattle so as to not hit them. How far behind the cattle should she push the bales out of the airplane?

Prior to rural electrification, barns were equipped with a vertical pulley and a horizontal track along which a bale of hay was guided manually. ==See also== *Baler Category:Agricultural machinery The act of throwing the bales up to a higher level is called "bucking". thumb|220px|right|A 1950s hay elevator A hay elevator is an elevator that hauls bales of hay or straw up to a hayloft, the section of a barn used for hay storage. thumb|Hay hooks stuck into a haystack Hay bucking, or "bucking hay", is a type of manual labor where rectangular hay bales, ranging in weight from about , are stacked by hand in a field, in a storage area such as a barn, or stacked on a vehicle for transportation, such as a flatbed trailer or semi truck for delivery to where the hay is needed. The work is very strenuous and physically demanding, and is dependent upon using a proper technique in order to not become fatigued and avoid injury. thumb|left|A mechanical hay stacker Large quantities of small square bales are sometimes gathered with mechanical equipment such as a hay stacker, which can hold up to about 100 hay bales. thumb|Tractor with a bale handling implement thumb|Tractor carrying bales A bale handler is a generic term describing a piece of farm implement used to transport hay or straw bales. Bale handlers with hooks are used to move large and small bales. A typical hay elevator includes an open skeletal frame, with a chain that has dull 3-inch spikes every few feet along the chain to grab bales and drag them along. Bale spears can often move both round and square large bales. The term hay elevator also includes machinery involved in the stacking and storage of bales. Because the work is so labor-intensive, many farmers have taken to making multiple ton bales that are moved with machines. Shipping live cattle by truck was much more economical, humane and offered more options in routing cattle to auctions, feeders, and processors. Hay elevators are either ramped conveyor belts that bales rest on, or a mechanized pair of chains that holds bales taut between them. Cattle trails were carefully chosen to minimize distance and maximize feed to sustain and fatten cattle. thumb|Dangerous proximity of a hot air balloon to an overhead line. They can pinch, spear, hook and fork the bales, one or several at a time. Monty Python's Cow Tossing is a catapult-physics game. Prime Cut: Livestock Raising and Meatpacking in the United States 1607–1983. Workers are usually paid by the ton or by the number of bales. An apparatus known as an elevator is used to move the bales, conveyor belt style, to levels too high to buck them. Livestock transportation is the movement of livestock, by road, rail, ship, or air. The type of bale handling attachment will be built to handle the particular size and type of bale.

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Consider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped to $1 / e$ of its initial value. Find the ratio of the frequency of the damped oscillator to its natural frequency.

For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient: : \zeta = \frac{c}{c_c} = \frac {\text{actual damping}} {\text{critical damping}}, where the system's equation of motion is : m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0 and the corresponding critical damping coefficient is : c_c = 2 \sqrt{k m} or : c_c = 2 m \sqrt{\frac{k}{m}} = 2m \omega_n where : \omega_n = \sqrt{\frac{k}{m}} is the natural frequency of the system. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next. The damping ratio is dimensionless, being the ratio of two coefficients of identical units. == Derivation == Using the natural frequency of a harmonic oscillator \omega_n = \sqrt{{k}/{m}} and the definition of the damping ratio above, we can rewrite this as: : \frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = 0. The damping ratio is a parameter, usually denoted by ζ (Greek letter zeta), that characterizes the frequency response of a second-order ordinary differential equation. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism). == Q factor and decay rate == The Q factor, damping ratio ζ, and exponential decay rate α are related such that : \zeta = \frac{1}{2 Q} = { \alpha \over \omega_n }. If the forced frequency is equal to the natural frequency, the vibrations' amplitude increases manyfold. The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. If the oscillating system is driven by an external force at the frequency at which the amplitude of its motion is greatest (close to a natural frequency of the system), this frequency is called resonant frequency. ==Overview== Free vibrations of an elastic body are called natural vibrations and occur at a frequency called the natural frequency. * Time constant: \tau = 1 / \lambda, the time for the amplitude to decrease by the factor of e. Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. * Q factor: Q = 1 / (2 \zeta) is another non-dimensional characterization of the amount of damping; high Q indicates slow damping relative to the oscillation. == Damping ratio definition == thumb|400px|upright=1.3|The effect of varying damping ratio on a second-order system. The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. In a mass–spring system, with mass m and spring stiffness k, the natural angular frequency can be calculated as: \omega _0 =\sqrt{\frac{k}{m}} In an electrical network, ω is a natural angular frequency of a response function f(t) if the Laplace transform F(s) of f(t) includes the term , where for a real σ, and is a constant. The cycle per second is a once-common English name for the unit of frequency now known as the hertz (Hz). * Damping ratio: \zeta is a non-dimensional characterization of the decay rate relative to the frequency, approximately \zeta = \lambda / \omega, or exactly \zeta = \lambda / \sqrt{\lambda^2 + \omega^2} < 1. For example, a high quality tuning fork, which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer. ==Logarithmic decrement== thumb|400px|right| For underdamped vibrations, the damping ratio is also related to the logarithmic decrement \delta. Other important parameters include: * Frequency: f = \omega / (2\pi), the number of cycles per time unit. In LC and RLC circuits, its natural angular frequency can be calculated as: \omega _0 =\frac{1}{\sqrt{LC}} ==See also== * Fundamental frequency ==Footnotes== ==References== * * * * Category:Waves Category:Oscillation The general equation for an exponentially damped sinusoid may be represented as: y(t) = A e^{-\lambda t} \cos(\omega t - \varphi) where: *y(t) is the instantaneous amplitude at time ; *A is the initial amplitude of the envelope; *\lambda is the decay rate, in the reciprocal of the time units of the independent variable ; *\varphi is the phase angle at ; *\omega is the angular frequency. The damping ratio is a system parameter, denoted by (zeta), that can vary from undamped (), underdamped () through critically damped () to overdamped (). It can be shown that the set of natural frequencies in a network can be obtained by calculating the poles of all impedance and admittance functions of the network. In electronics, the long-term stability of an oscillator is the degree of uniformity of frequency over time, when the frequency is measured under identical environmental conditions, such as supply voltage, load, and temperature.

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What is the minimum escape velocity of a spacecraft from the moon?

Escape velocity is the minimum speed an object without propulsion needs to have to move away indefinitely from the source of the gravity field. The Apollo 17 LRV traveled a cumulative distance of approximately in a total drive time of about four hours and twenty-six minutes; the greatest distance Cernan and Schmitt traveled from the lunar module was about . The return to lunar orbit took just over seven minutes. It struck the Moon just under 87 hours into the mission, triggering the seismometers from Apollo 12, 14, 15 and 16. Apollo 17 (December 7–19, 1972) was the final mission of NASA's Apollo program, the most recent time humans have set foot on the Moon or traveled beyond low Earth orbit. This remains the furthest distance any spacefarers have ever traveled away from the safety of a pressurizable spacecraft while on a planetary body, and also during an EVA of any type. thumb|right|Concept of LESS Lunar escape systems (LESS) were a series of emergency vehicles designed for never-flown long-duration Apollo missions. The mission broke several records for crewed spaceflight, including the longest crewed lunar landing mission (12 days, 14 hours), greatest distance from a spacecraft during an extravehicular activity of any type (7.6 kilometers or 4.7 miles), longest total duration of lunar-surface extravehicular activities (22 hours, 4 minutes), largest lunar-sample return (approximately 115 kg or 254 lb), longest time in lunar orbit (6 days, 4 hours), and greatest number of lunar orbits (75). == Crew and key Mission Control personnel == In 1969, NASA announced that the backup crew of Apollo 14 would be Gene Cernan, Ronald Evans, and former X-15 pilot Joe Engle. * George J. Hurt Jr, David B. Middleton, and Marion A. Wise, Development Of A Simulator For Studying Simplified Lunar Escape Systems, April 1971 * George J. Hurt Jr and David B. Middleton, Fixed-base Simulator Investigation Of Lightweight Vehicles For Lunar Escape To Orbit With Kinesthetic Attitude Control And Simplified Manual Guidance, June 1971 * David B. Middleton and George J. Hurt Jr, A Simulation Study Of Emergency Lunar Escape To Orbit Using Several Simplified Manual Guidance And Control Techniques, October 1971 ==References== ==External links== *False Steps - LESS: The Lunar Escape System *Simulation of the LESS on YouTube Category:Apollo program hardware It was a Luna E-8-5M spacecraft, the second of three to be launched. The Apollo 17 spacecraft reentered Earth's atmosphere and splashed down safely in the Pacific Ocean at 2:25 p.m. EST, from the recovery ship, . The CMP was given information regarding the lunar features he would overfly in the CSM and which he was expected to photograph. == Mission hardware and experiments == thumb|SA-512, Apollo 17's Saturn V rocket, on the launch pad awaiting liftoff, November 1972|alt=Saturn five rocket on a launch pat at dusk while cloudy outside. === Spacecraft and launch vehicle === The Apollo 17 spacecraft comprised CSM-114 (consisting of Command Module 114 (CM-114) and Service Module 114 (SM-114)); Lunar Module 12 (LM-12); a Spacecraft-Lunar Module Adapter (SLA) numbered SLA-21; and a Launch Escape System (LES). * Apollo 17 Mission Experiments Overview at the Lunar and Planetary Institute * Apollo 17 Voice Transcript Pertaining to the Geology of the Landing Site (PDF) by N. G. Bailey and G. E. Ulrich, United States Geological Survey, 1975 * "Apollo Program Summary Report" (PDF), NASA, JSC-09423, April 1975 * The Apollo Spacecraft: A Chronology NASA, NASA SP-4009 * * "The Final Flight" – Excerpt from the September 1973 issue of National Geographic magazine Category:Gene Cernan Category:Ronald Evans (astronaut) Category:Harrison Schmitt Category:1972 in the United States Category:Apollo program missions Category:Articles containing video clips Category:Extravehicular activity Category:Lunar rovers Category:Crewed missions to the Moon Category:Sample return missions Category:Soft landings on the Moon Category:Spacecraft launched in 1972 Category:Spacecraft which reentered in 1972 Category:Last events Category:December 1972 events Category:Spacecraft launched by Saturn rockets Category:1972 on the Moon On 20 March 2013, the asteroid passed 49 lunar distances or from Earth at a relative velocity of . Launched at 12:33 a.m. Eastern Standard Time (EST) on December 7, 1972, following the only launch-pad delay in the course of the whole Apollo program that was caused by a hardware problem, Apollo 17 was a "J-type" mission that included three days on the lunar surface, expanded scientific capability, and the use of the third Lunar Roving Vehicle (LRV). The Lunar Roving Vehicle allowed the astronauts to travel fairly quickly over a few miles, but an improved version of the LESS could allow rapid travel over much longer distances on rocket thrust. Luna E-8-5M No.412, also known as Luna Ye-8-5M No.412, and sometimes identified by NASA as Luna 1975A, was a Soviet spacecraft which was lost in a launch failure in 1975. (7888) 1993 UC is a near-Earth minor planet in the Apollo group. At approximately 160,000 nautical miles (184,000 mi; 296,000 km) from Earth, it was the third "deep space" EVA in history, performed at great distance from any planetary body. For one possible solution, NASA studied a number of low-cost, low-mass lunar escape systems (LESS) which could be carried on the lunar module as a backup, rather like a lifeboat on a ship. (7341) 1991 VK is a near-Earth minor planet in the Apollo group. At 3:46 a.m. EST, the S-IVB third stage was reignited for the 351-second trans-lunar injection burn to propel the spacecraft towards the Moon.

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Find the value of the integral $\int_S \mathbf{A} \cdot d \mathbf{a}$, where $\mathbf{A}=x \mathbf{i}-y \mathbf{j}+z \mathbf{k}$ and $S$ is the closed surface defined by the cylinder $c^2=x^2+y^2$. The top and bottom of the cylinder are at $z=d$ and 0 , respectively.

Let us note that the surface integral of this 2-form is the same as the surface integral of the vector field which has as components f_x, f_y and f_z. == Theorems involving surface integrals == Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem. == Dependence on parametrization == Let us notice that we defined the surface integral by using a parametrization of the surface S. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h. In other words, we have to integrate v with respect to the vector surface element \mathrm{d}\mathbf s = {\mathbf n} \mathrm{d}s, which is the vector normal to S at the given point, whose magnitude is \mathrm{d}s = \|\mathrm{d}{\mathbf s}\|. This formula defines the integral on the left (note the dot and the vector notation for the surface element). To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. Then, the surface integral of f on S is given by :\iint_D \left[ f_{z} ( \mathbf{r} (s,t)) \frac{\partial(x,y)}{\partial(s,t)} + f_{x} ( \mathbf{r} (s,t)) \frac{\partial(y,z)}{\partial(s,t)} + f_{y} ( \mathbf{r} (s,t))\frac{\partial(z,x)}{\partial(s,t)} \right]\, \mathrm ds\, \mathrm dt where :{\partial \mathbf{r} \over \partial s}\times {\partial \mathbf{r} \over \partial t}=\left(\frac{\partial(y,z)}{\partial(s,t)}, \frac{\partial(z,x)}{\partial(s,t)}, \frac{\partial(x,y)}{\partial(s,t)}\right) is the surface element normal to S. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, \Gamma = \oint_{\partial S} \mathbf{V}\cdot \mathrm{d}\mathbf{l} = \iint_S abla \times \mathbf{V} \cdot \mathrm{d}\mathbf{S}=\iint_S \boldsymbol{\omega} \cdot \mathrm{d}\mathbf{S} Here, the closed integration path is the boundary or perimeter of an open surface , whose infinitesimal element normal is oriented according to the right-hand rule. For integrals of scalar fields, the answer to this question is simple; the value of the surface integral will be the same no matter what parametrization one uses. thumb|right|An illustration of Stokes' theorem, with surface , its boundary and the normal vector . The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. If the Gaussian surface is chosen such that for every point on the surface the component of the electric field along the normal vector is constant, then the calculation will not require difficult integration as the constants which arise can be taken out of the integral. This is indeed how things work, but when integrating vector fields, one needs to again be careful how to choose the normal-pointing vector for each piece of the surface, so that when the pieces are put back together, the results are consistent. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal n to S at each point, which will give us a scalar field, and integrate the obtained field as above. The integral of v on S was defined in the previous section. The surface integral can also be expressed in the equivalent form : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \sqrt{g} \, \mathrm ds\, \mathrm dt where is the determinant of the first fundamental form of the surface mapping . Then, the surface integral is given by : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \left\|{\partial \mathbf{r} \over \partial s} \times {\partial \mathbf{r} \over \partial t}\right\| \mathrm ds\, \mathrm dt where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of , and is known as the surface element (which would, for example, yield a smaller value near the poles of a sphere. where the lines of longitude converge more dramatically, and latitudinal coordinates are more compactly spaced). The differential vector area is , on each surface a, b and c. right|frame|Closed surface in the form of a cylinder having line charge in the center and showing differential areas of all three surfaces. If, however, the normals for these parametrizations point in opposite directions, the value of the surface integral obtained using one parametrization is the negative of the one obtained via the other parametrization. The obvious solution is then to split that surface into several pieces, calculate the surface integral on each piece, and then add them all up. With the above notation, if \mathbf{F} is any smooth vector field on \R^3, thenRobert Scheichl, lecture notes for University of Bath mathematics course \oint_{\partial\Sigma} \mathbf{F}\, \cdot\, \mathrm{d}{\mathbf{\Gamma}} = \iint_{\Sigma} abla\times\mathbf{F}\, \cdot\, \mathrm{d}\mathbf{\Sigma}. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface.

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A rocket has an initial mass of $7 \times 10^4 \mathrm{~kg}$ and on firing burns its fuel at a rate of 250 $\mathrm{kg} / \mathrm{s}$. The exhaust velocity is $2500 \mathrm{~m} / \mathrm{s}$. If the rocket has a vertical ascent from resting on the earth, how long after the rocket engines fire will the rocket lift off? What is wrong with the design of this rocket?

# A two-stage rocket with a length of 6 metres and a takeoff thrust of 50 kN. # A three-stage rocket with a length of 12.8 metres, a diameter of 0.56 metres and a takeoff thrust of 50 kN. In rocketry, the Goddard problem is to optimize the peak altitude of a rocket, ascending vertically, and taking into account atmospheric drag and the gravitational field. This rocket was first launched on November 19, 1962, near Cuxhaven and reached a height of 40 km. For miniature black powder rocket motors (13 mm diameter), the maximum thrust is between 5 and 12 N, the total impulse is between .5 and 2.2 Ns, and the burn time is between .25 and 1 second. Some rockets (typically long thin rockets) are the proper proportions to safely glide to Earth tail-first. This rocket was first launched on February 7, 1963, and reached a height of 80 km. This rocket was first launched on May 2, 1963, with reduced fuel and reached an altitude of 110 km. thumb|The Rocket The Rocket (previously The Rising Sun) is a Grade II listed public house at 120 Euston Road, Euston, London NW1 2AL. thumb|250px|Picture sequence of a model rocket launch using a B4-4 engine thumb|250px|A typical model rocket during launch (16 times slower) A model rocket is a small rocket designed to reach low altitudes (e.g., for model) and be recovered by a variety of means. Many science fiction authors as well as depictions in popular culture showed rockets landing vertically, typically resting after landing on the space vehicle's fins. Kappa is a family of solid-fuel Japanese sounding rockets, which were built starting from 1956. ==Rockets== ===Kappa 1=== * Ceiling: 40 km * Takeoff thrust: 10.00 kN * Diameter: 0.13 m * Length: 2.70 m ===Kappa 2=== * Ceiling: 40 km * Mass: 300 kg * Diameter: 0.22 m * Length: 5 m ===Kappa 6 (in two stages)=== * Pay load: 20 kg * Ceiling: 60 km * Takeoff weight: 270 kg * Diameter: 0.25 m * Length: 5.61 m ===Kappa 7=== * Ceiling: 50 km * Diameter: 0.42 m * Length: 8.70 m ===Kappa 8 (in two stages)=== * Pay load: 50 kg * Ceiling: 160 km * Takeoff weight: 1500 kg * Diameter: 0.42 m * Length: 10.90 m ===Kappa 4=== * Ceiling: 80 km * Takeoff thrust: 105.00 kN * Diameter: 0.33 m * Length: 5.90 m ===Kappa 9L=== * Pay load: 15 kg * Ceiling: 350 km * Takeoff weight: 1550 kg * Diameter: 0.42 m * Length: 12.50 m ===Kappa 9M=== * Pay load: 50 kg * Ceiling: 350 km * Mass: 1500 kg * Diameter: 0.42 m * Length: 11.10 m ===Kappa 8L=== * Pay load: 25 kg * Ceiling: 200 km * Takeoff weight: 350 kg * Diameter: 0.25 m * Length: 7.30 m ===Kappa 10=== * Ceiling: 742 km ==See also== * R-25 Vulkan ==External links== * Kappa-Rocket Category:Solid- fuel rockets Category:Sounding rockets of Japan Category:Japanese inventions The spacecraft stopped mid-air again and, as the engines throttled back, began its successful vertical landing. For example, a heavier rocket would require an engine with more initial thrust to get it off of the launch pad, whereas a lighter rocket would need less initial thrust and would sustain a longer burn, reaching higher altitudes. ===Last number=== The last number is the delay in seconds between the end of the thrust phase and ignition of the ejection charge. University of New South Wales at the Australian Defence Force Academy. 2008.Measuring thrust and predicting trajectory in model rocketry M. Courtney and A. Courtney. Later with maximum fuel it reached a height of 150 km. They were # A single- stage rocket with a length of 3.4 metres and a takeoff thrust of 50 kN. *On July 20, 2021, Blue Origin's New Shepard rocket made its first-ever successful vertical landing following a crewed suborbital flight. All Seliger Rockets return to the ground by parachute. The D class 24 mm motors have a maximum thrust between 29.7 and 29.8 N, a total impulse between 16.7 and 16.85 Ns, and a burn time between 1.6 and 1.7 seconds. The E class 24 mm motors have a maximum thrust between 19.4 and 19.5 N, a total impulse between 28.45 and 28.6 Ns, and a burn time between 3 and 3.1 seconds. *On December 21, 2015, SpaceX's 20th Falcon 9 first stage made the first-ever successful vertical landing of an orbital-class booster after boosting 11 commercial satellites to low Earth orbit on Falcon 9 Flight 20.

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A spacecraft of mass $10,000 \mathrm{~kg}$ is parked in a circular orbit $200 \mathrm{~km}$ above Earth's surface. What is the minimum energy required (neglect the fuel mass burned) to place the satellite in a synchronous orbit (i.e., $\tau=24 \mathrm{hr}$ )?

The specific orbital energy associated with this orbit is −29.6MJ/kg: the potential energy is −59.2MJ/kg, and the kinetic energy 29.6MJ/kg. Thus, the total energy of the system remains unchanged though the mechanical energy of the system has reduced. ===Satellite=== thumb|plot of kinetic energy K, gravitational potential energy, U and mechanical energy E_\text{mechanical} versus distance away from centre of earth, r at R= Re, R= 2*Re, R=3*Re and lastly R = geostationary radius A satellite of mass m at a distance r from the centre of Earth possesses both kinetic energy, K, (by virtue of its motion) and gravitational potential energy, U, (by virtue of its position within the Earth's gravitational field; Earth's mass is M). Hence, mechanical energy E_\text{mechanical} of the satellite-Earth system is given by E_\text{mechanical} = U + K E_\text{mechanical} = - G \frac{M m}{r}\ + \frac{1}{2}\, m v^2 If the satellite is in circular orbit, the energy conservation equation can be further simplified into E_\text{mechanical} = - G \frac{M m}{2r} since in circular motion, Newton's 2nd Law of motion can be taken to be G \frac{M m}{r^2}\ = \frac{m v^2}{r} ==Conversion== Today, many technological devices convert mechanical energy into other forms of energy or vice versa. If the satellite's orbit is an ellipse the potential energy of the satellite, and its kinetic energy, both vary with time but their sum remains constant. For the Earth and a just little more than R the additional specific energy is (gR/2); which is the kinetic energy of the horizontal component of the velocity, i.e. \frac{1}{2}V^2 = \frac{1}{2}gR, V=\sqrt{gR}. ==Examples== ===ISS=== The International Space Station has an orbital period of 91.74 minutes (5504s), hence by Kepler's Third Law the semi-major axis of its orbit is 6,738km. For an elliptic orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity (parabolic orbit). This corresponds to the fact that for such orbits the total energy is one half of the potential energy, because the kinetic energy is minus one half of the potential energy. ==Additional energy== If the central body has radius R, then the additional specific energy of an elliptic orbit compared to being stationary at the surface is -\frac{\mu}{2a}+\frac{\mu}{R} = \frac{\mu(2a-R)}{2aR} The quantity 2a-R is the height the ellipse extends above the surface, plus the periapsis distance (the distance the ellipse extends beyond the center of the Earth). In the gravitational two-body problem, the specific orbital energy \varepsilon (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (\varepsilon_p) and their total kinetic energy (\varepsilon_k), divided by the reduced mass. Finally, a circularization burn is required to raise the perigee to the same altitude and remove any remaining inclination. ===Translunar or interplanetary spacecraft=== In order to reach the Moon or a planet at a desired time, the spacecraft must be launched within a limited range of times known as the launch window. A parking orbit is a temporary orbit used during the launch of a spacecraft. For an altitude of 100km (radius is 6471km): The energy is −30.8MJ/kg: the potential energy is −61.6MJ/kg, and the kinetic energy 30.8MJ/kg. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points. The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. Unless the launch site itself is quite close to the equator, it requires an impractically large amount of fuel to launch a spacecraft directly into such an orbit. Indian Mini Satellite (IMS) is a family of modular \- mini satellite buses developed by the Indian Space Research Organisation (ISRO). ==Variants== Indian Mini Satellite (Variants) Feature IMS-1 IMS-2 IMS-3 (Planned IMS-2 Derivative) Launch Mass Maximum bus mass Payload mass Propellant Design lifetime 2 years 5 years Raw bus voltage 28-33 Volts 28-33 Volts 28-42 Volts Solar Array Power 330 Watts (EOL) 675 Watts (EOL) 850 Watts (BOL) 850 Watts (BOL) Payload power 30 Watts (Continuous) 70 Watts (Duty Cycle) 250 Watts (Continuous) 600 Watts (Duty Cycle) 250 Watts (Continuous) 400 Watts (Duty Cycle) Attitude Control 3-axis stabilized Four Reaction Wheels Single 1N thruster 3-axis stabilized Four Reaction Wheels Mono-propellant RCS Four 1N thrusters Four 0.2N thrusters Pointing Accuracy ±0.1° (3σ) (all axes) ± 0.1° (all axes) ± 0.1° (all axes) SSR Storage 32 Gb 32 Gb (SDRAM) 256 Gb (Flash Memory) 32 Gb (SDRAM) 256 Gb (Flash Memory) Payload data storage ≤ 16 Gb ≤ 32 Gb Downlink ≤ 8 Mbit/s DL rate ≤ 105 Mbit/s DL rate ≤ 160 Mbit/s DL rate Missions IMS-1 Youthsat Microsat-TD SARAL \- ScatSat-1 EMISAT HySIS XPoSat (Planned) center|thumb|150x150px|IMS-1 ==See also== * Comparison of satellite buses ==References== Category:Indian Space Research Organisation Category:Satellite buses Thus, if orbital position vector (\mathbf{r}) and orbital velocity vector (\mathbf{v}) are known at one position, and \mu is known, then the energy can be computed and from that, for any other position, the orbital speed. ==Rate of change== For an elliptic orbit the rate of change of the specific orbital energy with respect to a change in the semi-major axis is \frac{\mu}{2a^2} where * \mu={G}(m_1 + m_2) is the standard gravitational parameter; *a\,\\! is semi- major axis of the orbit. The speed is 7.8km/s, the net delta-v to reach this orbit is 8.0km/s. The average speed is 7.7km/s, the net delta-v to reach this orbit is 8.1km/s (the actual delta-v is typically 1.5–2.0km/s more for atmospheric drag and gravity drag). A satellite is said to occupy an inclined orbit around Earth if the orbit exhibits an angle other than 0° to the equatorial plane. For a hyperbolic orbit, it is equal to the excess energy compared to that of a parabolic orbit. thumb|250px|An example of a mechanical system: A satellite is orbiting the Earth influenced only by the conservative gravitational force; its mechanical energy is therefore conserved. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time: \begin{align} \varepsilon &= \varepsilon_k + \varepsilon_p \\\ &= \frac{v^2}{2} - \frac{\mu}{r} = -\frac{1}{2} \frac{\mu^2}{h^2} \left(1 - e^2\right) = -\frac{\mu}{2a} \end{align} where *v is the relative orbital speed; *r is the orbital distance between the bodies; *\mu = {G}(m_1 + m_2) is the sum of the standard gravitational parameters of the bodies; *h is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass; *e is the orbital eccentricity; *a is the semi- major axis.

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A uniformly solid sphere of mass $M$ and radius $R$ is fixed a distance $h$ above a thin infinite sheet of mass density $\rho_5$ (mass/area). With what force does the sphere attract the sheet?

This force depends on the surface separation h. Since the hydrodynamic drag of a sphere close to a planar substrate is known theoretically, the spring constant of the cantilever can be deduced. In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. The mass distribution of a solid defines its center of gravity and influences its dynamical behaviour - e.g. the oscillations and eventual rotation. ==Mathematical modelling== A mass distribution can be modeled as a measure. Similar use of the equation can be made in the settling of fine particles in water or other fluids. === Terminal velocity of sphere falling in a fluid === At terminal (or settling) velocity, the excess force due to the difference between the weight and buoyancy of the sphere (both caused by gravity) is given by: :F_g = ( \rho_p - \rho_f)\, g\, \frac{4}{3}\pi\, R^3, where (in SI units): * is the mass density of the sphere [kg/m3] * is the mass density of the fluid [kg/m3] * is the gravitational acceleration [m/s] Requiring the force balance and solving for the velocity gives the terminal velocity . A sphere of known size and density is allowed to descend through the liquid. An analytical and closed- form solution for both 2D and 3D, mechanically stable, random packings of spheres has been found by A. Zaccone in 2022 using the assumption that the most random branch of jammed states (maximally random jammed packings, extending up to the fcc closest packing) undergo crowding in a way qualitatively similar to an equilibrium liquid. Random close packing (RCP) of spheres is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. thumb|right|280px|Scheme of the colloidal probe technique for direct force measurements in the sphere-plane and sphere-sphere geometries. The random close packing value is significantly below the maximum possible close-packing of same-size hard spheres into a regular crystalline arrangements, which is 74.04%.Modes of wall induced granular crystallisation in vibrational packing.Granular Matter, 21(2), 26 Both the face-centred cubic (fcc) and hexagonal close packed (hcp) crystal lattices have maximum densities equal to this upper limit, which can occur through the process of granular crystallisation. The distance to the sphere of influence must thus satisfy \frac{m_B}{m_A} \frac{r^3}{R^3} = \frac{m_A}{m_B} \frac{R^2}{r^2} and so r = R\left(\frac{m_A}{m_B}\right)^{2/5} is the radius of the sphere of influence of body A ==See also== * Hill sphere * Sphere of influence (black hole) ==References== == General references == * * * == External links == *Project Pluto Category:Astrodynamics Category:Orbits When the force is attractive, the cantilever is attracted to the surface and may become unstable. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.Batchelor (1967), p. 233. ==Statement of the law== The force of viscosity on a small sphere moving through a viscous fluid is given by: :F_{\rm d} = 6 \pi \mu R v where (in SI units): * is the frictional force – known as Stokes' drag – acting on the interface between the fluid and the particle (newtons, kg m s−2); * (some authors use the symbol ) is the dynamic viscosity (Pascal-seconds, kg m−1 s−1); * is the radius of the spherical object (meters); * is the flow velocity relative to the object (meters per second). A sphere of influence (SOI) in astrodynamics and astronomy is the oblate- spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. The problem of predicting theoretically the random close pack of spheres is difficult mainly because of the absence of a unique definition of randomness or disorder. In fluid dynamics, Stokes' law is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. The matrix represents the identity-matrix. :\boldsymbol{\sigma} = - p \cdot \mathbf{I} + \mu \cdot \left(( abla \mathbf{u}) + ( abla \mathbf{u})^T \right) The force acting on the sphere is to calculate by surface-integral, where represents the radial unit-vector of spherical-coordinates: :\begin{align} \mathbf{F} &= \iint_{\partial V}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\subset\\!\supset \;\boldsymbol{\sigma}\cdot \text{d}\mathbf{S} \\\\[4pt] &= \int_{0}^{\pi}\int_{0}^{2\pi} \boldsymbol{\sigma}\cdot \mathbf{e_r}\cdot R^2 \sin\theta \text{d}\varphi\text{d}\theta \\\\[4pt] &= \int_{0}^{\pi}\int_{0}^{2\pi} \frac{3\mu \cdot \mathbf{u}_{\infty}}{2 R}\cdot R^2 \sin\theta \text{d}\varphi\text{d}\theta \\\\[4pt] &= 6\pi\mu R \cdot \mathbf{u}_{\infty} \end{align} === Rotational flow around a sphere === thumb|434x434px|Stokes-Flow around sphere: \boldsymbol{\omega}_R = \begin{pmatrix} 0 & 0 & 2 \end{pmatrix}^T \; \text{Hz} , \mu = 1 \; \text{mPa} \cdot \text{s}, R = 1 \; \text{m} :\begin{align} \mathbf{u}(\mathbf{x}) &= - \;R^3 \cdot \frac{ \boldsymbol{\omega}_{R} \times \mathbf{x}}{\|\mathbf{x}\|^3} \\\\[8pt] \boldsymbol{\omega}(\mathbf{x}) &= \frac{R^3 \cdot \boldsymbol{\omega}_{R}}{\|\mathbf{x}\|^3} - \frac{3 R^3 \cdot (\boldsymbol{\omega}_{R} \cdot \mathbf{x})\cdot \mathbf{x}}{\|\mathbf{x}\|^5} \\\\[8pt] p(\mathbf{x}) &= 0 \\\\[8pt] \boldsymbol{\sigma} &= - p \cdot \mathbf{I} + \mu \cdot \left( ( abla \mathbf{u}) + ( abla \mathbf{u})^T \right) \\\\[8pt] \mathbf{T} &= \iint_{\partial V}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\subset\\!\supset \mathbf{x} \times \left( \boldsymbol{\sigma} \cdot \text{d}\boldsymbol{S} \right) \\\ &= \int_{0}^{\pi} \int_{0}^{2\pi} (R \cdot \mathbf{e_r}) \times \left( \boldsymbol{\sigma} \cdot \mathbf{e_r} \cdot R^2 \sin\theta \text{d}\varphi \text{d}\theta \right) \\\ &= 8\pi\mu R^3 \cdot \boldsymbol{\omega}_{R} \end{align} ==Other types of Stokes flow== Although the liquid is static and the sphere is moving with a certain velocity, with respect to the frame of sphere, the sphere is at rest and liquid is flowing in the opposite direction to the motion of the sphere. ==See also== * Einstein relation (kinetic theory) * Scientific laws named after people * Drag equation * Viscometry * Equivalent spherical diameter * Deposition (geology) == Sources == * * Originally published in 1879, the 6th extended edition appeared first in 1932. ==References== Category:Fluid dynamics In the case of a soft repulsive force, the cantilever is repelled from the surface and only slowly approaches the constant compliance region. Experiments and computer simulations have shown that the most compact way to pack hard perfect same- size spheres randomly gives a maximum volume fraction of about 64%, i.e., approximately 64% of the volume of a container is occupied by the spheres. It is also possible to study forces between colloidal particles by attaching another particle to the substrate and perform the measurement in the sphere-sphere geometry, see figure above. thumb|left|350px|Principle of the force measurements by the colloidal probe technique. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). From the previous two equations, and with the appropriate boundary conditions, for a far-field uniform-flow velocity in the –direction and a sphere of radius , the solution is found to beLamb (1994), §337, p. 598. : \psi(r,z) = - \frac{1}{2}\, u\, r^2\, \left[ 1 \- \frac{3}{2} \frac{R}{\sqrt{r^2+z^2}} \+ \frac{1}{2} \left( \frac{R}{\sqrt{r^2+z^2}} \right)^3\; \right]. thumb|420x420px|Stokes-Flow around sphere with parameters of Far-Field velocity \mathbf{u}_{\infty} = \begin{pmatrix} 6 & 0 & 6 \end{pmatrix}^T \text{m/s}, radius of sphere R = 1 \; \text{m}, viscosity of water (T = 20°C) \mu = 1 \; \text{mPa}\cdot \text{s} .

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Consider a thin rod of length $l$ and mass $m$ pivoted about one end. Calculate the moment of inertia.

* The moment of inertia of a thin rod with constant cross-section s and density \rho and with length \ell about a perpendicular axis through its center of mass is determined by integration. Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about one end. * The moment of inertia of the compound pendulum is now obtained by adding the moment of inertia of the rod and the disc around the pivot point P as, I_P = I_{C, \text{rod}} + M_\text{rod}\left(\frac{L}{2}\right)^2 + I_{C, \text{disc}} + M_\text{disc}(L + R)^2, where L is the length of the pendulum. The moment of inertia I_\mathbf{C} about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the polar moment of inertia. The quantity I = mr^2 is the moment of inertia of this single mass around the pivot point. Description Figure Moment(s) of inertia Point mass M at a distance r from the axis of rotation. File:moment of inertia thin cylinder.png I \approx m r^2 \,\\! For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. The moment of inertia is also defined as the ratio of the net angular momentum of a system to its angular velocity around a principal axis, that is I = \frac{L}{\omega}. File:moment of inertia thick cylinder h.svg I_z = \frac{1}{2} m \left(r_2^2 + r_1^2\right) = m r_2^2 \left(1-t+\frac{t^2}{2}\right) Classical Mechanics - Moment of inertia of a uniform hollow cylinder . Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point P so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (t), to obtain I_P = \frac{mgr}{\omega_\text{n}^2} = \frac{mgrt^2}{4\pi^2}, where t is the period (duration) of oscillation (usually averaged over multiple periods). ====Center of oscillation==== A simple pendulum that has the same natural frequency as a compound pendulum defines the length L from the pivot to a point called the center of oscillation of the compound pendulum. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems. For a simple pendulum, this definition yields a formula for the moment of inertia in terms of the mass of the pendulum and its distance from the pivot point as, I = mr^2. File:moment of inertia solid rectangular prism.png I_h = \frac{1}{12} m \left(w^2+d^2\right) I_w = \frac{1}{12} m \left(d^2+h^2\right) I_d = \frac{1}{12} m \left(w^2+h^2\right) Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal. Align the x-axis with the rod and locate the origin its center of mass at the center of the rod, then I_{C, \text{rod}} = \iiint_Q \rho\,x^2 \, dV = \int_{-\frac{\ell}{2}}^\frac{\ell}{2} \rho\,x^2 s\, dx = \left. \rho s\frac{x^3}{3}\right|_{-\frac{\ell}{2}}^\frac{\ell}{2} = \frac{\rho s}{3} \left(\frac{\ell^3}{8} + \frac{\ell^3}{8}\right) = \frac{m\ell^2}{12}, where m = \rho s \ell is the mass of the rod. Also, a point mass m at the end of a rod of length r has this same moment of inertia and the value r is called the radius of gyration. The moment of inertia of an arbitrarily shaped body is the sum of the values mr^2 for all of the elements of mass in the body. === Compound pendulums === A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. The stresses in a beam are calculated using the second moment of the cross-sectional area around either the x-axis or y-axis depending on the load. ==== Examples ==== thumb|right The moment of inertia of a compound pendulum constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centers of mass. In this case, the moment of inertia of the mass in this system is a scalar known as the polar moment of inertia. The moments of inertia of a mass have units of dimension ML2([mass] × [length]2). A point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved. frameless|upright I = M r^2 Two point masses, m1 and m2, with reduced mass μ and separated by a distance x, about an axis passing through the center of mass of the system and perpendicular to the line joining the two particles. frameless|upright I = \frac{ m_1 m_2 }{ m_1 \\! + \\! m_2 } x^2 = \mu x^2 Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about its center.

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A clown is juggling four balls simultaneously. Students use a video tape to determine that it takes the clown $0.9 \mathrm{~s}$ to cycle each ball through his hands (including catching, transferring, and throwing) and to be ready to catch the next ball. What is the minimum vertical speed the clown must throw up each ball?

175px|thumb|Ladder diagram In toss juggling, a cascade is the simplest juggling pattern achievable with an odd number of props. The ball is then thrown from above the 4 diagonally downward to the opposite hand. 150px|thumb|Ladder diagram for box: (4,2x)(2x,4) In toss juggling, the box is a juggling pattern for 3 objects, most commonly balls or bean bags. "Juggling with your arms up in the air above your head & looking up from underneath the pattern."Darbyshire (1993), p.22. and a virtually infinite number of cascade patterns such as 522, 720, 900, 72222, and so on (see article on Siteswap notation).Beever, Ben (2001), p.15. ==Shannon's theorem== thumb|An illustration of Shannon's juggling theorem for the cascade juggling pattern 175px|thumb|Cascade ladder suggested by Shannon's formula Claude Shannon, builder of the first juggling robot, developed a juggling theorem, relating the time balls spend in the air and in the hands: (F+D)H=(V+D)N, where F = time a ball spends in the air, D = time a ball spends in a hand/time a hand is full, V = time a hand is vacant, N = number of balls, and H = number of hands. ==Number of props== ===Three-ball=== For the three-ball cascade the juggler starts with two balls in one hand and the third ball in the other hand. Before catching this ball the juggler must throw the ball in the receiving hand, in a similar arc, to the first hand. The simplest juggling pattern is the three-ball cascade,Bernstein, Nicholai A. (1996). 150px|thumb|Cascade flash: 3 throws & 3 catches 150px|thumb|Mills mess flash: 6 throws & 6 catches In toss juggling, a flash is either a form of numbers juggling where each ball in a juggling pattern is only thrown and caught once or it is a juggling trick where every prop is simultaneously in the air and both hands are empty."Three ball flash", TWJC.co.uk. Two balls are dedicated to a specific hand with vertical throws, and the third ball is thrown horizontally between the two hands. For some tricks the number of throws and catches to complete a juggling cycle for that trick is not simply a multiple of the number of objects being juggled. Juggling, p.23. thumb|400px|right|Racking a game of three-ball with the standard fifteen-ball triangle rack. However, in order to keep the number of props in the juggler's hands to a minimum, it is necessary to begin the pattern by throwing, from alternating hands, all but one prop (in the same hand as the first throw, which started with one more prop than the other) before any catches are made. ==Reverse cascade== thumb|right|upright|An illustration of the three-ball reverse cascade. One ball is thrown from the first hand in an arc to the other hand. One juggles, "a cascade with two balls while the 'tennis' ball is thrown [back and forth] over the top."Darbyshire (1993), p.23. The goal is to () the three object balls in as few shots as possible.PoolSharp's "Three-Ball Rules". "The cascade is the simplest three ball juggling pattern." Juggling, p.26. "In the cascade...the crossing of the balls between the hands demands that one hand catches at the same rate that the other hand throws . Higher numbers require the balls to be tossed higher into the air in order to allow more time for a complete cycle of throws. Title Description Demonstration The Shuffle In a shuffle throw, the vamp ball begins above and outside the vertical path of one of the box's "side balls" and is thrown diagonally downward, caught below the opposite "side ball". Charlie Dancey's Encyclopædia of Ball Juggling p98.

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A billiard ball of initial velocity $u_1$ collides with another billiard ball (same mass) initially at rest. The first ball moves off at $\psi=45^{\circ}$. For an elastic collision, what are the velocities of both balls after the collision?

Indeed, to derive the equations, one may first change the frame of reference so that one of the known velocities is zero, determine the unknown velocities in the new frame of reference, and convert back to the original frame of reference. ====Examples==== ;Before collision: :Ball 1: mass = 3 kg, velocity = 4 m/s :Ball 2: mass = 5 kg, velocity = −6 m/s ;After collision: :Ball 1: velocity = −8.5 m/s :Ball 2: velocity = 1.5 m/s Another situation: frame|center|Elastic collision of unequal masses. The conservation of the total momentum before and after the collision is expressed by: m_{1}u_{1}+m_{2}u_{2} \ =\ m_{1}v_{1} + m_{2}v_{2}. In physics, an elastic collision is an encounter (collision) between two bodies in which the total kinetic energy of the two bodies remains the same. Consider particles 1 and 2 with masses m1, m2, and velocities u1, u2 before collision, v1, v2 after collision. In an elastic collision these magnitudes do not change. The final velocities can then be calculated from the two new component velocities and will depend on the point of collision. (To get the and velocities of the second ball, one needs to swap all the '1' subscripts with '2' subscripts.) In the center of momentum frame where the total momentum equals zero, \begin{align} p_1 &= - p_2 \\\ p_1^2 &= p_2^2 \\\ E &= \sqrt {m_1^2c^4 + p_1^2c^2} + \sqrt {m_2^2c^4 + p_2^2c^2} = E \\\ p_1 &= \pm \frac{\sqrt{E^4 - 2E^2m_1^2c^4 - 2E^2m_2^2c^4 + m_1^4c^8 - 2m_1^2m_2^2c^8 + m_2^4c^8}}{2cE} \\\ u_1 &= -v_1. \end{align} Here m_1, m_2 represent the rest masses of the two colliding bodies, u_1, u_2 represent their velocities before collision, v_1, v_2 their velocities after collision, p_1, p_2 their momenta, c is the speed of light in vacuum, and E denotes the total energy, the sum of rest masses and kinetic energies of the two bodies. In the case of macroscopic bodies, perfectly elastic collisions are an ideal never fully realized, but approximated by the interactions of objects such as billiard balls. Once v_1 is determined, v_2 can be found by symmetry. ====Center of mass frame==== With respect to the center of mass, both velocities are reversed by the collision: a heavy particle moves slowly toward the center of mass, and bounces back with the same low speed, and a light particle moves fast toward the center of mass, and bounces back with the same high speed. The magnitudes of the velocities of the particles after the collision are: \begin{align} v'_1 &= v_1\frac{\sqrt{m_1^2+m_2^2+2m_1m_2\cos \theta}}{m_1+m_2} \\\ v'_2 &= v_1\frac{2m_1}{m_1+m_2}\sin \frac{\theta}{2}. \end{align} ===Two-dimensional collision with two moving objects=== The final x and y velocities components of the first ball can be calculated as: \begin{align} v'_{1x} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\cos(\varphi)+v_{1}\sin(\theta_1-\varphi)\cos(\varphi + \tfrac{\pi}{2}) \\\\[0.8em] v'_{1y} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\sin(\varphi)+v_{1}\sin(\theta_1-\varphi)\sin(\varphi + \tfrac{\pi}{2}), \end{align} where and are the scalar sizes of the two original speeds of the objects, and are their masses, and are their movement angles, that is, v_{1x} = v_1\cos\theta_1,\; v_{1y}=v_1\sin\theta_1 (meaning moving directly down to the right is either a −45° angle, or a 315° angle), and lowercase phi () is the contact angle. A useful special case of elastic collision is when the two bodies have equal mass, in which case they will simply exchange their momenta. Assuming that the second particle is at rest before the collision, the angles of deflection of the two particles, \theta_1 and \theta_2, are related to the angle of deflection \theta in the system of the center of mass by \tan \theta_1=\frac{m_2 \sin \theta}{m_1+m_2 \cos \theta},\qquad \theta_2=\frac{{\pi}-{\theta}}{2}. To see this, consider the center of mass at time t before collision and time t' after collision: \begin{align} \bar{x}(t) &= \frac{m_{1} x_{1}(t)+m_{2} x_{2}(t)}{m_{1}+m_{2}} \\\ \bar{x}(t') &= \frac{m_{1} x_{1}(t')+m_{2} x_{2}(t')}{m_{1}+m_{2}}. \end{align} Hence, the velocities of the center of mass before and after collision are: \begin{align} v_{ \bar{x} } &= \frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}} \\\ v_{ \bar{x} }' &= \frac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}. \end{align} The numerators of v_{ \bar{x} } and v_{ \bar{x} }' are the total momenta before and after collision. The overall velocity of each body must be split into two perpendicular velocities: one tangent to the common normal surfaces of the colliding bodies at the point of contact, the other along the line of collision. These equations may be solved directly to find v_1,v_2 when u_1,u_2 are known: \begin{array}{ccc} v_1 &=& \dfrac{m_1-m_2}{m_1+m_2} u_1 + \dfrac{2m_2}{m_1+m_2} u_2 \\\\[.5em] v_2 &=& \dfrac{2m_1}{m_1+m_2} u_1 + \dfrac{m_2-m_1}{m_1+m_2} u_2. \end{array} If both masses are the same, we have a trivial solution: \begin{align} v_{1} &= u_{2} \\\ v_{2} &= u_{1}. \end{align} This simply corresponds to the bodies exchanging their initial velocities to each other. The following illustrate the case of equal mass, m_1=m_2. frame|center|Elastic collision of equal masses frame|center|Elastic collision of masses in a system with a moving frame of reference In the limiting case where m_1 is much larger than m_2, such as a ping-pong paddle hitting a ping-pong ball or an SUV hitting a trash can, the heavier mass hardly changes velocity, while the lighter mass bounces off, reversing its velocity plus approximately twice that of the heavy one. In the case of a large u_{1}, the value of v_{1} is small if the masses are approximately the same: hitting a much lighter particle does not change the velocity much, hitting a much heavier particle causes the fast particle to bounce back with high speed. At any instant, half the collisions are, to a varying extent, inelastic collisions (the pair possesses less kinetic energy in their translational motions after the collision than before), and half could be described as “super-elastic” (possessing more kinetic energy after the collision than before). Studies of two-dimensional collisions are conducted for many bodies in the framework of a two-dimensional gas. frame|center|Two-dimensional elastic collision In a center of momentum frame at any time the velocities of the two bodies are in opposite directions, with magnitudes inversely proportional to the masses. It can be shown that v_c is given by: v_c = \frac{p_T c^2}{E} Now the velocities before the collision in the center of momentum frame u_1 ' and u_2 ' are: \begin{align} u_1' &= \frac{u_1 - v_c}{1- \frac{u_1 v_c}{c^2}} \\\ u_2' &= \frac{u_2 - v_c}{1- \frac{u_2 v_c}{c^2}} \\\ v_1' &= -u_1' \\\ v_2' &= -u_2' \\\ v_1 &= \frac{v_1' + v_c}{1+ \frac{v_1' v_c}{c^2}} \\\ v_2 &= \frac{v_2' + v_c}{1+ \frac{v_2' v_c}{c^2}} \end{align} When u_1 \ll c and u_2 \ll c\,, \begin{align} p_T &\approx m_1 u_1 + m_2 u_2 \\\ v_c &\approx \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2} \\\ u_1' &\approx u_1 - v_c \approx \frac {m_1 u_1 + m_2 u_1 - m_1 u_1 - m_2 u_2}{m_1 + m_2} = \frac {m_2 (u_1 - u_2)}{m_1 + m_2} \\\ u_2' &\approx \frac {m_1 (u_2 - u_1)}{m_1 + m_2} \\\ v_1' &\approx \frac {m_2 (u_2 - u_1)}{m_1 + m_2} \\\ v_2' &\approx \frac {m_1 (u_1 - u_2)}{m_1 + m_2} \\\ v_1 &\approx v_1' + v_c \approx \frac {m_2 u_2 - m_2 u_1 + m_1 u_1 + m_2 u_2}{m_1 + m_2} = \frac{u_1 (m_1 - m_2) + 2m_2 u_2}{m_1 + m_2} \\\ v_2 &\approx \frac{u_2 (m_2 - m_1) + 2m_1 u_1}{m_1 + m_2} \end{align} Therefore, the classical calculation holds true when the speed of both colliding bodies is much lower than the speed of light (~300,000 kilometres per second). ===Relativistic derivation using hyperbolic functions=== Using the so-called parameter of velocity s (usually called the rapidity), \frac{v}{c}=\tanh(s), we get \sqrt{1-\frac{v^2}{c^2}}=\operatorname{sech}(s). While the assumptions of separate impacts is not actually valid (the balls remain in close contact with each other during most of the impact), this model will nonetheless reproduce experimental results with good agreement, and is often used to understand more complex phenomena such as the core collapse of supernovae, or gravitational slingshot manoeuvres. ==Sport regulations== Several sports governing bodies regulate the bounciness of a ball through various ways, some direct, some indirect.

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A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed $14.9 \mathrm{~km} / \mathrm{s}$ collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass of the neutron. If the deuteron is scattered through a $\mathrm{LAB}$ angle $\psi=10^{\circ}$, what is the final speed of the deuteron?

The charge radius of the deuteron is . This suggests that the state of the deuterium is indeed to a good approximation , state, which occurs with both nucleons spinning in the same direction, but their magnetic moments subtracting because of the neutron's negative moment. While the order of magnitude is reasonable, since the deuterium radius is of order of 1 femtometer (see below) and its electric charge is e, the above model does not suffice for its computation. The Prout is an obsolete unit of energy, whose value is: 1 Prout = 2.9638 \times 10^{-14} J This is equal to one twelfth of the binding energy of the deuteron. The nucleus of a deuterium atom, called a deuteron, contains one proton and one neutron, whereas the far more common protium has no neutrons in the nucleus. In the first case the deuteron is a spin triplet, so that its total spin s is 1. The latter contribution is dominant in the absence of a pure contribution, but cannot be calculated without knowing the exact spatial form of the nucleons wavefunction inside the deuterium. In the second case the deuteron is a spin singlet, so that its total spin s is 0. For hydrogen, this amount is about , or 1.000545, and for deuterium it is even smaller: , or 1.0002725. The deuteron, composed of a proton and a neutron, is one of the simplest nuclear systems. The measured electric quadrupole of the deuterium is . The name deuterium is derived from the Greek , meaning "second", to denote the two particles composing the nucleus. This is a nucleus with one proton and one neutron, i.e. a deuterium nucleus. The proton and neutron making up deuterium can be dissociated through neutral current interactions with neutrinos. But the slightly lower experimental number than that which results from simple addition of proton and (negative) neutron moments shows that deuterium is actually a linear combination of mostly , state with a slight admixture of , state. The energies of spectroscopic lines for deuterium and light hydrogen (hydrogen-1) therefore differ by the ratios of these two numbers, which is 1.000272. In this case, the exchange of the two nucleons will multiply the deuterium wavefunction by (−1) from isospin exchange, (+1) from spin exchange and (+1) from parity (location exchange), for a total of (−1) as needed for antisymmetry. This is about 17% of the terrestrial deuterium-to-hydrogen ratio of 156 deuterium atoms per million hydrogen atoms. The deuteron, being an isospin singlet, is antisymmetric under nucleons exchange due to isospin, and therefore must be symmetric under the double exchange of their spin and location. In this theory, the deuterium nucleus with mass two and charge one would contain two protons and one nuclear electron. thumb|upright=0.8|The deuterium–tritium fusion reaction Deuterium–tritium fusion (sometimes abbreviated D+T) is a type of nuclear fusion in which one deuterium nucleus fuses with one tritium nucleus, giving one helium nucleus, one free neutron, and 17.6 MeV of energy. This situation is known as the deuterium bottleneck.

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A mass $m$ moves in one dimension and is subject to a constant force $+F_0$ when $x<0$ and to a constant force $-F_0$ when $x>0$. Describe the motion by constructing a phase diagram. Calculate the period of the motion in terms of $m, F_0$, and the amplitude $A$ (disregard damping) .

The phase is zero at the start of each period; that is :\phi(t_0 + kT) = 0\quad\quad{} for any integer k. The initial conditions are x(0)=0 and \dot{x}(0)=0. *Damped harmonic motion, see animation (right). thumb|Phase portrait of damped oscillator, with increasing damping strength. The equation of motion is \ddot x + 2\gamma \dot x + \omega^2 x = 0. A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stable steady states (with dots) and unstable steady states (with circles) in a phase space. The axes are of state variables. ==Examples== 400px|thumbnail|Illustration of how a phase portrait would be constructed for the motion of a simple pendulum. A motion diagram represents the motion of an object by displaying its location at various equally spaced times on the same diagram. The phase for each argument value, relative to the start of the cycle, is shown at the bottom, in degrees from 0° to 360° and in radians from 0 to 2π. In physics and mathematics, the phase of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. Then, F is said to be "at the same phase" at two argument values t_1 and t_2 (that is, \phi(t_1) = \phi(t_2)) if the difference between them is a whole number of periods. thumb|350px|right|A plot of f(y) (left) and its phase line (right). The term phase can refer to several different things: * It can refer to a specified reference, such as \textstyle \cos(2 \pi f t), in which case we would say the phase of \textstyle x(t) is \textstyle \varphi, and the phase of \textstyle y(t) is \textstyle \varphi - \frac{\pi}{2}. The red dots in the phase portraits are at times t which are an integer multiple of the period T=2\pi/\omega.Based on the examples shown in . ==References== ===Inline=== ===Historical=== * ===Other=== *. *. *. *. ==External links== *Duffing oscillator on Scholarpedia *MathWorld page * Category:Ordinary differential equations Category:Chaotic maps Category:Nonlinear systems Category:Articles containing video clips Motion diagrams. Motion diagrams. To get the phase as an angle between -\pi and +\pi, one uses instead :\phi(t) = 2\pi\left(\left[\\!\\!\left[\frac{t - t_0}{T} + \frac{1}{2}\right]\\!\\!\right] - \frac{1}{2}\right) The phase expressed in degrees (from 0° to 360°, or from −180° to +180°) is defined the same way, except with "360°" in place of "2π". ===Consequences=== With any of the above definitions, the phase \phi(t) of a periodic signal is periodic too, with the same period T: :\phi(t + T) = \phi(t)\quad\quad{} for all t. *Simple harmonic oscillator where the phase portrait is made up of ellipses centred at the origin, which is a fixed point. In this case the phase difference is increasing, indicating that the test signal is lower in frequency than the reference. ==Formula for phase of an oscillation or a periodic signal== The phase of an oscillation or signal refers to a sinusoidal function such as the following: :\begin{align} x(t) &= A\cdot \cos( 2 \pi f t + \varphi ) \\\ y(t) &= A\cdot \sin( 2 \pi f t + \varphi ) = A\cdot \cos\left( 2 \pi f t + \varphi - \tfrac{\pi}{2}\right) \end{align} where \textstyle A, \textstyle f, and \textstyle \varphi are constant parameters called the amplitude, frequency, and phase of the sinusoid. In mathematics, a phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. A Campbell diagram plot represents a system's response spectrum as a function of its oscillation regime. The formula above gives the phase as an angle in radians between 0 and 2\pi.

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A student drops a water-filled balloon from the roof of the tallest building in town trying to hit her roommate on the ground (who is too quick). The first student ducks back but hears the water splash $4.021 \mathrm{~s}$ after dropping the balloon. If the speed of sound is $331 \mathrm{~m} / \mathrm{s}$, find the height of the building, neglecting air resistance.

Integrating the internal air pressure over one hemisphere of the balloon then gives : P_\mathrm{in} - P_\mathrm{out} \equiv P = \frac{f_t}{\pi r^2} = \frac{C}{r_0^2r} \left[1-\left(\frac{r_0}{r}\right)^6 \right] where r0 is the balloon's uninflated radius. When air is first added to the balloon, the pressure rises rapidly to a peak. This is easy to verify by squeezing the air back and forth between two interconnected balloons. ==Non-ideal balloons== At large extensions, the pressure inside a natural rubber balloon once again goes up. The air flow ceases when the two balloons have equal pressure, with one on the left branch of the pressure curve (r < rp) and one on the right branch (r > rp). This result is surprising, since most people assume that the two balloons will have equal sizes after exchanging air. If the total quantity of air in both balloons is less than Np, defined as the number of molecules in both balloons if they both sit at the peak of the pressure curve, then both balloons settle down to the left of the pressure peak with the same radius, r < rp. thumb|upright=1.25|The pub in 2007 The Air Balloon was a public house and road junction at Birdlip, Gloucestershire, England and closed in 2022 as part of road improvements. Pressure curve for an ideal rubber balloon. It becomes smaller, and the larger balloon becomes larger. Two balloons are connected via a hollow tube. For many starting conditions, the smaller balloon then gets smaller and the balloon with the larger diameter inflates even more. Two identical balloons are inflated to different diameters and connected by means of a tube. The lower pressure balloon will expand. thumb|upright=1.5|Observation balloon being shot down by a German biplane Balloon busters were military pilots known for destroying enemy observation balloons. Amer., 62, 1129-35. ,and Mackenzie.Mackenzie, K.V. (1981) Nine-term equation for sound speed in the oceans. So, when the valve is opened, the smaller balloon pushes air into the larger balloon. The simplest way to do this is to imagine that the balloon is made up of a large number of small rubber patches, and to analyze how the size of a patch is affected by the force acting on it. Bio-physical models suggest that this process is effectively similar to the behavior of the balloons in the two-balloon experiment . For a balloon of radius r, a fixed volume of rubber means that r2t is constant, or equivalently : t \propto \frac{1}{r^2} hence : \frac{t}{t_0} = \left(\frac{r_0}{r}\right)^2 and the radial force equation becomes : p = \frac{1}{C_2} \left(\frac{r_0}{r}\right)^4 The equation for the tangential force ft (where Lt \propto r) then becomes : f_t \propto (r/r_0^2)\left[1-(r_0/r)^6\right]. thumb|left|300px|Fig. thumb|Dangerous proximity of a hot air balloon to an overhead line. Figure 2 (above left) shows a typical initial configuration: The smaller balloon has the higher pressure because of the sum of pressure of elastic force Fe which is proportional to pressure (P=Fe/S) plus air pressure in small balloon is greater than air pressure in big balloon. Although balloons were occasionally shot down by small-arms fire, generally it was difficult to shoot down a balloon with solid bullets, particularly at the distances and altitude involved.

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A thin disk of mass $M$ and radius $R$ lies in the $(x, y)$ plane with the $z$-axis passing through the center of the disk. Calculate the gravitational potential $\Phi(z)$.

For this the gravitational force, i.e. the gradient of the potential, must be computed. For a non-pointlike object of continuous mass distribution, each mass element dm can be treated as mass distributed over a small volume, so the volume integral over the extent of object 2 gives: = - Gm_1 \int\limits_V \frac{\rho_2 }{r^2}\mathbf{\hat{r}}\,dx\,dy\,dz |}} with corresponding gravitational potential where ρ = ρ(x, y, z) is the mass density at the volume element and of the direction from the volume element to point mass 1. u is the gravitational potential energy per unit mass. ===The case of a homogeneous sphere=== In the special case of a sphere with a spherically symmetric mass density then ρ = ρ(s); i.e., density depends only on the radial distance :s = \sqrt{x^2 + y^2 + z^2} \,. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. This leaves an ordinary differential equation in terms only of the radius, r, which determines the eigenstates for the particular potential, V(r). == Structure of the eigenfunctions == The eigenstates of the system have the form: \psi(r, \theta, \phi) = R(r)\Theta(\theta)\Phi(\phi) in which the spherical angles \theta and \phi represent the polar and azimuthal angle, respectively. In the literature it is common to introduce some arbitrary "reference radius" R close to Earth's radius and to work with the dimensionless coefficients :\begin{align} \tilde{J_n} &= -\frac{J_n}{\mu\ R^n}, & \tilde{C_{n}^m} &= -\frac{C_{n}^m}{\mu\ R^n}, & \tilde{S_{n}^m} &= -\frac{S_{n}^m}{\mu\ R^n} \end{align} and to write the potential as {{NumBlk|:| u = -\frac{\mu }{r} \left(1 + \sum_{n=2}^{N_z} \frac{\tilde{J_n} P^0_n(\sin\theta) }{{(\frac{r}{R})}^n} + \sum_{n=2}^{N_t} \sum_{m=1}^n \frac{ P^m_n(\sin\theta) (\tilde{C_{n}^m} \cos m\varphi + \tilde{S_{n}^m} \sin m\varphi)}{{(\frac{r}{R})}^n}\right) |}} ===Largest terms=== The dominating term (after the term −μ/r) in () is the "J2 coefficient", representing the oblateness of Earth: :u = \frac{J_2\ P^0_2(\sin\theta)}{r^3} = J_2 \frac{1}{r^3} \frac{1}{2} (3\sin^2\theta -1) = J_2 \frac{1}{r^5} \frac{1}{2} (3 z^2 -r^2) Relative the coordinate system thumb|right|Figure 1: The unit vectors. If this shape were perfectly known together with the exact mass density ρ = ρ(x, y, z), the integrals () and () could be evaluated with numerical methods to find a more accurate model for Earth's gravitational field. From the definition of φ as a spherical coordinate it is clear that Φ(φ) must be periodic with the period 2π and one must therefore have that and for some integer m as the family of solutions to () then are With the variable substitution :x=\sin \theta equation () takes the form From () follows that in order to have a solution \phi with :R(r) = \frac{1}{r^{n+1}} one must have that :\lambda = n (n + 1) If Pn(x) is a solution to the differential equation one therefore has that the potential corresponding to m = 0 :\phi = \frac{1}{r^{n+1}}\ P_n(\sin\theta) which is rotationally symmetric around the z-axis is a harmonic function If P_{n}^{m}(x) is a solution to the differential equation {dx}\right)\ +\ \left(n(n + 1) - \frac{m^2}{1 - x^2} \right)\ P_{n}^{m}\ =\ 0 |}} with m ≥ 1 one has the potential \ P_{n}^{m}(\sin\theta)\ (a\ \cos m\varphi\ +\ b\ \sin m\varphi) |}} where a and b are arbitrary constants is a harmonic function that depends on φ and therefore is not rotationally symmetric around the z-axis The differential equation () is the Legendre differential equation for which the Legendre polynomials defined are the solutions. The differential equation which characterizes the function R(r) is called the radial equation. == Derivation of the radial equation == The kinetic energy operator in spherical polar coordinates is:\frac{\hat{p}^2}{2m_0} = -\frac{\hbar^2}{2m_0} abla^2 = \- \frac{\hbar^2}{2m_0\,r^2} \left[ \frac{\partial}{\partial r} \left(r^2 \frac{\partial}{\partial r}\right) - \hat{L}^2 \right].The spherical harmonics satisfy \hat{L}^2 Y_{\ell m}(\theta,\phi)\equiv \left\\{ -\frac{1}{\sin^2\theta} \left[ \sin\theta \frac{\partial}{\partial\theta} \left(\sin\theta \frac{\partial}{\partial\theta}\right) +\frac{\partial^2}{\partial \phi^2}\right]\right\\} Y_{\ell m}(\theta,\phi) = \ell(\ell+1)Y_{\ell m}(\theta,\phi). There should be a theta, not lambda \hat{\varphi}\ ,\ \hat{\theta}\ ,\ \hat{r} illustrated in figure 1 the components of the force caused by the "J2 term" are In the rectangular coordinate system (x, y, z) with unit vectors (x̂ ŷ ẑ) the force components are: The components of the force corresponding to the "J3 term" :u = \frac{J_3 P^0_3(\sin\theta) }{r^4} = J_3 \frac{1}{r^4} \frac{1}{2} \sin\theta \left(5\sin^2\theta - 3\right) = J_3 \frac{1}{r^7} \frac{1}{2} z \left(5 z^2 - 3 r^2\right) are and The exact numerical values for the coefficients deviate (somewhat) between different Earth models but for the lowest coefficients they all agree almost exactly. This is the shell theorem saying that in this case: with corresponding potential where M = ∫Vρ(s)dxdydz is the total mass of the sphere. ==Spherical harmonics representation== In reality, Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. By straightforward calculations one gets that for any function f Introducing the expression () in () one gets that As the term :\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) only depends on the variable r and the sum :\frac{1}{\Theta\cos\theta}\frac{d}{d\theta}\left(\cos\theta \frac{d\Theta}{d\theta}\right) + \frac{1}{\Phi\cos^2\theta}\frac{d^2\Phi}{d\varphi^2} only depends on the variables θ and φ. thumb|right|300px|A contour plot of the effective potential of a two-body system due to gravity and inertia at one point in time. For the same reason, the solution will be of this kind inside the sphere:R(r) = A j_\ell\left(\sqrt{\frac{2 m_0 (E-V_0)}{\hbar^2}}r\right), \qquad r < r_0.Note that for bound states, V_0 < E < 0. Several other definitions are in use, and so care must be taken in comparing different sources.Wolfram Mathworld, spherical coordinates == Cylindrical coordinate system == === Vector fields === Vectors are defined in cylindrical coordinates by (ρ, φ, z), where * ρ is the length of the vector projected onto the xy- plane, * φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), * z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by: \begin{bmatrix} \rho \\\ \phi \\\ z \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2} \\\ \operatorname{arctan}(y / x) \\\ z \end{bmatrix},\ \ \ 0 \le \phi < 2\pi, thumb or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} \rho\cos\phi \\\ \rho\sin\phi \\\ z \end{bmatrix}. Let the points have position vectors \textbf{r} and \textbf{r}' , then the Laplace expansion is : \frac{1}{|\mathbf{r}-\mathbf{r}'|} = \sum_{\ell=0}^\infty \frac{4\pi}{2\ell+1} \sum_{m=-\ell}^{\ell} (-1)^m \frac{r_^\ell }{r_{\scriptscriptstyle>}^{\ell+1} } Y^{-m}_\ell(\theta, \varphi) Y^m_\ell(\theta', \varphi'). The first spherical harmonics with n = 0, 1, 2, 3 are presented in the table below. : n Spherical harmonics 0 \frac{1}{r} 1 \frac{1}{r^2} P^0_1(\sin\theta) = \frac{1}{r^2} \sin\theta \frac{1}{r^2} P^1_1(\sin\theta) \cos\varphi= \frac{1}{r^2} \cos\theta \cos\varphi \frac{1}{r^2} P^1_1(\sin\theta) \sin\varphi= \frac{1}{r^2} \cos\theta \sin\varphi 2 \frac{1}{r^3} P^0_2(\sin\theta) = \frac{1}{r^3} \frac{1}{2} (3\sin^2\theta - 1) \frac{1}{r^3} P^1_2(\sin\theta) \cos\varphi = \frac{1}{r^3} 3 \sin\theta \cos\theta\ \cos\varphi \frac{1}{r^3} P^1_2(\sin\theta) \sin\varphi = \frac{1}{r^3} 3 \sin\theta \cos\theta \sin\varphi \frac{1}{r^3} P^2_2(\sin\theta) \cos2\varphi = \frac{1}{r^3} 3 \cos^2 \theta\ \cos2\varphi \frac{1}{r^3} P^2_2(\sin\theta) \sin2\varphi = \frac{1}{r^3} 3 \cos^2 \theta \sin 2\varphi 3 \frac{1}{r^4} P^0_3(\sin\theta) = \frac{1}{r^4} \frac{1}{2} \sin\theta\ (5\sin^2\theta -3) \frac{1}{r^4} P^1_3(\sin\theta) \cos\varphi = \frac{1}{r^4} \frac{3}{2}\ (5\ \sin^2\theta - 1) \cos\theta \cos\varphi \frac{1}{r^4} P^1_3(\sin\theta) \sin\varphi = \frac{1}{r^4} \frac{3}{2}\ (5\ \sin^2\theta - 1) \cos\theta \sin\varphi \frac{1}{r^4} P^2_3(\sin\theta) \cos 2\varphi = \frac{1}{r^4} 15 \sin\theta \cos^2 \theta \cos 2\varphi \frac{1}{r^4} P^2_3(\sin\theta) \sin 2\varphi = \frac{1}{r^4} 15 \sin\theta \cos^2 \theta \sin 2\varphi \frac{1}{r^4} P^3_3(\sin\theta) \cos 3\varphi = \frac{1}{r^4} 15 \cos^3 \theta \cos 3\varphi \frac{1}{r^4} P^3_3(\sin\theta) \sin 3\varphi = \frac{1}{r^4} 15 \cos^3 \theta \sin 3\varphi ===Application=== The model for Earth's gravitational potential is a sum \+ \sum_{n=2}^{N_t} \sum_{m=1}^n \frac{ P^m_n(\sin\theta) \left(C_n^m \cos m\varphi + S_n^m \sin m\varphi\right)}{r^{n+1}} |}} where \mu = GM and the coordinates () are relative to the standard geodetic reference system extended into space with origin in the center of the reference ellipsoid and with z-axis in the direction of the polar axis. In physics, the Laplace expansion of potentials that are directly proportional to the inverse of the distance ( 1/r ), such as Newton's gravitational potential or Coulomb's electrostatic potential, expresses them in terms of the spherical Legendre polynomials. In quantum mechanics, a particle in a spherically symmetric potential is a system with a potential that depends only on the distance between the particle and a center. Inclination to the invariable plane for the giant planets Year Jupiter Saturn Uranus Neptune 2009 0.32° 0.93° 1.02° 0.72° 142400 (produced with Solex 10) 0.48° 0.79° 1.04° 0.55° 168000 (produced with Solex 10) 0.23° 1.01° 1.12° 0.55° The invariable plane of a planetary system, also called Laplace's invariable plane, is the plane passing through its barycenter (center of mass) perpendicular to its angular momentum vector. From () then follows that :\frac{1}{\Theta}\ \cos\theta\ \frac{d}{d\theta}\left(\cos\theta \frac{d\Theta}{d\theta}\right)\ + \lambda\ \cos^2\theta\ +\ \frac{1}{\Phi}\frac{d^2\Phi}{d\varphi^2}\ =\ 0 The first two terms only depend on the variable \theta and the third only on the variable \varphi. They take the forms: P^m_n(\sin \theta) \cos m\varphi \,,& 0 &\le m \le n \,,& n &= 0, 1, 2, \dots \\\ h(x, y, z) &= \frac{1}{r^{n+1}} P^m_n(\sin \theta) \sin m\varphi \,,& 1 &\le m \le n \,,& n &= 1, 2, \dots \end{align}|}} where spherical coordinates (r, θ, φ) are used, given here in terms of cartesian (x, y, z) for reference: also P0n are the Legendre polynomials and Pmn for are the associated Legendre functions.

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A steel ball of velocity $5 \mathrm{~m} / \mathrm{s}$ strikes a smooth, heavy steel plate at an angle of $30^{\circ}$ from the normal. If the coefficient of restitution is 0.8 , at what velocity does the steel ball bounce off the plate?

The motion of a ball is generally described by projectile motion (which can be affected by gravity, drag, the Magnus effect, and buoyancy), while its impact is usually characterized through the coefficient of restitution (which can be affected by the nature of the ball, the nature of the impacting surface, the impact velocity, rotation, and local conditions such as temperature and pressure). A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. This implies that the ball would bounce to 9 times its original height.Since conservation of mechanical energy implies \textstyle \frac{1}{2}mv_\text{f}^2 = mgH_\text{f}, then \textstyle H_\text{f} is proportional to v^2_\text{f}. Composites Science and Technology 64:35-54. . is as follows: V_b=\frac{\pi\,\Gamma\,\sqrt{\rho_t\,\sigma_e}\,D^2\,T}{4\,m} \left [1+\sqrt{1+\frac{8\,m}{\pi\,\Gamma^2\,\rho_t\,D^2\,T}}\, \right ] where *V_b\, is the ballistic limit *\Gamma\, is a projectile constant determined experimentally *\rho_t\, is the density of the laminate *\sigma_e\, is the static linear elastic compression limit *D\, is the diameter of the projectile *T\, is the thickness of the laminate *m\, is the mass of the projectile Additionally, the ballistic limit for small-caliber into homogeneous armor by TM5-855-1 is: V_1= 19.72 \left [ \frac{7800 d^3 \left [ \left ( \frac{e_h}{d} \right) \sec \theta \right ]^{1.6}}{W_T} \right ]^{0.5} where *V_1 is the ballistic limit velocity in fps *d is the caliber of the projectile, in inches *e_h is the thickness of the homogeneous armor (valid from BHN 360 - 440) in inches *\theta is the angle of obliquity *W_T is the weight of the projectile, in lbs == References == Category:Ballistics A pair of colliding spheres of different sizes but of the same material have the same coefficient as below, but multiplied by \left(\frac{R_1}{R_2}\right)^{{3}/{8}} Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material. * e = coefficient of restitution * Sy = dynamic yield strength (dynamic "elastic limit") * E′ = effective elastic modulus * ρ = density * v = velocity at impact * μ = Poisson's ratio e = 3.1 \left(\frac{S_\text{y}}{1}\right)^{5/8} \left(\frac{1}{E'}\right)^{1/2} \left(\frac{1}{v}\right)^{1/4} \left(\frac{1}{\rho}\right)^{1/8} E' = \frac{E}{1-\mu^2} This equation overestimates the actual COR. The International Table Tennis Federation specifies that the ball shall bounce up 24–26 cm when dropped from a height of 30.5 cm on to a standard steel block thereby having a COR of 0.887 to 0.923. The ball's angular velocity will be reduced after impact, but its horizontal velocity will be increased. The ball's angular velocity will be increased after impact, but its horizontal velocity will be decreased. To analyze the vertical and horizontal components of the motion, the COR is sometimes split up into a normal COR (ey), and tangential COR (ex), defined as :e_\text{y} = -\frac{v_\text{yf} - u_\text{yf}}{v_\text{yi} - u_\text{yi}}, :e_\text{x} = -\frac{(v_\text{xf}-r\omega_\text{f})-(u_\text{xf}-R\Omega_\text{f})}{(v_\text{xi}-r\omega_\text{i})-(u_\text{xi}-R\Omega_\text{i})}, where r and ω denote the radius and angular velocity of the ball, while R and Ω denote the radius and angular velocity the impacting surface (such as a baseball bat). It gives the following theoretical coefficient of restitution for solid spheres dropped 1 meter (v = 4.5 m/s). Depending on the ball's alignment at impact, the normal force can act ahead or behind the centre of mass of the ball, and friction from the ground will depend on the alignment of the ball, as well as its rotation, spin, and impact velocity. The ball's angular velocity will be reduced after impact, as will its horizontal velocity, and the ball is propelled upwards, possibly even exceeding its original height. A comprehensive study of coefficients of restitution in dependence on material properties (elastic moduli, rheology), direction of impact, coefficient of friction and adhesive properties of impacting bodies can be found in Willert (2020). === Predicting from material properties === The COR is not a material property because it changes with the shape of the material and the specifics of the collision, but it can be predicted from material properties and the velocity of impact when the specifics of the collision are simplified. More specifically, if the ball is bounced at an angle θ with the ground, the motion in the x- and y-axes (representing horizontal and vertical motion, respectively) is described by x-axis y-axis :\begin{align} a_\text{x} & = 0, \\\ v_\text{x} & = v_0 \cos \left(\theta \right), \\\ x & = x_0 + v_0 \cos \left( \theta \right) t, \end{align} :\begin{align} a_\text{y} & = -g, \\\ v_\text{y} & = v_0 \sin \left(\theta \right) -gt, \\\ y & = y_0 + v_0 \sin \left( \theta \right) t -\frac{1}{2}gt^2. \end{align} The equations imply that the maximum height (H) and range (R) and time of flight (T) of a ball bouncing on a flat surface are given by :\begin{align} H & = \frac{v_0^2}{2g}\sin^2\left(\theta\right), \\\ R &= \frac{v_0^2}{g}\sin\left(2\theta\right),~\text{and} \\\ T &= \frac{2v_0}{g} \sin \left(\theta \right). \end{align} Further refinements to the motion of the ball can be made by taking into account air resistance (and related effects such as drag and wind), the Magnus effect, and buoyancy. For an object bouncing off a stationary target, C_R is defined as the ratio of the object's speed after the impact to that prior to impact: C_R = \frac{v}{u}, where *v is the speed of the object after impact *u is the speed of the object before impact In a case where frictional forces can be neglected and the object is dropped from rest onto a horizontal surface, this is equivalent to: C_R = \sqrt{\frac{h}{H}}, where *h is the bounce height *H is the drop height The coefficient of restitution can be thought of as a measure of the extent to which mechanical energy is conserved when an object bounces off a surface. In particular rω is the tangential velocity of the ball's surface, while RΩ is the tangential velocity of the impacting surface. According to one article (addressing COR in tennis racquets), "[f]or the Benchmark Conditions, the coefficient of restitution used is 0.85 for all racquets, eliminating the variables of string tension and frame stiffness which could add or subtract from the coefficient of restitution." If the ball moves horizontally at impact, friction will have a 'translational' component in the direction opposite to the ball's motion. This roughly corresponds to a COR of 0.727 to 0.806.Calculated using \textstyle e = \sqrt{\frac{H_\text{f}}{H_\text{i}}} and (if applicable) the diameter of the ball. For a straight drop on the ground with no rotation, with only the force of gravity acting on the ball, the COR can be related to several other quantities by: :e = \left|\frac{v_\text{f}}{v_\text{i}}\right| = \sqrt{\frac{K_\text{f}}{K_\text{i}}} = \sqrt{\frac{U_\text{f}}{U_\text{i}}} = \sqrt{\frac{H_\text{f}}{H_\text{i}}} = \frac{T_\text{f}}{T_\text{i}} =\sqrt{\frac{gT^2_\text{f}}{8H_\text{i}}}. In general, the ball will deform more at higher impact velocities and will accordingly lose more of its energy, decreasing its COR. ===Spin and angle of impact=== Upon impacting the ground, some translational kinetic energy can be converted to rotational kinetic energy and vice versa depending on the ball's impact angle and angular velocity. This energy loss is usually characterized (indirectly) through the coefficient of restitution (or COR, denoted e):Here, v and u are not just the magnitude of velocities, but include also their direction (sign). :e = -\frac{v_\text{f} - u_\text{f}}{v_\text{i} - u_\text{i}}, where vf and vi are the final and initial velocities of the ball, and uf and ui are the final and initial velocities impacting surface, respectively.

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Include air resistance proportional to the square of the ball's speed in the previous problem. Let the drag coefficient be $c_W=0.5$, the softball radius be $5 \mathrm{~cm}$ and the mass be $200 \mathrm{~g}$. Find the initial speed of the softball needed now to clear the fence.

The study concludes that, assuming average observed values for lift coefficient, a 65mph rise ball must have at least a three degree launch angle in order to pass the strike zone at a point higher than the release point (the bottom of the strike zone and release point are the same at 1.5 feet).Clark, J.M., Greer, M.L. & Semon, M.D. Modeling pitch trajectories in fastpitch softball. The most appropriate are the Reynolds number, given by \mathrm{Re} = \frac{u\sqrt{A}}{ u} and the drag coefficient, given by c_{\rm d} = \frac{F_{\rm d}}{\frac12 \rho A u^2}. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. Another study utilizes a theoretical physics approach to modelling the trajectories of various softball pitches, including the rise ball. Because the only unknown in the above equation is the drag force Fd, it is possible to express it as \begin{align} \frac{F_{\rm d}}{\frac12 \rho A u^2} &= f_c\left(\frac{u \sqrt{A}}{ u} \right) \\\ F_{\rm d} &= \tfrac12 \rho A u^2 f_c(\mathrm{Re}) \\\ c_{\rm d} &= f_c(\mathrm{Re}) \end{align} Thus the force is simply ½ ρ A u2 times some (as-yet-unknown) function fc of the Reynolds number Re – a considerably simpler system than the original five-argument function given above. The general rules for the softball throw parallel those of the javelin throw when conducted in a formal environment, but the implement being thrown is a standard softball, which resembles the size of a standard shot but is considerably lighter. One image appears to show that the ball follows an increasingly upward trajectory; however, this image was taken of a particular type of training ball known as a JUGS LITE-FLITE ball, which has “one third of the mass (59.5g) of a regulation softball (181.71g)”. A similar image shown of a regulation softball pitched at the same speed (70mph) seems to show a decreasing upward trajectory, although the author describes the outcome nebulously as “the rise is not apparent”. Alongside the Olympic discipline of fastpitch softball, which is the most popular variation of softball, there is also modified fastpitch softball and slow-pitch softball. === Baseball5 === thumb|A B5 batter hitting the ball into play. In softball, a pitch is the act of throwing a ball underhand by using a windmill motion. The authors consider the effects of gravity, drag and the Magnus Effect using Newton’s laws of motion to calculate the position of the ball at different points in time, allowing them to model the trajectory of the ball in 3 dimensions. At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0. From Stokes' solution, the drag force acting on the sphere of diameter d can be obtained as where the Reynolds number, Re = \frac{\rho d}{\mu} V. If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. Using the formula for terminal velocity V_t = \sqrt\frac{2mg}{\rho A C_d} the equation can be rewritten as v = V_t \tanh \left(t \frac{g}{V_t}\right). Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes v = \sqrt\frac{2mg}{\rho A C_d} \tanh \left(t \sqrt{\frac{g \rho A C_d }{2m}}\right). Air is 1.293 kg/m3 at 0°C and 1 atmosphere *u is the flow velocity relative to the object, *A is the reference area, and *c_{\rm d} is the drag coefficient – a dimensionless coefficient related to the object's geometry and taking into account both skin friction and form drag. As the speed of an object increases, so does the drag force acting on it, which also depends on the substance it is passing through (for example air or water). That this is so becomes apparent when the drag force Fd is expressed as part of a function of the other variables in the problem: f_a(F_{\rm d}, u, A, \rho, u) = 0. The softball throw is a track and field event used as a substitute for more technical throwing events in competitions involving Youth, Paralympic, Special Olympics and Senior competitors. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach.

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A child slides a block of mass $2 \mathrm{~kg}$ along a slick kitchen floor. If the initial speed is 4 $\mathrm{m} / \mathrm{s}$ and the block hits a spring with spring constant $6 \mathrm{~N} / \mathrm{m}$, what is the maximum compression of the spring?

If a spring has a rate of 100 then the spring would compress 1 inch (2.54 cm) with of load. == Types== Types of coil spring are: * Tension/extension coil springs, designed to resist stretching. This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. The effective mass of the spring can be determined by finding its kinetic energy. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. The force an ideal spring would exert is exactly proportional to its extension or compression. \, The mass of the spring is small in comparison to the mass of the attached mass and is ignored. As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). ; Compression spring: Designed to operate with a compression load, so the spring gets shorter as the load is applied to it. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. The manufacture normally specifies the spring rate. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. These springs are compression springs and can differ greatly in strength and in size depending on application. The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *"The Effective Mass of an Oscillating Spring" Am. J. Phys., 38, 98 (1970) *"Effective Mass of an Oscillating Spring" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. Springs can store energy when compressed. In a real spring–mass system, the spring has a non-negligible mass m.

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If the field vector is independent of the radial distance within a sphere, find the function describing the density $\rho=\rho(r)$ of the sphere.

In the plane \textstyle \textbf{R}^2, this expression simplifies to : H_s(r)=1-e^{-\lambda \pi r^2}. ==Relationship to other functions== ===Nearest neighbour function=== In general, the spherical contact distribution function and the corresponding nearest neighbour function are not equal. For three dimensions, this normalization is the number density of the system ( \rho ) multiplied by the volume of the spherical shell, which symbolically can be expressed as \rho \, 4\pi r^2 dr. Taking particle 0 as fixed at the origin of the coordinates, \textstyle \rho g(\mathbf{r}) d^3r = \mathrm{d} n (\mathbf{r}) is the average number of particles (among the remaining N-1) to be found in the volume \textstyle d^3r around the position \textstyle \mathbf{r}. thumb|250px|right|calculation of g(r) In statistical mechanics, the radial distribution function, (or pair correlation function) g(r) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle. More specifically, a spherical contact distribution function is defined as probability distribution of the radius of a sphere when it first encounters or makes contact with a point in a point process. The radial distribution function is usually determined by calculating the distance between all particle pairs and binning them into a histogram. In probability and statistics, a spherical contact distribution function, first contact distribution function,D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. Given a potential energy function, the radial distribution function can be computed either via computer simulation methods like the Monte Carlo method, or via the Ornstein-Zernike equation, using approximative closure relations like the Percus-Yevick approximation or the Hypernetted Chain Theory. The radial distribution function may also be inverted to predict the potential energy function using the Ornstein-Zernike equation or structure-optimized potential refinement. ==Definition== Consider a system of N particles in a volume V (for an average number density \rho =N/V) and at a temperature T (let us also define \textstyle \beta = \frac{1}{kT}). In other words, spherical contact distribution function is the probability there are no points from the point process located in a hyper- sphere of radius r. ===Contact distribution function=== The spherical contact distribution function can be generalized for sets other than the (hyper-)sphere in \textstyle \textbf{R}^{ d}. That is, ƒ is radial if and only if :f\circ \rho = f\, for all , the special orthogonal group in n dimensions. The radial distribution function is of fundamental importance since it can be used, using the Kirkwood–Buff solution theory, to link the microscopic details to macroscopic properties. The spherical contact function is also referred to as the contact distribution function, but some authors define the contact distribution function in relation to a more general set, and not simply a sphere as in the case of the spherical contact distribution function. In mathematics, a radial function is a real-valued function defined on a Euclidean space Rn whose value at each point depends only on the distance between that point and the origin. For a non-interacting gas, it is independent of the position \textstyle \mathbf{r}_1 and equal to the overall number density, \rho, of the system. In this way, the radial distribution function has been determined for a wide variety of systems, ranging from liquid metals to charged colloids. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. For distances r such that u(r) is significant, the mean local density will differ from the mean density \rho, depending on the sign of u(r) (higher for negative interaction energy and lower for positive u(r)). For fewer positions, we integrate over extraneous arguments, and include a correction factor to prevent overcounting, \begin{align} \rho^{(n)}(\mathbf{r}_1,\ldots,\mathbf{r}_n) &=\frac{1}{(N-n)!}\left(\prod_{i=n+1}^N\int\mathrm{d}^3\mathbf{r}_i\right)\sum_{\pi\in S_N} P^{(N)}(\mathbf{r}_{\pi (1)},\ldots,\mathbf{r}_{\pi (N)}) \\\ \end{align} This quantity is called the n-particle density function. To wit, :\phi(x) = \frac{1}{\omega_{n-1}}\int_{S^{n-1}} f(rx')\,dx' where ωn−1 is the surface area of the (n−1)-sphere Sn−1, and , . Spherical contact distribution functions are used in the study of point processesD. The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated.

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An Earth satellite has a perigee of $300 \mathrm{~km}$ and an apogee of $3,500 \mathrm{~km}$ above Earth's surface. How far is the satellite above Earth when it has rotated $90^{\circ}$ around Earth from perigee?

Objects orbiting the Earth must be within this radius, otherwise, they may become unbound by the gravitational perturbation of the Sun. Orbital characteristics epoch J2000.0 aphelion 1.0167 AU perihelion 0.98329 AU semimajor axis 1.0000010178 AU eccentricity 0.0167086 inclination 7.155° to Sun's equator 1.578690° to invariable plane longitude of the ascending node 174.9° longitude of perihelion 102.9° argument of periapsis 288.1° period daysThe figure appears in multiple references, and is derived from the VSOP87 elements from section 5.8.3, p. 675 of the following: average orbital speed speed at aphelion speed at perihelion The following diagram shows the relation between the line of the solstice and the line of apsides of Earth's elliptical orbit. Theta Persei (Theta Per, θ Persei, θ Per) is a star system 37 light years away from Earth, in the constellation Perseus. A satellite is said to occupy an inclined orbit around Earth if the orbit exhibits an angle other than 0° to the equatorial plane. thumb|upright=1.5|Earth at seasonal points in its orbit (not to scale) thumb|Earth orbit (yellow) compared to a circle (gray) Earth orbits the Sun at an average distance of 149.60 million km (92.96 million mi) in a counterclockwise direction as viewed from above the Northern Hemisphere. A full orbit has 360°. Beta angle can be controlled to keep a satellite as cool as possible (for instruments that require low temperatures, such as infrared cameras) by keeping the beta angle as close to zero as possible, or, conversely, to keep a satellite in sunlight as much as possible (for conversion of sunlight by its solar panels, for solar stability of sensors, or to study the Sun) by maintaining a beta angle as close to +90 or -90 as possible. ==Determination and application of beta angles== The value of a solar beta angle for a satellite in Earth orbit can be found using the equation \beta=\sin^{-1}[\cos(\Gamma)\sin(\Omega)\sin(i)-\sin(\Gamma)\cos(\epsilon)\cos(\Omega)\sin(i)+\sin(\Gamma)\sin(\epsilon)\cos(i)] where \Gamma is the ecliptic true solar longitude, \Omega is the right ascension of ascending node (RAAN), i is the orbit's inclination, and \epsilon is the obliquity of the ecliptic (approximately 23.45 degrees for Earth at present). thumb|300px|Beta angle (\boldsymbol{\beta}) In orbital spaceflight, the beta angle (\boldsymbol{\beta}) is the angle between a satellite's orbital plane around Earth and the geocentric position of the sun. 9 Persei is a single variable star in the northern constellation Perseus, located around 4,300 light years away from the Sun. At a LEO of 280 kilometers, the object is in sunlight through 59% of its orbit (approximately 53 minutes in Sunlight, and 37 minutes in shadow.) The Satellite () is a small rock peak rising to 1,100 m, protruding slightly above the ice sheet 3 nautical miles (6 km) southwest of Pearce Peak and 8 nautical miles (15 km) east of Baillieu Peak. The changing Earth-Sun distance results in an increase of about 7% in total solar energy reaching the Earth at perihelion relative to aphelion. This angle is called the orbit's inclination. The beta angle varies between +90° and −90°, and the direction in which the satellite orbits its primary body determines whether the beta angle sign is positive or negative. It is radiating over 12,000 times the luminosity of the Sun from its swollen photosphere at an effective temperature of 9,840 K. 9 Persei has one visual companion, designated component B, at an angular separation of and magnitude 12.0. One complete orbit takes days (1 sidereal year), during which time Earth has traveled 940 million km (584 million mi).Jean Meeus, Astronomical Algorithms 2nd ed, (Richmond, VA: Willmann-Bell, 1998) 238. Since this value is close to zero, the center of the orbit is relatively close to the center of the Sun (relative to the size of the orbit). That same satellite also will have a beta angle with respect to the Sun, and in fact it has a beta angle for any celestial object one might wish to calculate one for: any satellite orbiting a body (i.e. the Earth) will be in that body's shadow with respect to a given celestial object (like a star) some of the time, and in its line-of-sight the rest of the time. A satellite in such an orbit spends at least 59% of its orbital period in sunlight. ==Light and shadow== The degree of orbital shadowing an object in LEO experiences is determined by that object's beta angle. The above discussion defines the beta angle of satellites orbiting the Earth, but a beta angle can be calculated for any orbiting three body system: the same definition can be applied to give the beta angle of other objects. The Hill sphere (gravitational sphere of influence) of the Earth is about 1,500,000 kilometers (0.01 AU) in radius, or approximately four times the average distance to the Moon.For the Earth, the Hill radius is :R_H = a \left(\frac{m}{3M}\right)^{1/3}, where m is the mass of the Earth, a is an astronomical unit, and M is the mass of the Sun. The orbital ellipse goes through each of the six Earth images, which are sequentially the perihelion (periapsis—nearest point to the Sun) on anywhere from January 2 to January 5, the point of March equinox on March 19, 20, or 21, the point of June solstice on June 20, 21, or 22, the aphelion (apoapsis—the farthest point from the Sun) on anywhere from July 3 to July 5, the September equinox on September 22, 23, or 24, and the December solstice on December 21, 22, or 23. So the radius in AU is about \left(\frac{1}{3 \cdot 332\,946}\right)^{1/3} \approx 0.01.

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Two masses $m_1=100 \mathrm{~g}$ and $m_2=200 \mathrm{~g}$ slide freely in a horizontal frictionless track and are connected by a spring whose force constant is $k=0.5 \mathrm{~N} / \mathrm{m}$. Find the frequency of oscillatory motion for this system.

As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the "effective total mass" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon "external" factors such as the acceleration of gravity along it. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *"The Effective Mass of an Oscillating Spring" Am. J. Phys., 38, 98 (1970) *"Effective Mass of an Oscillating Spring" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. Many clocks keep time by mechanical resonance in a balance wheel, pendulum, or quartz crystal. ==Description== The natural frequency of a simple mechanical system consisting of a weight suspended by a spring is: :f = {1\over 2 \pi} \sqrt {k\over m} where m is the mass and k is the spring constant. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. The effective mass of the spring can be determined by finding its kinetic energy. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. thumb|The second pendulum, with a period of two seconds so each swing takes one second A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.Seconds pendulum == Pendulum == A pendulum is a weight suspended from a pivot so that it can swing freely. The resonance frequency of a pendulum, the only frequency at which it will vibrate, is given approximately, for small displacements, by the equation:Mechanical resonance :f = {1\over 2 \pi} \sqrt {g\over L} where g is the acceleration due to gravity (about 9.8 m/s2 near the surface of Earth), and L is the length from the pivot point to the center of mass. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. Note that, in this approximation, the frequency does not depend on mass. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. To relate v to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring: :T=\frac{1}{2}m(\dot x^2+(l_0+x)^2\dot \theta^2) So the Lagrangian becomes: :L = T -V_k - V_g :L[x,\dot x,\theta, \dot \theta] = \frac{1}{2}m(\dot x^2+(l_0+x)^2\dot \theta^2) -\frac{1}{2}kx^2 + gm(l_0+x)\cos \theta ===Equations of motion=== With two degrees of freedom, for x and \theta, the equations of motion can be found using two Euler-Lagrange equations: :{\partial L\over\partial x}-{\operatorname d \over \operatorname dt} {\partial L\over\partial \dot x}=0 :{\partial L\over\partial \theta}-{\operatorname d \over \operatorname dt} {\partial L\over\partial \dot \theta}=0 For x: :m(l_0+x)\dot \theta^2 -kx + gm\cos \theta-m \ddot x=0 \ddot x isolated: :\ddot x =(l_0+x)\dot \theta^2 -\frac{k}{m}x + g\cos \theta And for \theta: :-gm(l_0+x)\sin \theta - m(l_0+x)^2\ddot \theta- 2m(l_0+x)\dot x \dot \theta=0 \ddot \theta isolated: :\ddot \theta=-\frac{g}{l_0+x}\sin \theta-\frac{2\dot x}{l_0+x}\dot \theta The elastic pendulum is now described with two coupled ordinary differential equations. In a real spring–mass system, the spring has a non-negligible mass m. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. If the system is excited (pushed) with a period between pushes equal to the inverse of the pendulum's natural frequency, the swing will swing higher and higher, but if excited at a different frequency, it will be difficult to move. For a hardening spring oscillator (\alpha>0 and large enough positive \beta>\beta_{c+}>0) the frequency response overhangs to the high- frequency side, and to the low-frequency side for the softening spring oscillator (\alpha>0 and \beta<\beta_{c-}<0). The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations. ==Analysis and interpretation== thumb|300px|2 DOF elastic pendulum with polar coordinate plots. Furthermore, one can use analytical methods to study the intriguing phenomenon of order-chaos-order in this system. ==See also== * Double pendulum * Duffing oscillator * Pendulum (mathematics) * Spring-mass system == References == == Further reading == * * ==External links== * Holovatsky V., Holovatska Y. (2019) "Oscillations of an elastic pendulum" (interactive animation), Wolfram Demonstrations Project, published February 19, 2019.

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A particle moves with $v=$ const. along the curve $r=k(1+\cos \theta)$ (a cardioid). Find $\ddot{\mathbf{r}} \cdot \mathbf{e}_r=\mathbf{a} \cdot \mathbf{e}_r$.

Its main use is in the analysis of almost circular orbits, as perturbing (non-Keplerian) forces on an actual orbit will cause the osculating eccentricity vector to change continuously as opposed to the eccentricity and argument of periapsis parameters for which eccentricity zero (circular orbit) corresponds to a singularity. ==Calculation== The eccentricity vector \mathbf{e} \, is: : \mathbf{e} = {\mathbf{v}\times\mathbf{h}\over{\mu}} - {\mathbf{r}\over{\left|\mathbf{r}\right|}} = \left ( {\mathbf{\left |v \right |}^2 \over {\mu} }- {1 \over{\left|\mathbf{r}\right|}} \right ) \mathbf{r} - {\mathbf{r} \cdot \mathbf{v} \over{\mu}} \mathbf{v} which follows immediately from the vector identity: : \mathbf{v}\times \left ( \mathbf{r}\times \mathbf{v} \right ) = \left ( \mathbf{v} \cdot \mathbf{v} \right ) \mathbf{r} - \left ( \mathbf{r} \cdot \mathbf{v} \right ) \mathbf{v} where: *\mathbf{r}\,\\! is position vector *\mathbf{v}\,\\! is velocity vector *\mathbf{h}\,\\! is specific angular momentum vector (equal to \mathbf{r}\times\mathbf{v}) *\mu\,\\! is standard gravitational parameter ==See also== *Kepler orbit *Orbit *Eccentricity *Laplace–Runge–Lenz vector ==References== Category:Orbits Category:Vectors (mathematics and physics) Together with the electric potential φ, the magnetic vector potential can be used to specify the electric field E as well. Then, A is a vector potential for , that is, abla \times \mathbf{A} =\mathbf{v}. By analogy with Biot-Savart's law, the following \boldsymbol{A}(\textbf{x}) is also qualify as a vector potential for v. :\boldsymbol{A}(\textbf{x}) =\int_\Omega \frac{\boldsymbol{v}(\boldsymbol{y}) \times (\boldsymbol{x} - \boldsymbol{y})}{4 \pi |\boldsymbol{x} - \boldsymbol{y}|^3} d^3 \boldsymbol{y} Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law. The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. That is, A' below is also a vector potential of v; \mathbf{A'} (\mathbf{x}) = \frac{1}{4 \pi} \int_{\Omega} \frac{ abla_y \times \mathbf{v} (\mathbf{y})}{\left\|\mathbf{x} -\mathbf{y} \right\|} \, d^3\mathbf{y}. For this purpose Newton's notation will be used for the time derivative (\dot{\mathbf{A}}). The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. The direction from r′ to r does not enter into the equation. For Kepler orbits the eccentricity vector is a constant of motion. * The position of r, the point at which values for ϕ and A are found, only enters the equation as part of the scalar distance from r′ to r. * The equation for A is a vector equation. In vector calculus, a vector potential is a vector field whose curl is a given vector field. Substituting curl[v] for the current density j of the retarded potential, you will get this formula. The drag curve or drag polar is the relationship between the drag on an aircraft and other variables, such as lift, the coefficient of lift, angle-of- attack or speed. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: abla \times \mathbf{A} = \mathbf{B}. They are given by: \begin{align} \boldsymbol{\dot{\hat r}} &= \dot\theta \boldsymbol{\hat\theta} + \dot\phi\sin\theta \boldsymbol{\hat\phi} \\\ \boldsymbol{\dot{\hat\theta}} &= - \dot\theta \boldsymbol{\hat r} + \dot\phi\cos\theta \boldsymbol{\hat\phi} \\\ \boldsymbol{\dot{\hat\phi}} &= - \dot\phi\sin\theta \boldsymbol{\hat{r}} - \dot\phi\cos\theta \boldsymbol{\hat\theta} \end{align} Thus the time derivative becomes: \mathbf{\dot A} = \boldsymbol{\hat r} \left(\dot A_r - A_\theta \dot\theta - A_\phi \dot\phi \sin\theta \right) \+ \boldsymbol{\hat\theta} \left(\dot A_\theta + A_r \dot\theta - A_\phi \dot\phi \cos\theta\right) \+ \boldsymbol{\hat\phi} \left(\dot A_\phi + A_r \dot\phi \sin\theta + A_\theta \dot\phi \cos\theta\right) == See also == * Del in cylindrical and spherical coordinates for the specification of gradient, divergence, curl, and Laplacian in various coordinate systems. ==References== Category:Vector calculus Category:Coordinate systems This means that \mathbf{A} = \mathbf{P} = \rho \mathbf{\hat \rho} + z \mathbf{\hat z}. Hence the charge and current densities affecting the electric and magnetic potential at r and t, from remote location r′ must also be at some prior time t′. The forces involved are obtained from the coefficients by multiplication with , where ρ is the density of the atmosphere at the flight altitude, is the wing area and is the speed. The vector potential A is used when studying the Lagrangian in classical mechanics and in quantum mechanics (see Schrödinger equation for charged particles, Dirac equation, Aharonov–Bohm effect).

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Calculate the minimum $\Delta v$ required to place a satellite already in Earth's heliocentric orbit (assumed circular) into the orbit of Venus (also assumed circular and coplanar with Earth). Consider only the gravitational attraction of the Sun.

The formula by Ramanujan is accurate enough. giving an average orbital speed of . ==Conjunctions and transits== When the geocentric ecliptic longitude of Venus coincides with that of the Sun, it is in conjunction with the Sun – inferior if Venus is nearer and superior if farther. That said, Venus and Earth still have the lowest gravitational potential difference between them than to any other planet, needing the lowest delta-v to transfer between them, than to any other planet from them. A heliocentric orbit (also called circumsolar orbit) is an orbit around the barycenter of the Solar System, which is usually located within or very near the surface of the Sun. It is necessary for a spacecraft, like Gaia, to follow a Lissajous orbit or a halo orbit around L2 in order for its solar panels to get full sun. ===L3=== The location of L3 is the solution to the following equation, gravitation providing the centripetal force: \frac{M_1}{\left(R-r\right)^2}+\frac{M_2}{\left(2R-r\right)^2}=\left(\frac{M_2}{M_1+M_2}R+R-r\right)\frac{M_1+M_2}{R^3} with parameters M1, M2, and R defined as for the L1 and L2 cases, and r now indicates the distance of L3 from the position of the smaller object, if it were rotated 180 degrees about the larger object, while positive r implying L3 is closer to the larger object than the smaller object. Because the range of heliocentric distances is greater for the Earth than for Venus, the closest approaches come near Earth's perihelion. The 3.4° inclination of Venus's orbit is great enough to usually prevent the inferior planet from passing directly between the Sun and Earth at inferior conjunction. thumb|right|300 px|Representation of Venus (yellow) and Earth (blue) circling around the Sun. Venus has an orbit with a semi-major axis of , and an eccentricity of 0.007.Jean Meeus, Astronomical Formulæ for Calculators, by Jean Meeus. Elements by Simon Newcomb The low eccentricity and comparatively small size of its orbit give Venus the least range in distance between perihelion and aphelion of the planets: 1.46 million km. The orbit is now known to sub-kilometer accuracy. ==Table of orbital parameters== No more than five significant figures are presented here, and to this level of precision the numbers match very well the VSOP87 elements and calculations derived from them, Standish's (of JPL) 250-year best fit,Standish and Williams(2012) p 27 Newcomb's, and calculations using the actual positions of Venus over time. distances au Million km semimajor axis 0.72333 108.21 perihelion 0.71843 107.48 aphelion 0.7282 108.94 averageAverage distance over times. Sun orbit may refer to: * Heliocentric orbit, around the sun * Orbit of the sun around the Galactic Center The longitudes of perihelion were only 29 degrees apart at J2000, so the smallest distances, which come when inferior conjunction happens near Earth's perihelion, occur when Venus is near perihelion. A satellite is said to occupy an inclined orbit around Earth if the orbit exhibits an angle other than 0° to the equatorial plane. Calculations assume the two bodies orbit in a perfect circle with separation equal to the semimajor axis and no other bodies are nearby. It is at the point that the combined pull of Earth and Sun causes the object to orbit with the same period as Earth, in effect orbiting an Earth+Sun mass with the Earth-Sun barycenter at one focus of its orbit. === and points=== thumb|right|200px|Gravitational accelerations at The and points lie at the third vertices of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies 60° ahead of () or behind () the smaller mass with regard to its orbit around the larger mass. ===Stability=== The triangular points ( and ) are stable equilibria, provided that the ratio of is greater than 24.96.Actually (25 + 3)/2 ≈ This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system. Now the orbit estimates are dominated by observations of the Venus Express spacecraft. The first spacecraft to be put in a heliocentric orbit was Luna 1 in 1959. The heliocentric longitude of Earth advances by 0.9856° per day, and after 2919.6 days, it has advanced by 2878°, only 2° short of eight revolutions (2880°). Two important Lagrange points in the Sun-Earth system are L1, between the Sun and Earth, and L2, on the same line at the opposite side of the Earth; both are well outside the Moon's orbit. Lagrangian points in Solar System Body pair Semimajor axis, SMA (×109 m) L1 (×109 m) 1 − L1/SMA (%) L2 (×109 m) L2/SMA − 1 (%) L3 (×109 m) 1 + L3/SMA (%) Earth–Moon Sun–Mercury Sun–Venus Sun–Earth Sun–Mars Sun–Jupiter Sun–Saturn Sun–Uranus Sun–Neptune ==Spaceflight applications== ===Sun–Earth=== Sun–Earth is suited for making observations of the Sun–Earth system. This distance can be described as being such that the orbital period, corresponding to a circular orbit with this distance as radius around M2 in the absence of M1, is that of M2 around M1, divided by ≈ 1.73: T_{s,M_2}(r) = \frac{T_{M_2,M_1}(R)}{\sqrt{3}}. ====== The location of L2 is the solution to the following equation, gravitation providing the centripetal force: \frac{M_1}{(R+r)^2}+\frac{M_2}{r^2}=\left(\frac{M_1}{M_1+M_2}R+r\right)\frac{M_1+M_2}{R^3} with parameters defined as for the L1 case. Such solar transits of Venus rarely occur, but with great predictability and interest.Venus transit page. by Aldo Vitagliano, creator of SolexWilliam Sheehan, John Westfall The Transits of Venus (Prometheus Books, 2004) ==Close approaches to Earth and Mercury== In this current era, the nearest that Venus comes to Earth is just under 40 million km. The distance between Venus and Earth varies from about 42 million km (at inferior conjunction) to about 258 million km (at superior conjunction).

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C

Find the ratio of the radius $R$ to the height $H$ of a right-circular cylinder of fixed volume $V$ that minimizes the surface area $A$.

Since the area of a circle of radius r\,, which is the base of the cylinder, is given by B = \pi r^2 it follows that: * V = \pi r^2 h or even * V = \pi r^2 g . == Equilateral cylinder == thumb|Illustration of a cylinder circumscribed by a sphere of radius r. Where: * L\,represents the lateral surface area of the cylinder; * \pi\,is approximately 3.14; * r\,is the distance between the lateral surface of the cylinder and the axis, i.e. it is the value of the radius of the base; * h\,is the height of the cylinder; * 2 \pi r is the length of the circumference of the base, since \pi = \frac{C}{2r}, that is, C = 2\pi r. Simply substitute the radius and height measurements defined earlier into the volume formula for a straight circular cylinder: * V = \pi r^2 \cdot h \Rightarrow V = \pi r^2 \cdot 2r \Rightarrow V = 2\pi r^3 == Meridian section == It is the intersection between a plane containing the axis of the cylinder and the cylinder. For a right circular cylinder of radius and height , the lateral area is the area of the side surface of the cylinder: . To calculate the total area of a right circular cylinder, you simply add the lateral area to the area of the two bases: * A = L + 2 \cdot B. Replacing L = 2 \pi r h and B = \pi r^2, we have: * A=2\pi rh + 2\pi r^2 \Rightarrow A = 2 \pi r (h + r) or even * A = 2 \pi r (g + r) . == Volume == thumb|Illustration of a cylinder and a prism, both with height h. Then, assuming that the radius of the base of an equilateral cylinder is r\, then the diameter of the base of this cylinder is 2r\, and its height is 2r\,. Its lateral area can be obtained by replacing the height value by 2r: * L = 2 \pi r \cdot 2r \Rightarrow L = 4 \pi r^2 . The result can be obtained in a similar way for the total area: * T = 2 \pi r (h + r) \Rightarrow T = 2 \pi r (2r + r) \Rightarrow T = 2 \pi r \cdot 3 r \Rightarrow T = 6 \pi r^2 . Therefore, the lateral surface area is given by: * L=2\pi rh. For the equilateral cylinder it is possible to obtain a simpler formula to calculate the volume. Note that in the case of the right circular cylinder, the height and the generatrix have the same measure, so the lateral area can also be given by: * L = 2 \pi r g . Therefore, simply multiply the area of the base by the height: * V = B \cdot h. The area of the base of a cylinder is the area of a circle (in this case we define that the circle has a radius with measure r): * B = \pi r^2. Note that the cylinder is equilateral. Fixing g as the side on which the revolution takes place, we obtain that the side r, perpendicular to g, will be the measure of the radius of the cylinder. It can be obtained by the product between the length of the circumference of the base and the height of the cylinder. However, the head of the radius is not perfectly cylindrical but slightly oval. Through Cavalieri's principle, which defines that if two solids of the same height, with congruent base areas, are positioned on the same plane, such that any other plane parallel to this plane sections both solids, determining from this section two polygons with the same area, then the volume of the two solids will be the same, we can determine the volume of the cylinder. This lateral surface area can be calculated by multiplying the perimeter of the base by the height of the prism. The surface to volume ratio for this cube is thus :\mbox{SA:V} = \frac{6~\mbox{cm}^2}{1~\mbox{cm}^3} = 6~\mbox{cm}^{-1}. This is because the volume of a cylinder can be obtained in the same way as the volume of a prism with the same height and the same area of the base. For example, the volume of the torus with minor radius r and major radius R is V = (\pi r^2)(2\pi R) = 2\pi^2 R r^2.

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E

Find the dimension of the parallelepiped of maximum volume circumscribed by a sphere of radius $R$.

If the radius of the sphere is called , the radii of the spherical segment bases are and and the height of the segment (the distance from one parallel plane to the other) called , then the volume of the spherical segment is : V = \frac{\pi h}{6} \left(3 r_1^2 + 3 r_2^2 + h^2\right). The volume can also be expressed in terms of A_n, the area of the unit -sphere. == Formulas == The first volumes are as follows: Dimension Volume of a ball of radius Radius of a ball of volume 0 1 (all 0-balls have volume 1) 1 2R \frac{V}{2}=0.5\times V 2 \pi R^2 \approx 3.142\times R^2 \frac{V^{1/2}}{\sqrt{\pi}}\approx 0.564\times V^{\frac{1}{2}} 3 \frac{4\pi}{3} R^3 \approx 4.189\times R^3 \left(\frac{3V}{4\pi}\right)^{1/3}\approx 0.620\times V^{1/3} 4 \frac{\pi^2}{2} R^4 \approx 4.935\times R^4 \frac{(2V)^{1/4}}{\sqrt{\pi}}\approx 0.671\times V^{1/4} 5 \frac{8\pi^2}{15} R^5\approx 5.264\times R^5 \left(\frac{15V}{8\pi^2}\right)^{1/5}\approx 0.717\times V^{1/5} 6 \frac{\pi^3}{6} R^6 \approx 5.168 \times R^6 \frac{(6V)^{1/6}}{\sqrt{\pi}}\approx 0.761\times V^{1/6} 7 \frac{16\pi^3}{105} R^7 \approx 4.725\times R^7 \left(\frac{105V}{16\pi^3}\right)^{1/7}\approx 0.801\times V^{1/7} 8 \frac{\pi^4}{24} R^8 \approx 4.059\times R^8 \frac{(24V)^{1/8}}{\sqrt{\pi}}\approx 0.839\times V^{1/8} 9 \frac{32\pi^4}{945} R^9 \approx 3.299\times R^9 \left(\frac{945V}{32\pi^4}\right)^{1/9}\approx 0.876\times V^{1/9} 10 \frac{\pi^5}{120} R^{10} \approx 2.550\times R^{10} \frac{(120V)^{1/10}}{\sqrt{\pi}}\approx 0.911\times V^{1/10} 11 \frac{64\pi^5}{10395} R^{11} \approx 1.884\times R^{11} \left(\frac{10395V}{64\pi^5}\right)^{1/11}\approx 0.944\times V^{1/11} 12 \frac{\pi^6}{720} R^{12} \approx 1.335\times R^{12} \frac{(720V)^{1/12}}{\sqrt{\pi}}\approx 0.976\times V^{1/12} 13 \frac{128\pi^6}{135135} R^{13} \approx 0.911\times R^{13} \left(\frac{135135V}{128\pi^6}\right)^{1/13}\approx 1.007\times V^{1/13} 14 \frac{\pi^7}{5040} R^{14} \approx 0.599\times R^{14} \frac{(5040V)^{1/14}}{\sqrt{\pi}}\approx 1.037\times V^{1/14} 15 \frac{256\pi^7}{2027025} R^{15} \approx 0.381\times R^{15} \left(\frac{2027025V}{256\pi^7}\right)^{1/15}\approx 1.066\times V^{1/15} n Vn(R) Rn(V) === Two-dimension recurrence relation === As is proved below using a vector-calculus double integral in polar coordinates, the volume of an -ball of radius can be expressed recursively in terms of the volume of an -ball, via the interleaved recurrence relation: : V_n(R) = \begin{cases} 1 &\text{if } n=0,\\\\[0.5ex] 2R &\text{if } n=1,\\\\[0.5ex] \dfrac{2\pi}{n}R^2 \times V_{n-2}(R) &\text{otherwise}. \end{cases} This allows computation of in approximately steps. === Closed form === The -dimensional volume of a Euclidean ball of radius in -dimensional Euclidean space is:Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/5.19#E4, Release 1.0.6 of 2013-05-06. It is the three-dimensional analogue of the sector of a circle. ==Volume== If the radius of the sphere is denoted by and the height of the cap by , the volume of the spherical sector is V = \frac{2\pi r^2 h}{3}\,. The volume of an odd-dimensional ball is :V_{2k+1}(R) = \frac{2(2\pi)^k}{(2k + 1)!!}R^{2k+1}. The volume of the sector is related to the area of the cap by: V = \frac{rA}{3}\,. ==Area== The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is A = 2\pi rh\,. The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If is the surface area of an -sphere of radius , then: :A_{n-1}(r) = r^{n-1} A_{n-1}(1). It is the region of a ball between two concentric spheres of differing radii. ==Volume== The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : V=\frac{4}{3}\pi R^3- \frac{4}{3}\pi r^3 : V=\frac{4}{3}\pi \left(R^3-r^3\right) : V=\frac{4}{3}\pi (R-r)\left(R^2+Rr+r^2\right) where is the radius of the inner sphere and is the radius of the outer sphere. ==Approximation== An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell: : V \approx 4 \pi r^2 t, when is very small compared to (t \ll r). To derive the volume of an -ball of radius from this formula, integrate the surface area of a sphere of radius for and apply the functional equation : :V_n(R) = \int_0^R \frac{2\pi^{n/2}}{\Gamma\bigl(\tfrac n2\bigr)} \,r^{n-1}\,dr = \frac{2\pi^{n/2}}{n\,\Gamma\bigl(\tfrac n2\bigr)}R^n = \frac{\pi^{n/2}}{\Gamma\bigl(\tfrac n2 + 1\bigr)}R^n. === Geometric proof === The relations V_{n+1}(R) = \frac{R}{n+1}A_n(R) and A_{n+1}(R) = (2\pi R)V_n(R) and thus the volumes of n-balls and areas of n-spheres can also be derived geometrically. The volume of a -ball of radius is R^nV_n, where V_n is the volume of the unit -ball, the -ball of radius . The volume of an ball of radius is :V^p_n(R) = \frac{\Bigl(2\,\Gamma\bigl(\tfrac1p + 1\bigr)\Bigr)^n}{\Gamma\bigl(\tfrac np + 1\bigr)}R^n. The total surface area of the spherical shell is 4 \pi r^2. == See also == * Spherical pressure vessel * Ball * Solid torus * Bubble * Sphere == References == Category:Elementary geometry Category:Geometric shapes Category:Spherical geometry The -sphere is the -dimensional boundary (surface) of the -dimensional ball of radius , and the sphere's hypervolume and the ball's hypervolume are related by: :A_{n-1}(R) = \frac{d}{dR} V_{n}(R) = \frac{n}{R}V_{n}(R). Let denote the distance between a point in the plane and the center of the sphere, and let denote the azimuth. Also, A_{n-1}(R) = \frac{dV_n(R)}{dR} because a ball is a union of concentric spheres and increasing radius by ε corresponds to a shell of thickness ε. The volume of the ball can therefore be written as an iterated integral of the volumes of the -balls over the possible radii and azimuths: :V_n(R) = \int_0^{2\pi} \int_0^R V_{n-2}\\!\left(\sqrt{R^2 - r^2}\right) r\,dr\,d\theta, The azimuthal coordinate can be immediately integrated out. As with the two-dimension recursive formula, the same technique can be used to give an inductive proof of the volume formula. === Direct integration in spherical coordinates === The volume of the n-ball V_n(R) can be computed by integrating the volume element in spherical coordinates. The spherical volume element is: :dV = r^{n-1}\sin^{n-2}(\varphi_1)\sin^{n-3}(\varphi_2) \cdots \sin(\varphi_{n-2})\,dr\,d\varphi_1\,d\varphi_2 \cdots d\varphi_{n-1}, and the volume is the integral of this quantity over between 0 and and all possible angles: :V_n(R) = \int_0^R \int_0^\pi \cdots \int_0^{2\pi} r^{n-1}\sin^{n-2}(\varphi_1) \cdots \sin(\varphi_{n-2})\,d\varphi_{n-1} \cdots d\varphi_1\,dr. Instead of expressing the volume of the ball in terms of its radius , the formulas can be inverted to express the radius as a function of the volume: :\begin{align} R_n(V) &= \frac{\Gamma\bigl(\tfrac n2 + 1\bigr)^{1/n}}{\sqrt{\pi}}V^{1/n} \\\ &= \left(\frac{n!! These are: :\begin{align} V_{2k}(R) &= \frac{\pi^k}{k!}R^{2k}, \\\ V_{2k+1}(R) &= \frac{2(k!)(4\pi)^k}{(2k + 1)!}R^{2k+1}. \end{align} The volume can also be expressed in terms of double factorials. thumb|A spherical segment In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. In computational geometry, the largest empty sphere problem is the problem of finding a hypersphere of largest radius in d-dimensional space whose interior does not overlap with any given obstacles. ==Two dimensions== The largest empty circle problem is the problem of finding a circle of largest radius in the plane whose interior does not overlap with any given obstacles. 300px|thumb|spherical shell, right: two halves In geometry, a spherical shell is a generalization of an annulus to three dimensions.

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B

A potato of mass $0.5 \mathrm{~kg}$ moves under Earth's gravity with an air resistive force of $-k m v$. (a) Find the terminal velocity if the potato is released from rest and $k=$ $0.01 \mathrm{~s}^{-1}$. (b) Find the maximum height of the potato if it has the same value of $k$, but it is initially shot directly upward with a student-made potato gun with an initial velocity of $120 \mathrm{~m} / \mathrm{s}$.

Note that d has its maximum value when : \sin 2\theta=1 , which necessarily corresponds to : 2\theta=90^\circ , or : \theta=45^\circ . thumb|350px|Trajectories of projectiles launched at different elevation angles but the same speed of 10 m/s in a vacuum and uniform downward gravity field of 10 m/s2. In air, which has a kinematic viscosity around 0.15\,\mathrm{cm^2/s}, this means that the drag force becomes quadratic in v when the product of speed and diameter is more than about 0.015\,\mathrm{m^2/s}, which is typically the case for projectiles. Terminal velocity is the maximum velocity (speed) attainable by an object as it falls through a fluid (air is the most common example). The magnitude of the velocity (under the Pythagorean theorem, also known as the triangle law): : v = \sqrt{v_x^2 + v_y^2 } . === Displacement === thumb|250px|Displacement and coordinates of parabolic throwing At any time t , the projectile's horizontal and vertical displacement are: : x = v_0 t \cos(\theta) , : y = v_0 t \sin(\theta) - \frac{1}{2}gt^2 . If the starting point is at height y0 with respect to the point of impact, the time of flight is: : t = \frac{d}{v \cos\theta} = \frac{v \sin \theta + \sqrt{(v \sin \theta)^2 + 2gy_0}}{g} As above, this expression can be reduced to : t = \frac{v\sin{\theta} + \sqrt{(v\sin{\theta})^{2}}}{g} = \frac{v\sin{\theta} + v\sin{\theta}}{g} = \frac{2v\sin{\theta}}{g} = \frac{2v\sin{(45)}}{g} = \frac{2v\frac{\sqrt{2}}{2}}{g} = \frac{\sqrt{2}v}{g} if θ is 45° and y0 is 0. === Time of flight to the target's position === As shown above in the Displacement section, the horizontal and vertical velocity of a projectile are independent of each other. The following assumptions are made: * Constant gravitational acceleration * Air resistance is given by the following drag formula, ::\mathbf{F_D} = -\tfrac{1}{2} c \rho A\, v\,\mathbf{v} ::Where: ::*FD is the drag force ::*c is the drag coefficient ::*ρ is the air density ::*A is the cross sectional area of the projectile ::*μ = k/m = cρA/(2m) ==== Special cases ==== Even though the general case of a projectile with Newton drag cannot be solved analytically, some special cases can. If h = R : \theta = \arctan(4)\approx 76.0^\circ === Maximum distance of projectile === thumb|250px|The maximum distance of projectile The range and the maximum height of the projectile does not depend upon its mass. Mathematically, it is given as t=U \sin\theta/g where g = acceleration due to gravity (app 9.81 m/s²), U = initial velocity (m/s) and \theta = angle made by the projectile with the horizontal axis. 2\. *Vertical motion downward: ::\dot{v}_y(t) = -g+\mu\,v_y^2(t) ::v_y(t) = -v_\infty \tanh\frac{t-t_{\mathrm{peak}}}{t_f} ::y(t) = y_{\mathrm{peak}} - \frac{1}{\mu}\ln\left(\cosh\frac{t-t_{\mathrm{peak}}}{t_f}\right) :After a time t_f, the projectile reaches almost terminal velocity -v_\infty. ==== Numerical solution ==== A projectile motion with drag can be computed generically by numerical integration of the ordinary differential equation, for instance by applying a reduction to a first-order system. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. * Stokes drag: \mathbf{F_{air}} = -k_{\mathrm{Stokes}}\cdot\mathbf{v}\qquad (for Re \lesssim 1000) * Newton drag: \mathbf{F_{air}} = -k\,|\mathbf{v}|\cdot\mathbf{v}\qquad (for Re \gtrsim 1000) right|thumb|320px|Free body diagram of a body on which only gravity and air resistance acts The free body diagram on the right is for a projectile that experiences air resistance and the effects of gravity. The vertical motion of the projectile is the motion of a particle during its free fall. An object with a large projected area relative to its mass, such as a parachute, has a lower terminal velocity than one with a small projected area relative to its mass, such as a dart. At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0. Hence range and maximum height are equal for all bodies that are thrown with the same velocity and direction. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. Because of this, we can find the time to reach a target using the displacement formula for the horizontal velocity: x = v_0 t \cos(\theta) \frac{x}{t}=v_0\cos(\theta) t=\frac{x}{v_0\cos(\theta)} This equation will give the total time t the projectile must travel for to reach the target's horizontal displacement, neglecting air resistance. === Maximum height of projectile === thumb|250px|Maximum height of projectile The greatest height that the object will reach is known as the peak of the object's motion. Attack of the Killer Potatoes is a 1997 science-fiction children's story by Peter Lerangis. When the terminal velocity is reached the weight of the object is exactly balanced by the upward buoyancy force and drag force. The mass of the projectile will be denoted by m, and \mu:=k/m. For the vertical displacement of the maximum height of the projectile: : h = v_0 t_h \sin(\theta) - \frac{1}{2} gt^2_h : h = \frac{v_0^2 \sin^2(\theta)}{2g} The maximum reachable height is obtained for θ=90°: : h_{\mathrm{max}} = \frac{v_0^2}{2g} If the projectile's position (x,y) and launch angle (θ) are known, the maximum height can be found by solving for h in the following equation: :h=\frac{(x\tan\theta)^2}{4(x\tan\theta-y)}. === Relation between horizontal range and maximum height === The relation between the range d on the horizontal plane and the maximum height h reached at \frac{t_d}{2} is: : h = \frac{d\tan\theta}{4} h = \frac{v_0^2\sin^2\theta}{2g} : d = \frac{v_0^2\sin2\theta}{g} : \frac{h}{d} = \frac{v_0^2\sin^2\theta}{2g} × \frac{g}{v_0^2\sin2\theta} : \frac{h}{d} = \frac{\sin^2\theta}{4\sin\theta\cos\theta} h = \frac{d\tan\theta}{4} . The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object.

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C

A particle of mass $m$ and velocity $u_1$ makes a head-on collision with another particle of mass $2 m$ at rest. If the coefficient of restitution is such to make the loss of total kinetic energy a maximum, what are the velocities $v_1$ after the collision?

It can be more than 1 if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity. \text{Coefficient of restitution } (e) = \frac{\left | \text{Relative velocity after collision} \right |}{\left | \text{Relative velocity before collision}\right |} The mathematics were developed by Sir Isaac Newton in 1687. The conservation of the total momentum before and after the collision is expressed by: m_{1}u_{1}+m_{2}u_{2} \ =\ m_{1}v_{1} + m_{2}v_{2}. Since the total energy and momentum of the system are conserved and their rest masses do not change, it is shown that the momentum of the colliding body is decided by the rest masses of the colliding bodies, total energy and the total momentum. Likewise, the conservation of the total kinetic energy is expressed by: \tfrac12 m_1u_1^2+\tfrac12 m_2u_2^2 \ =\ \tfrac12 m_1v_1^2 +\tfrac12 m_2v_2^2. Consider particles 1 and 2 with masses m1, m2, and velocities u1, u2 before collision, v1, v2 after collision. A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. The final velocities can then be calculated from the two new component velocities and will depend on the point of collision. Let v_1, v_2 be the final velocity of object 1 and object 2 respectively. \begin{cases} \frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 \\\ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \end{cases} From the first equation, m_1 \left(u_1^2 - v_1^2\right) = m_2 \left(v_2^2 - u_2^2\right) m_1 \left(u_1 + v_1\right) \left(u_1 - v_1\right) = m_2 \left(v_2 + u_2\right) \left(v_2 - u_2\right) From the second equation, m_1 \left(u_1 - v_1\right) = m_2 \left(v_2 - u_2\right) After division, u_1+v_1=v_2+u_2 u_1-u_2 = -(v_1-v_2) \frac{\left | v_1-v_2 \right |}{\left | u_1-u_2 \right |} = 1 The equation above is the restitution equation, and the coefficient of restitution is 1, which is a perfectly elastic collision. ===Sports equipment=== Thin-faced golf club drivers utilize a "trampoline effect" that creates drives of a greater distance as a result of the flexing and subsequent release of stored energy which imparts greater impulse to the ball. In the center of momentum frame where the total momentum equals zero, \begin{align} p_1 &= - p_2 \\\ p_1^2 &= p_2^2 \\\ E &= \sqrt {m_1^2c^4 + p_1^2c^2} + \sqrt {m_2^2c^4 + p_2^2c^2} = E \\\ p_1 &= \pm \frac{\sqrt{E^4 - 2E^2m_1^2c^4 - 2E^2m_2^2c^4 + m_1^4c^8 - 2m_1^2m_2^2c^8 + m_2^4c^8}}{2cE} \\\ u_1 &= -v_1. \end{align} Here m_1, m_2 represent the rest masses of the two colliding bodies, u_1, u_2 represent their velocities before collision, v_1, v_2 their velocities after collision, p_1, p_2 their momenta, c is the speed of light in vacuum, and E denotes the total energy, the sum of rest masses and kinetic energies of the two bodies. Indeed, to derive the equations, one may first change the frame of reference so that one of the known velocities is zero, determine the unknown velocities in the new frame of reference, and convert back to the original frame of reference. ====Examples==== ;Before collision: :Ball 1: mass = 3 kg, velocity = 4 m/s :Ball 2: mass = 5 kg, velocity = −6 m/s ;After collision: :Ball 1: velocity = −8.5 m/s :Ball 2: velocity = 1.5 m/s Another situation: frame|center|Elastic collision of unequal masses. To see this, consider the center of mass at time t before collision and time t' after collision: \begin{align} \bar{x}(t) &= \frac{m_{1} x_{1}(t)+m_{2} x_{2}(t)}{m_{1}+m_{2}} \\\ \bar{x}(t') &= \frac{m_{1} x_{1}(t')+m_{2} x_{2}(t')}{m_{1}+m_{2}}. \end{align} Hence, the velocities of the center of mass before and after collision are: \begin{align} v_{ \bar{x} } &= \frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}} \\\ v_{ \bar{x} }' &= \frac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}. \end{align} The numerators of v_{ \bar{x} } and v_{ \bar{x} }' are the total momenta before and after collision. When vehicles collide, the damage increases with the relative velocity of the vehicles, the damage increasing as the square of the velocity since it is the impact kinetic energy (1/2 mv2) which is the variable of importance. The coefficient of restitution (COR, also denoted by e), is the ratio of the final to initial relative speed between two objects after they collide. During the collision of small objects, kinetic energy is first converted to potential energy associated with a repulsive or attractive force between the particles (when the particles move against this force, i.e. the angle between the force and the relative velocity is obtuse), then this potential energy is converted back to kinetic energy (when the particles move with this force, i.e. the angle between the force and the relative velocity is acute). Assuming that the second particle is at rest before the collision, the angles of deflection of the two particles, \theta_1 and \theta_2, are related to the angle of deflection \theta in the system of the center of mass by \tan \theta_1=\frac{m_2 \sin \theta}{m_1+m_2 \cos \theta},\qquad \theta_2=\frac{{\pi}-{\theta}}{2}. The overall velocity of each body must be split into two perpendicular velocities: one tangent to the common normal surfaces of the colliding bodies at the point of contact, the other along the line of collision. Using the notation from above where u represents the velocity before the collision and v after, yields: \begin{align} & m_\text{a} u_\text{a} + m_\text{b} u_\text{b} = m_\text{a} v_\text{a} + m_\text{b} v_\text{b} \\\ & C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |} \\\ \end{align} Solving the momentum conservation equation for v_\text{a} and the definition of the coefficient of restitution for v_\text{b} yields: \begin{align} & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} - m_\text{b} v_\text{b}}{m_\text{a}} = v_\text{a} \\\ & v_\text{b} = C_R(u_\text{a} - u_\text{b}) + v_\text{a} \\\ \end{align} Next, substitution into the first equation for v_\text{b} and then resolving for v_\text{a} gives: \begin{align} & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} - m_\text{b} C_R(u_\text{a} - u_\text{b}) - m_\text{b} v_\text{a}}{m_\text{a}} = v_\text{a} \\\ & \\\ & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b} - u_\text{a})}{m_\text{a}} = v_\text{a} \left[ 1 + \frac{m_\text{b}}{m_\text{a}} \right] \\\ & \\\ & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b} - u_\text{a})}{m_\text{a} + m_\text{b}} = v_\text{a} \\\ \end{align} A similar derivation yields the formula for v_\text{b}. === COR variation due to object shape and off-center collisions === When colliding objects do not have a direction of motion that is in-line with their centers of gravity and point of impact, or if their contact surfaces at that point are not perpendicular to that line, some energy that would have been available for the post-collision velocity difference will be lost to rotation and friction. In this case, the change in Ek is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus: C_R = \sqrt{\frac{E_\text{p, (at bounce height)}}{E_\text{p, (at drop height)}}} = \sqrt{\frac{mgh}{mgH}} = \sqrt{\frac{h}{H}} == Speeds after impact == The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions, as well, and every possibility in between. v_\text{a} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b}-u_\text{a})}{m_\text{a}+m_\text{b}} and v_\text{b} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{a} C_R(u_\text{a}-u_\text{b})}{m_\text{a}+m_\text{b}} where *v_\text{a} is the final velocity of the first object after impact *v_\text{b} is the final velocity of the second object after impact *u_\text{a} is the initial velocity of the first object before impact *u_\text{b} is the initial velocity of the second object before impact *m_\text{a} is the mass of the first object *m_\text{b} is the mass of the second object === Derivation === The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). A comprehensive study of coefficients of restitution in dependence on material properties (elastic moduli, rheology), direction of impact, coefficient of friction and adhesive properties of impacting bodies can be found in Willert (2020). === Predicting from material properties === The COR is not a material property because it changes with the shape of the material and the specifics of the collision, but it can be predicted from material properties and the velocity of impact when the specifics of the collision are simplified. For two- and three-dimensional collisions of rigid bodies, the velocities used are the components perpendicular to the tangent line/plane at the point of contact, i.e. along the line of impact. The elastic range can be exceeded at higher velocities because all the kinetic energy is concentrated at the point of impact. In the case of a large u_{1}, the value of v_{1} is small if the masses are approximately the same: hitting a much lighter particle does not change the velocity much, hitting a much heavier particle causes the fast particle to bounce back with high speed.

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The height of a hill in meters is given by $z=2 x y-3 x^2-4 y^2-18 x+28 y+12$, where $x$ is the distance east and $y$ is the distance north of the origin. What is the $x$ distance of the top of the hill?

Nimbus Hills () is a rugged line of hills and peaks about 14 nautical miles (26 km) long, forming the southeast part of Pioneer Heights in the Heritage Range, Ellsworth Mountains. Target Hill () is a prominent hill which rises 1,010 m above the level of Larsen Ice Shelf. Gerdkooh ancient hills (Persian: تپه باستان گردکوه) consists of three hills, the tallest of which is 26 m in height. Vantages Hill () (Adam Hayat) is a flat-topped hill, over 2,000 m above sea level and 300 m above the surrounding plateau, standing 10 nautical miles (18 km) southwest of Mount Henderson in the western part of Britannia Range. Sistenup Peak () is a low peak at the northeast end of the Kirwan Escarpment, about 5 nautical miles (9 km) north of Sistefjell Mountain, in Queen Maud Land. Powell Hill () is a rounded, ice-covered prominence 6 nautical miles (11 km) west-southwest of Mount Christmas, overlooking the head of Algie Glacier. Named by Advisory Committee on Antarctic Names (US-ACAN) after the National Aeronautics and Space Administration weather satellite, Nimbus, which took photographs of Antarctica (including the Ellsworth Mountains) from approximately 500 nautical miles (900 km) above earth on September 13, 1964. ==See also== Geographical features include: ===Samuel Nunataks=== ===Other features=== * Flanagan Glacier * Mount Capley * Warren Nunatak Category:Hills of Ellsworth Land The hill was the most westerly point reached by the Falkland Islands Dependencies Survey (FIDS) survey party in 1955; it was visible to the party as a target upon which to steer from the summit of Richthofen Pass. Category:Hills of Oates Land The hills are located in Qaem Shahr in Mazandaran Province. It stands 6 nautical miles (11 km) west of Mount Fritsche on the south flank of Leppard Glacier in eastern Graham Land. There is evidence that the hills at the time of the Sasanian Empire and Muslim conquest of Persia were part of a Castle. == See also == *Castles in Iran *Tepe Sialk ==References== == External links == * video link * Persian Wikipedia article Category:Archaeological sites in Iran Category:Geography of Mazandaran Province Category:Castles in Iran Category:Buildings and structures in Mazandaran Province Category:Tourist attractions in Mazandaran Province Category:National works of Iran Category:Mountains of Queen Maud Land Category:Princess Martha Coast Category:Hills of Graham Land Category:Oscar II Coast Mapped by Norwegian cartographers from surveys and air photos by Norwegian-British-Swedish Antarctic Expedition (NBSAE) (1949–52) and air photos by the Norwegian exp (1958–59) and named Sistenup (last peak). Category:Hills of the Ross Dependency Category:Shackleton Coast Mapped by United States Geological Survey (USGS) from ground surveys and U.S. Navy air photos, 1961–66. This is the most southerly point reached by the Darwin Glacier Party of the Commonwealth Trans-Antarctic Expedition (1957–58), who gave it this name because of the splendid view it afforded. Named by Advisory Committee on Antarctic Names (US-ACAN) for Lieutenant Commander James A. Powell, U.S. Navy, communications officer at McMurdo Station during U.S. Navy Operation Deepfreeze 1963 and 1964. In exploring this area, a 4500-year-old grave has been found, as well as objects such as disposable tableware dishes related to the Parthian Empire and Sasanian Empire. Their history has been estimated to date back to the Iron Age. There is evidence that the hills at the time of the Sasanian Empire and Muslim conquest of Persia were part of a Castle. == See also == *Castles in Iran *Tepe Sialk ==References== == External links == * video link * Persian Wikipedia article Category:Archaeological sites in Iran Category:Geography of Mazandaran Province Category:Castles in Iran Category:Buildings and structures in Mazandaran Province Category:Tourist attractions in Mazandaran Province Category:National works of Iran

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Shot towers were popular in the eighteenth and nineteenth centuries for dropping melted lead down tall towers to form spheres for bullets. The lead solidified while falling and often landed in water to cool the lead bullets. Many such shot towers were built in New York State. Assume a shot tower was constructed at latitude $42^{\circ} \mathrm{N}$, and the lead fell a distance of $27 \mathrm{~m}$. In what direction and how far did the lead bullets land from the direct vertical?

Large shot which could not be made by the shot tower was made by tumbling pieces of cut lead sheet in a barrel until round.. Molten lead at the top of the tower was poured through a sieve or mesh, forming uniform spherical shot before falling into a large vat of water at the bottom of the tower. A shot tower with a 40-meter drop can produce up to #6 shot (nominally 2.4mm in diameter) while an 80-meter drop can produce #2 shot (nominally 3.8mm in diameter). thumb|100px|How a shot tower works A shot tower is a tower designed for the production of small-diameter shot balls by free fall of molten lead, which is then caught in a water basin. Shot towers work on the principle that molten lead forms perfectly round balls when poured from a high place. The "wind tower" method, which used a blast of cold air to dramatically shorten the drop necessary and was patented in 1848 by the T.O LeRoy Company of New York City,, Lynne Belluscio, LeRoy Penny Saver NewsHistory of the American Shot Tower meant that tall shot towers became unnecessary, but many were still constructed into the late 1880s, and two surviving examples date from 1916 and 1969. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. It can be seen looking west from I-95. 1870 Panorama from Sparks Shot Tower.jpg|1870 photo from the top of the tower toward the Delaware River 1880 survey Sparks Shot Tower.png|1880 Hexamer General Survey page on the tower Sparks Shot Tower2.jpg|1973 Historic American Buildings Survey photo Sparks Shot Tower Historical Marker 129-131 Carpenter St Philadelphia PA (DSC 3814).jpg|Historical Marker ==See also== *Lead shot *Phoenix Shot Tower *Shotgun shell ==References== ==External links== *Listing and photographs at the Historic American Buildings Survey *Sparks Shot Tower at USHistory.org *Waymark *Listing and photograph at Philadelphia Buildings and Architects Category:Industrial buildings completed in 1808 Category:Towers completed in 1808 Category:Buildings and structures in Philadelphia Category:Shot towers Category:South Philadelphia The shot is primarily used for projectiles in shotguns, and for ballast, radiation shielding, and other applications for which small lead balls are useful. == Shot making == === Process === In a shot tower, lead is heated until molten, then dropped through a copper sieve high in the tower. Use of shot towers replaced earlier techniques of casting shot in moulds, which was expensive, or of dripping molten lead into water barrels, which produced insufficiently spherical balls. In ballistics, the elevation is the angle between the horizontal plane and the axial direction of the barrel of a gun, mortar or heavy artillery. Originally used to produce shot for hunters, the tower produced ammunition during the War of 1812 and the Civil War. The Sparks Shot Tower is a historic shot tower located at 129-131 Carpenter Street in Philadelphia, Pennsylvania. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. thumb|upright|Shortly after completion thumb|upright|Shortly before demolition The Tower Building was a structure in the Financial District of Manhattan, New York City, located at 50-52 Broadway on a lot that extended east to New Street. Originally, elevation was a linear measure of how high the gunners had to physically lift the muzzle of a gun up from the gun carriage to compensate for projectile drop and hit targets at a certain distance. ==Until WWI== Though early 20th-century firearms were relatively easy to fire, artillery was not. With advances in the 21st century, it has become easy to determine how much elevation a gun needed to hit a target. thumb|right|360px|Lamoka projectile points from central New York State. At the start of the War of 1812, the federal government became their major customer, buying war munitions, and Quaker John Bishop sold his part of the company to Thomas Sparks.Sparks Shot Tower, 1808, in John Mayer, Workshop of the World (Oliver Evans Press, 1990) Before the use of shot towers, shot was made in wooden molds, which resulted in unevenly formed, low quality shot. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time.

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Perform an explicit calculation of the time average (i.e., the average over one complete period) of the potential energy for a particle moving in an elliptical orbit in a central inverse-square-law force field. Express the result in terms of the force constant of the field and the semimajor axis of the ellipse.

Conclusions: *The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (a\,\\!), *For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law). ==Energy== Under standard assumptions, the specific orbital energy (\epsilon) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form: :{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0 where: *v\, is the orbital speed of the orbiting body, *r\, is the distance of the orbiting body from the central body, *a\, is the length of the semi-major axis, *\mu\, is the standard gravitational parameter. Using the virial theorem we find: *the time-average of the specific potential energy is equal to −2ε **the time-average of r−1 is a−1 *the time-average of the specific kinetic energy is equal to ε === Energy in terms of semi major axis === It can be helpful to know the energy in terms of the semi major axis (and the involved masses). The empty focus (\mathbf{F2} = \left(f_x,f_y\right)) can be found by first determining the Eccentricity vector: :\mathbf{e} = \frac{\mathbf{r}}{|\mathbf{r}|} - \frac{\mathbf{v}\times \mathbf{h}}{\mu} Where \mathbf{h} is the specific angular momentum of the orbiting body: :\mathbf{h} = \mathbf{r} \times \mathbf{v} Then :\mathbf{F2} = -2a\mathbf{e} ====Using XY Coordinates==== This can be done in cartesian coordinates using the following procedure: The general equation of an ellipse under the assumptions above is: : \sqrt{ \left(f_x - x\right)^2 + \left(f_y - y\right)^2} + \sqrt{ x^2 + y^2 } = 2a \qquad\mid z=0 Given: :r_x, r_y \quad the initial position coordinates :v_x, v_y \quad the initial velocity coordinates and :\mu = Gm_1 \quad the gravitational parameter Then: :h = r_x v_y - r_y v_x \quad specific angular momentum :r = \sqrt{r_x^2 + r_y^2} \quad initial distance from F1 (at the origin) :a = \frac{\mu r}{2\mu - r \left(v_x^2 + v_y^2 \right)} \quad the semi-major axis length :e_x = \frac{r_x}{r} - \frac{h v_y}{\mu} \quad the Eccentricity vector coordinates :e_y = \frac{r_y}{r} + \frac{h v_x}{\mu} \quad Finally, the empty focus coordinates :f_x = - 2 a e_x \quad :f_y = - 2 a e_y \quad Now the result values fx, fy and a can be applied to the general ellipse equation above. == Orbital parameters == The state of an orbiting body at any given time is defined by the orbiting body's position and velocity with respect to the central body, which can be represented by the three-dimensional Cartesian coordinates (position of the orbiting body represented by x, y, and z) and the similar Cartesian components of the orbiting body's velocity. The solution of this equation is u(\varphi) = -\frac{\alpha}{mh^{2}} \left[ 1 + e \cos \left( \varphi - \varphi_{0}\right) \right] which shows that the orbit is a conic section of eccentricity e; here, φ0 is the initial angle, and the center of force is at the focus of the conic section. The total energy of the orbit is given by :E = \frac{1}{2}m v^2 - G \frac{Mm}{r}. The total energy of the orbit is given by :E = - G \frac{M m}{2a}, where a is the semi major axis. ==== Derivation ==== Since gravity is a central force, the angular momentum is constant: :\dot{\mathbf{L}} = \mathbf{r} \times \mathbf{F} = \mathbf{r} \times F(r)\mathbf{\hat{r}} = 0 At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore: :L = r p = r m v. Adopting the radial distance r and the azimuthal angle φ as the coordinates, the Hamilton-Jacobi equation for a central-force problem can be written \frac{1}{2m} \left( \frac{dS_{r}}{dr} \right)^{2} + \frac{1}{2m r^{2}} \left( \frac{dS_{\varphi}}{d\varphi} \right)^{2} + U(r) = E_{\mathrm{tot}} where S = Sφ(φ) + Sr(r) − Etott is Hamilton's principal function, and Etot and t represent the total energy and time, respectively. Assume that a particle is moving under an arbitrary central force F1(r), and let its radius r and azimuthal angle φ be denoted as r(t) and φ1(t) as a function of time t. Four line segments go out from the left focus to the ellipse, forming two shaded pseudo-triangles with two straight sides and the third side made from the curved segment of the intervening ellipse.|As for all central forces, the particle in the Kepler problem sweeps out equal areas in equal times, as illustrated by the two blue elliptical sectors. A central-force problem is said to be "integrable" if this final integration can be solved in terms of known functions. ===Orbit of the particle=== The total energy of the system Etot equals the sum of the potential energy and the kinetic energyGoldstein, p. For an attractive force (α < 0), the orbit is an ellipse, a hyperbola or parabola, depending on whether u1 is positive, negative, or zero, respectively; this corresponds to an eccentricity e less than one, greater than one, or equal to one. The eccentricity e is related to the total energy E (cf. the Laplace–Runge–Lenz vector) : e = \sqrt{1 + \frac{2EL^2}{k^2 m}} Comparing these formulae shows that E<0 corresponds to an ellipse (all solutions which are closed orbits are ellipses), E=0 corresponds to a parabola, and E>0 corresponds to a hyperbola. In particular, E=-\frac{k^2 m}{2L^2} for perfectly circular orbits (the central force exactly equals the centripetal force requirement, which determines the required angular velocity for a given circular radius). If the force is the sum of an inverse quadratic law and a linear term, i.e., if F(r) = \frac{a}{r^2} + cr, the problem also is solved explicitly in terms of Weierstrass elliptic functions.Izzo and Biscani ==References== ==Bibliography== * * Category:Classical mechanics If the eccentricity is less than 1 then the equation of motion describes an elliptical orbit. Thus, the areal velocity is constant for a particle acted upon by any type of central force; this is Kepler's second law.Goldstein, p. 73; Landau and Lifshitz, p. 31; Sommerfeld, p. 39; Symon, p. 135. 250px|thumb|Animation of Orbit by eccentricity In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. This includes the radial elliptic orbit, with eccentricity equal to 1. Conclusions: *For a given semi-major axis the specific orbital energy is independent of the eccentricity. The center of force is located at one of the foci of the elliptical orbit. where u1 and u2 are constants, with u2 larger than u1. In that case, :n = \frac{d}{2\pi}\sqrt{\frac{ G( M + m ) }{a^3}} = d\sqrt{\frac{ G( M + m ) }{4\pi^2 a^3}}\,\\! where *d is the quantity of time in a day, *G is the gravitational constant, *M and m are the masses of the orbiting bodies, *a is the length of the semi-major axis. By Newton's second law of motion, the magnitude of F equals the mass m of the particle times the magnitude of its radial accelerationGoldstein, p.

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A simple harmonic oscillator consists of a 100-g mass attached to a spring whose force constant is $10^4 \mathrm{dyne} / \mathrm{cm}$. The mass is displaced $3 \mathrm{~cm}$ and released from rest. Calculate the natural frequency $\nu_0$.

As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). In a mass–spring system, with mass m and spring stiffness k, the natural angular frequency can be calculated as: \omega _0 =\sqrt{\frac{k}{m}} In an electrical network, ω is a natural angular frequency of a response function f(t) if the Laplace transform F(s) of f(t) includes the term , where for a real σ, and is a constant. By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the "effective total mass" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon "external" factors such as the acceleration of gravity along it. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *"The Effective Mass of an Oscillating Spring" Am. J. Phys., 38, 98 (1970) *"Effective Mass of an Oscillating Spring" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. If the forced frequency is equal to the natural frequency, the vibrations' amplitude increases manyfold. If the oscillating system is driven by an external force at the frequency at which the amplitude of its motion is greatest (close to a natural frequency of the system), this frequency is called resonant frequency. ==Overview== Free vibrations of an elastic body are called natural vibrations and occur at a frequency called the natural frequency. In LC and RLC circuits, its natural angular frequency can be calculated as: \omega _0 =\frac{1}{\sqrt{LC}} ==See also== * Fundamental frequency ==Footnotes== ==References== * * * * Category:Waves Category:Oscillation The effective mass of the spring can be determined by finding its kinetic energy. thumb|Scale of harmonics on C. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Natural vibrations are different from forced vibrations which happen at the frequency of an applied force (forced frequency). thumb|119x119px|Energy level scheme of half-harmonic generation process. In a real spring–mass system, the spring has a non-negligible mass m. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. For instance: the frequency ratio 5:4 is equal to of the string length and is the complement of , the position of the fifth harmonic (and the fourth overtone). It can be shown that the set of natural frequencies in a network can be obtained by calculating the poles of all impedance and admittance functions of the network.

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Find the center of mass of a uniformly solid cone of base diameter $2a$ and height $h$

If the chosen ogive radius of a secant ogive is greater than the ogive radius of a tangent ogive with the same and , then the resulting secant ogive appears as a tangent ogive with a portion of the base truncated. :\rho > {R^2 + L^2 \over 2R} and \alpha = \arccos \left({\sqrt{L^2 + R^2} \over 2\rho}\right)-\arctan \left({R \over L}\right) Then the radius at any point as varies from to is: :y = \sqrt{\rho^2 - (\rho\cos(\alpha) - x)^2} - \rho\sin(\alpha) If the chosen is less than the tangent ogive and greater than half the length of the nose cone, then the result will be a secant ogive that bulges out to a maximum diameter that is greater than the base diameter. The profile of this shape is formed by a segment of a circle such that the rocket body is tangent to the curve of the nose cone at its base, and the base is on the radius of the circle. The radius of the circle that forms the ogive is called the ogive radius, , and it is related to the length and base radius of the nose cone as expressed by the formula: :\rho = {R^2 + L^2\over 2R} The radius at any point , as varies from to is: :y = \sqrt{\rho^2 - (L - x)^2}+R - \rho The nose cone length, , must be less than or equal to . The center of the spherical nose cap, , can be found from: : x_o = x_t + \sqrt{ r_n^2 - y_t^2} And the apex point, can be found from: : x_a = x_o - r_n === Bi-conic === A bi-conic nose cone shape is simply a cone with length stacked on top of a frustum of a cone (commonly known as a conical transition section shape) with length , where the base of the upper cone is equal in radius to the top radius of the smaller frustum with base radius . For 0 \le K' \le 1 : y = R\left({2 \left({x \over L}\right) - K'\left({x \over L}\right)^2 \over 2 - K'}\right) can vary anywhere between and , but the most common values used for nose cone shapes are: Parabola Type Value Cone Half Three Quarter Full For the case of the full parabola () the shape is tangent to the body at its base, and the base is on the axis of the parabola. The failure is governed by crack growth in concrete, which forms a typical cone shape having the anchor's axis as revolution axis. ==Mechanical models== ===ACI 349-85=== Under tension loading, the concrete cone failure surface has 45° inclination. The tangency point where the sphere meets the cone can be found from: : x_t = \frac{L^2}{R} \sqrt{ \frac{r_n^2}{R^2 + L^2} } : y_t = \frac{x_t R}{L} where is the radius of the spherical nose cap. :L=L_1+L_2 :For 0 \le x \le L_1 : y = {xR_1 \over L_1} :For L_1 \le x \le L : y = R_1 + {(x - L_1)(R_2-R_1)\over L_2} Half angles: :\phi_1 = \arctan \left({R_1 \over L_1}\right) and y = x \tan(\phi_1)\; :\phi_2 = \arctan \left({R_2 - R_1 \over L_2}\right) and y = R_1 + (x - L_1) \tan(\phi_2)\; === Tangent ogive === Next to a simple cone, the tangent ogive shape is the most familiar in hobby rocketry. alt=Two-dimensional drawing of an elliptical nose cone with dimensions added to show how L is the total length of the nose cone, R is the radius at the base, and y is the radius at a point x distance from the tip.|thumb|300x300px|General parameters used for constructing nose cone profiles. Finally, the apex point can be found from: : x_a = x_o - r_n === Secant ogive === The profile of this shape is also formed by a segment of a circle, but the base of the shape is not on the radius of the circle defined by the ogive radius. For , Haack nose cones bulge to a maximum diameter greater than the base diameter. The full body of revolution of the nose cone is formed by rotating the profile around the centerline . For many applications, such a task requires the definition of a solid of revolution shape that experiences minimal resistance to rapid motion through such a fluid medium. == Nose cone shapes and equations == === General dimensions === In all of the following nose cone shape equations, is the overall length of the nose cone and is the radius of the base of the nose cone. is the radius at any point , as varies from , at the tip of the nose cone, to . The sides of a conic profile are straight lines, so the diameter equation is simply: : y = {xR \over L} Cones are sometimes defined by their half angle, : : \phi = \arctan \left({R \over L}\right) and y = x \tan(\phi)\; ==== Spherically blunted conic ==== In practical applications such as re-entry vehicles, a conical nose is often blunted by capping it with a segment of a sphere. Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium (such as a rocket or aircraft, missile, shell or bullet), an important problem is the determination of the nose cone geometrical shape for optimum performance. right|thumb|Concrete Cone Model Concrete cone is one of the failure modes of anchors in concrete, loaded by a tensile force. The length/diameter relation is also often called the caliber of a nose cone. Wallis's Conical Edge with right|thumb|600px| Figure 2. thumb|right|400px|The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The concrete cone failure load N_0 of a single anchor in uncracked concrete unaffected by edge influences or overlapping cones of neighboring anchors is given by: N_0 = k \sqrt{f_{cc}} {h_{ef}}^{1.5} [N] , Where: k \- 13.5 for post-installed fasteners, 15.5 for cast-in-site fasteners f_{cc} \- Concrete compressive strength measured on cubes [MPa] {h_{ef}} \- Embedment depth of the anchor [mm] The model is based on fracture mechanics theory and takes into account the size effect, particularly for the factor {h_{ef}}^{1.5} which differentiates from {h_{ef}}^{2} expected from the first model. The concrete cone failure load N_0 of a single anchor in uncracked concrete unaffected by edge influences or overlapping cones of neighboring anchors is given by: N_0 = f_{ct} {A_{N}} [N] Where: f_{ct} \- tensile strength of concrete A_{N} \- Cone's projected area === Concrete capacity design (CCD) approach for fastening to concrete=== Under tension loading, the concrete capacity of a single anchor is calculated assuming an inclination between the failure surface and surface of the concrete member of about 35°. The rocket body will not be tangent to the curve of the nose at its base.

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A particle is projected with an initial velocity $v_0$ up a slope that makes an angle $\alpha$ with the horizontal. Assume frictionless motion and find the time required for the particle to return to its starting position. Find the time for $v_0=2.4 \mathrm{~m} / \mathrm{s}$ and $\alpha=26^{\circ}$.

Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one. \begin{align} v & = at+v_0 & [1]\\\ r & = r_0 + v_0 t + \tfrac12 {a}t^2 & [2]\\\ r & = r_0 + \tfrac12 \left( v+v_0 \right )t & [3]\\\ v^2 & = v_0^2 + 2a\left( r - r_0 \right) & [4]\\\ r & = r_0 + vt - \tfrac12 {a}t^2 & [5]\\\ \end{align} where: * is the particle's initial position * is the particle's final position * is the particle's initial velocity * is the particle's final velocity * is the particle's acceleration * is the time interval Equations [1] and [2] are from integrating the definitions of velocity and acceleration, subject to the initial conditions and ; \begin{align} \mathbf{v} & = \int \mathbf{a} dt = \mathbf{a}t+\mathbf{v}_0 \,, & [1] \\\ \mathbf{r} & = \int (\mathbf{a}t+\mathbf{v}_0) dt = \frac{\mathbf{a}t^2}{2}+\mathbf{v}_0t +\mathbf{r}_0 \,, & [2] \\\ \end{align} in magnitudes, \begin{align} v & = at+v_0 \,, & [1] \\\ r & = \frac{{a}t^2}{2}+v_0t +r_0 \,. thumb|360px|v vs t graph for a moving particle under a non-uniform acceleration a. Again these axial vectors must all be parallel to the axis of rotation, so only the magnitudes of the vectors are necessary, \begin{align} \omega & = \omega_0 + \alpha t \\\ \theta &= \theta_0 + \omega_0t + \tfrac12\alpha t^2 \\\ \theta & = \theta_0 + \tfrac12(\omega_0 + \omega)t \\\ \omega^2 & = \omega_0^2 + 2\alpha(\theta - \theta_0) \\\ \theta & = \theta_0 + \omega t - \tfrac12\alpha t^2 \\\ \end{align} where is the constant angular acceleration, is the angular velocity, is the initial angular velocity, is the angle turned through (angular displacement), is the initial angle, and is the time taken to rotate from the initial state to the final state. ===General planar motion=== These are the kinematic equations for a particle traversing a path in a plane, described by position . Suppose that C is the curve traced out by P and s is the arc length of C corresponding to time t. The velocity vector of the particle is : \mathbf{v} = \frac{d \mathbf{r}}{dt} = \dot{s}\mathbf{e}_t = v\mathbf{e}_t , where et is the unit tangent vector to C. Define the angular momentum of P as : \mathbf{h} = \mathbf{r} \times m\mathbf{v} = h\mathbf{k}, where k = i x j. Then the acceleration vector of P can be expressed as : \mathbf{a} = -\frac{\kappa v^2r}{p} \mathbf{e}_r + \left( v \frac{dv}{ds} + \frac{\kappa v^2q}{p} \right) \mathbf{e}_t . Given initial velocity , one can calculate how high the ball will travel before it begins to fall. According to Siacci's theorem, the acceleration a of P can be expressed as : \mathbf{a} = -\frac{\kappa v^2r}{p} \mathbf{e}_r + \frac{(h^2)'}{2p^2} \mathbf{e}_t = S_r \mathbf{e}_r + S_t \mathbf{e}_t . where the prime denotes differentiation with respect to the arc length s, and κ is the curvature function of the curve C. Let a particle P of mass m move in a two-dimensional Euclidean space (planar motion). The Merton school proved that the quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion. Let there be m variables that govern the forward- kinematics equation, i.e. the position function. The position of the particle is \mathbf{r} =\mathbf{r}\left ( r(t),\theta(t) \right ) = r \mathbf{\hat{e}}_r where and are the polar unit vectors. right|thumb|200px|The forward kinematics equations define the trajectory of the end-effector of a PUMA robot reaching for parts. Using equation [4] in the set above, we have: s= \frac{v^2 - u^2}{-2g}. Only Domingo de Soto, a Spanish theologian, in his commentary on Aristotle's Physics published in 1545, after defining "uniform difform" motion (which is uniformly accelerated motion) – the word velocity was not used – as proportional to time, declared correctly that this kind of motion was identifiable with freely falling bodies and projectiles, without his proving these propositions or suggesting a formula relating time, velocity and distance. Let p_0 = p(x_0) give the initial position of the system, and :p_1 = p(x_0 + \Delta x) be the goal position of the system. In kinematics, the acceleration of a particle moving along a curve in space is the time derivative of its velocity. Differentiating with respect to time again obtains the acceleration \mathbf{a} =\left ( \frac{d^2 r}{dt^2} - r\omega^2\right )\mathbf{\hat{e}}_r + \left ( r \alpha + 2 \omega \frac{dr}{dt} \right )\mathbf{\hat{e}}_\theta which breaks into the radial acceleration , centripetal acceleration , Coriolis acceleration , and angular acceleration . The initial position, initial velocity, and acceleration vectors need not be collinear, and take an almost identical form. The solution to the equation of motion, with specified initial values, describes the system for all times after . Thus, let C be a space curve traced out by P and s is the arc length of C corresponding to time t. From the instantaneous position , instantaneous meaning at an instant value of time , the instantaneous velocity and acceleration have the general, coordinate-independent definitions; \mathbf{v} = \frac{d \mathbf{r}}{d t} \,, \quad \mathbf{a} = \frac{d \mathbf{v}}{d t} = \frac{d^2 \mathbf{r}}{d t^2} Notice that velocity always points in the direction of motion, in other words for a curved path it is the tangent vector.

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Use the function described in Example 4.3, $x_{n+1}=\alpha x_n\left(1-x_n^2\right)$ where $\alpha=2.5$. Consider two starting values of $x_1$ that are similar, 0.9000000 and 0.9000001 . Make a plot of $x_n$ versus $n$ for the two starting values and determine the lowest value of $n$ for which the two values diverge by more than $30 \%$.

In mathematics, the reciprocal difference of a finite sequence of numbers (x_0, x_1, ..., x_n) on a function f(x) is defined inductively by the following formulas: :\rho_1(x_1, x_2) = \frac{x_1 - x_2}{f(x_1) - f(x_2)} :\rho_2(x_1, x_2, x_3) = \frac{x_1 - x_3}{\rho_1(x_1, x_2) - \rho_1(x_2, x_3)} + f(x_2) :\rho_n(x_1,x_2,\ldots,x_{n+1})=\frac{x_1-x_{n+1}}{\rho_{n-1}(x_1,x_2,\ldots,x_{n})-\rho_{n-1}(x_2,x_3,\ldots,x_{n+1})}+\rho_{n-2}(x_2,\ldots,x_{n}) ==See also== *Divided differences ==References== * * Category:Finite differences Finally, the sequence :(d_k) = \left\\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \ldots, \frac{1}{k + 1}, \ldots \right\\} converges sublinearly and logarithmically. thumb|alt=Plot showing the different rates of convergence for the sequences ak, bk, ck and dk.|Linear, linear, superlinear (quadratic), and sublinear rates of convergence|600px|center ==Convergence speed for discretization methods== A similar situation exists for discretization methods designed to approximate a function y = f(x), which might be an integral being approximated by numerical quadrature, or the solution of an ordinary differential equation (see example below). More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative argument, which reaches to linear growth of slope 1/4 for an argument near 0, then approaches 1 with an exponentially decaying gap. In particular, iterating a point x0 in [0, 1] gives rise to a sequence x_n: :x_{n+1} = f_\mu(x_n) = \begin{cases} \mu x_n & \mathrm{for}~~ x_n < \frac{1}{2} \\\ \mu (1-x_n) & \mathrm{for}~~ \frac{1}{2} \le x_n \end{cases} where μ is a positive real constant. In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. From top to bottom. 10, 100, 1000, 10000 points. ===Additive recurrence=== For any irrational \alpha, the sequence : s_n = \\{s_0 + n\alpha\\} has discrepancy tending to 1/N. Note that the sequence can be defined recursively by : s_{n+1} = (s_n + \alpha)\bmod 1 \;. Convergence with order * q = 1 is called linear convergence if \mu \in (0, 1), and the sequence is said to converge Q-linearly to L. * q = 2 is called quadratic convergence. * q = 3 is called cubic convergence. * etc. ==== Order estimation ==== A practical method to calculate the order of convergence for a sequence is to calculate the following sequence, which converges to q: :q \approx \frac{\log \left|\frac{x_{k+1} - x_k}{x_k - x_{k-1}}\right|}{\log \left|\frac{x_k - x_{k-1}}{x_{k-1} - x_{k-2}}\right|}. ==== Q-convergence definitions ==== In addition to the previously defined Q-linear convergence, a few other Q-convergence definitions exist. The difference of the approximations, 2\,\text{cm}, is in error by 100% of the magnitude of the difference of the true values, 1\,\text{cm}. Roughly speaking, the discrepancy of a sequence is low if the proportion of points in the sequence falling into an arbitrary set B is close to proportional to the measure of B, as would happen on average (but not for particular samples) in the case of an equidistributed sequence. thumb|right|Graph of tent map function 300px|thumb|right|Example of iterating the initial condition x0 = 0.4 over the tent map with μ = 1.9. The value of c with lowest discrepancy is the fractional part of the golden ratio: : c = \frac{\sqrt{5}-1}{2} = \varphi - 1 \approx 0.618034. We can solve this equation using the Forward Euler scheme for numerical discretization: : \frac{y_{n+1} - y_n}{h} = -\kappa y_{n}, which generates the sequence : y_{n+1} = y_n(1 - h\kappa). thumb|320px|right|Standard logistic function where L=1,k=1,x_0=0 A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation f(x) = \frac{L}{1 + e^{-k(x-x_0)}}, where For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the graph of f approaching L as x approaches +\infty and approaching zero as x approaches -\infty. If the points are chosen as elements of a low-discrepancy sequence, this is the quasi-Monte Carlo method. Category:Numerical analysis Convergence This means running a Monte-Carlo analysis with e.g. s=20 variables and N=1000 points from a low-discrepancy sequence generator may offer only a very minor accuracy improvement. ===Random numbers=== Sequences of quasirandom numbers can be generated from random numbers by imposing a negative correlation on those random numbers. The important parameter here for the convergence speed to y = f(x) is the grid spacing h, inversely proportional to the number of grid points, i.e. the number of points in the sequence required to reach a given value of x. In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. The relative errors of `x` from 1.000000000000001 and of `y` from 1.000000000000002 are both below 10^{-15} = 0.0000000000001\%, and the floating-point subtraction `y - x` is computed exactly by the Sterbenz lemma. This sequence converges with order 1 according to the convention for discretization methods.

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A gun fires a projectile of mass $10 \mathrm{~kg}$ of the type to which the curves of Figure 2-3 apply. The muzzle velocity is $140 \mathrm{~m} / \mathrm{s}$. Through what angle must the barrel be elevated to hit a target on the same horizontal plane as the gun and $1000 \mathrm{~m}$ away? Compare the results with those for the case of no retardation.

The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. In ballistics, the elevation is the angle between the horizontal plane and the axial direction of the barrel of a gun, mortar or heavy artillery. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. With advances in the 21st century, it has become easy to determine how much elevation a gun needed to hit a target. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. The projectile trajectory is affected by atmospheric conditions, the velocity of the projectile, the difference in altitude between the firer and the target, and other factors. The second solution is the useful one for determining the range of the projectile. Let tg be any time when the height of the projectile is equal to its initial value. : 0 = v t \sin \theta - \frac{1} {2} g t^2 By factoring: : t = 0 or : t = \frac{2 v \sin \theta} {g} but t = T = time of flight : T = \frac{2 v \sin \theta} {g} The first solution corresponds to when the projectile is first launched. The laser rangefinder and computer-based FCS make guns highly accurate. ==Superelevation== When firing a missile such as a MANPADS at an aircraft target, superelevation is an additional angle of elevation above the angle sighted on which corrects for the effect of gravity on the missile. ==See also== *Altitude (astronomy) *Pitching moment ==References== * Gunnery Instructions, U.S. Navy (1913), Register No. 4090 * Gunnery And Explosives For Artillery Officers (1911) * Fire Control Fundamentals, NAVPERS 91900 (1953), Part C: The Projectile in Flight - Exterior Ballistics * FM 6-40, Tactics, Techniques, and Procedures for Field Artillery Manual Cannon Gunnery (23 April 1996), Chapter 3 - Ballistics; Marine Corps Warfighting Publication No. 3-1.6.19 * FM 23-91, Mortar Gunnery (1 March 2000), Chapter 2 Fundamentals of Mortar Gunnery * Fundamentals of Naval Weapons Systems: Chapter 19 (Weapons and Systems Engineering Department United States Naval Academy) * Naval Ordnance and Gunnery (Vol.1 - Naval Ordnance) NAVPERS 10797-A (1957) * Naval Ordnance and Gunnery (Vol.2 - Fire Control) NAVPERS 10798-A (1957) * Naval Ordnance and Gunnery Category:Ballistics Category:Angle Originally, elevation was a linear measure of how high the gunners had to physically lift the muzzle of a gun up from the gun carriage to compensate for projectile drop and hit targets at a certain distance. ==Until WWI== Though early 20th-century firearms were relatively easy to fire, artillery was not. thumb|right|upright=1.35|Cutaway view of M1128 round The M1128 "Insensitive Munition High Explosive Base Burn Projectile" is a 155 mm boosted artillery round designed to achieve a maximum range of 30–40 km. The surface of the projectile also must be considered: a smooth projectile will face less air resistance than a rough-surfaced one, and irregularities on the surface of a projectile may change its trajectory if they create more drag on one side of the projectile than on the other. There are two dimensions in aiming a weapon: * In the horizontal plane (azimuth); and * In the vertical plane (elevation), which is governed by the distance (range) to the target and the energy of the propelling charge. Projectile and propellant gases act on barrel along barrel centerline A. Forces are resisted by shooter contact with gun at grips and stock B. Height difference between barrel centerline and average point of contact is height C. Forces A and B operating over moment arm / height C create torque or moment D, which rotates the firearm's muzzle up as illustrated at E. Muzzle rise, muzzle flip or muzzle climb refers to the tendency of a firearm's or airgun's muzzle (front end of the barrel) to rise up after firing.Recoil management: how you hold makes all the difference, Guns Magazine, Oct 2006 by Dave Anderson It more specifically refers to the seemingly unpredictable "jump" of the firearm's muzzle, caused by combined recoil from multiple shots being fired in quick succession. thumb|upright=1.35|Indirect fire trajectories for rockets, howitzers, field guns and mortars Indirect fire is aiming and firing a projectile without relying on a direct line of sight between the gun and its target, as in the case of direct fire. Originally "zero", meaning 6400 mils, 360 degrees or their equivalent, was set at whatever the direction the oriented gun was pointed. File:AKM and MP5K.JPEG|An AKM assault rifle asymmetric slant cut muzzle fixture designed to counteract muzzle rise (and muzzle climb) during (automatic) firing. Thus, the solution is : t = \frac {v \sin \theta} {g} + \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} Solving for the range once again : d = \frac {v \cos \theta} {g} \left ( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2 g y_0} \right) To maximize the range at any height : \theta = \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} Checking the limit as y_0 approaches 0 : \lim_{y_0 \to 0} \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} = \frac {\pi} {4} ====Angle of impact==== The angle ψ at which the projectile lands is given by: : \tan \psi = \frac {-v_y(t_d)} {v_x(t_d)} = \frac {\sqrt { v^2 \sin^2 \theta + 2 g y_0 }} { v \cos \theta} For maximum range, this results in the following equation: : \tan^2 \psi = \frac { 2 g y_0 + v^2 } { v^2 } = C+1 Rewriting the original solution for θ, we get: : \tan^2 \theta = \frac { 1 - \cos^2 \theta } { \cos^2 \theta } = \frac { v^2 } { 2 g y_0 + v^2 } = \frac { 1 } { C + 1 } Multiplying with the equation for (tan ψ)^2 gives: : \tan^2 \psi \, \tan^2 \theta = \frac { 2 g y_0 + v^2 } { v^2 } \frac { v^2 } { 2 g y_0 + v^2 } = 1 Because of the trigonometric identity : \tan (\theta + \psi) = \frac { \tan \theta + \tan \psi } { 1 - \tan \theta \tan \psi } , this means that θ + ψ must be 90 degrees. == Actual projectile motion == In addition to air resistance, which slows a projectile and reduces its range, many other factors also have to be accounted for when actual projectile motion is considered. === Projectile characteristics === Generally speaking, a projectile with greater volume faces greater air resistance, reducing the range of the projectile.

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A spacecraft is placed in orbit $200 \mathrm{~km}$ above Earth in a circular orbit. Calculate the minimum escape speed from Earth.

For the Earth at perihelion, the value is: : \sqrt {1.327 \times 10^{20} ~\text{m}^3 \text{s}^{-2} \cdot \left({2 \over 1.471 \times 10^{11} ~\text{m}} - {1 \over 1.496 \times 10^{11} ~\text{m}}\right)} \approx 30,300 ~\text{m}/\text{s} which is slightly faster than Earth's average orbital speed of , as expected from Kepler's 2nd Law. == Planets == The closer an object is to the Sun the faster it needs to move to maintain the orbit. This is true for r being the closest / furthest distance so we get two simultaneous equations which we solve for E: :E = - G \frac{Mm}{r_1 + r_2} Since r_1 = a + a \epsilon and r_2 = a - a \epsilon, where epsilon is the eccentricity of the orbit, we finally have the stated result. ==Flight path angle== The flight path angle is the angle between the orbiting body's velocity vector (= the vector tangent to the instantaneous orbit) and the local horizontal. Orbital velocities of the Planets Planet Orbital velocity Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Halley's Comet on an eccentric orbit that reaches beyond Neptune will be moving 54.6 km/s when from the Sun, 41.5 km/s when 1 AU from the Sun (passing Earth's orbit), and roughly 1 km/s at aphelion from the Sun., where r is the distance from the Sun, and a is the major semi-axis. Conclusions: *The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (a\,\\!), *For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law). ==Energy== Under standard assumptions, the specific orbital energy (\epsilon) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form: :{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0 where: *v\, is the orbital speed of the orbiting body, *r\, is the distance of the orbiting body from the central body, *a\, is the length of the semi-major axis, *\mu\, is the standard gravitational parameter. Objects passing Earth's orbit going faster than 42.1 km/s have achieved escape velocity and will be ejected from the Solar System if not slowed down by a gravitational interaction with a planet. The total energy of the orbit is given by :E = \frac{1}{2}m v^2 - G \frac{Mm}{r}. Escape velocity is the minimum speed an object without propulsion needs to have to move away indefinitely from the source of the gravity field. The total energy of the orbit is given by :E = - G \frac{M m}{2a}, where a is the semi major axis. ==== Derivation ==== Since gravity is a central force, the angular momentum is constant: :\dot{\mathbf{L}} = \mathbf{r} \times \mathbf{F} = \mathbf{r} \times F(r)\mathbf{\hat{r}} = 0 At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore: :L = r p = r m v. When one of the bodies is not of considerably lesser mass see: Gravitational two-body problem So, when one of the masses is almost negligible compared to the other mass, as the case for Earth and Sun, one can approximate the orbit velocity v_o as: :v_o \approx \sqrt{\frac{GM}{r}} or assuming equal to the radius of the orbit :v_o \approx \frac{v_e}{\sqrt{2}} Where is the (greater) mass around which this negligible mass or body is orbiting, and is the escape velocity. This can be used to obtain a more accurate estimate of the average orbital speed: : v_o = \frac{2\pi a}{T}\left[1-\frac{1}{4}e^2-\frac{3}{64}e^4 -\frac{5}{256}e^6 -\frac{175}{16384}e^8 - \cdots \right] The mean orbital speed decreases with eccentricity. ==Instantaneous orbital speed== For the instantaneous orbital speed of a body at any given point in its trajectory, both the mean distance and the instantaneous distance are taken into account: : v = \sqrt {\mu \left({2 \over r} - {1 \over a}\right)} where is the standard gravitational parameter of the orbited body, is the distance at which the speed is to be calculated, and is the length of the semi-major axis of the elliptical orbit. Being closest to the Sun and having the most eccentric orbit, Mercury's orbital speed varies from about 59 km/s at perihelion to 39 km/s at aphelion. Under standard assumptions of the conservation of angular momentum the flight path angle \phi satisfies the equation: :h\, = r\, v\, \cos \phi where: * h\, is the specific relative angular momentum of the orbit, * v\, is the orbital speed of the orbiting body, * r\, is the radial distance of the orbiting body from the central body, * \phi \, is the flight path angle \psi is the angle between the orbital velocity vector and the semi-major axis. u is the local true anomaly. \phi = u + \frac{\pi}{2} - \psi, therefore, :\cos \phi = \sin(\psi - u) = \sin\psi\cos u - \cos\psi\sin u = \frac{1 + e\cos u}{\sqrt{1 + e^2 + 2e\cos u}} :\tan \phi = \frac{e\sin u}{1 + e\cos u} where e is the eccentricity. Velocities of better-known numbered objects that have perihelion close to the Sun Object Velocity at perihelion Velocity at 1 AU (passing Earth's orbit) 322P/SOHO 181 km/s @ 0.0537 AU 37.7 km/s 96P/Machholz 118 km/s @ 0.124 AU 38.5 km/s 3200 Phaethon 109 km/s @ 0.140 AU 32.7 km/s 1566 Icarus 93.1 km/s @ 0.187 AU 30.9 km/s 66391 Moshup 86.5 km/s @ 0.200 AU 19.8 km/s 1P/Halley 54.6 km/s @ 0.586 AU 41.5 km/s ==See also== *Escape velocity *Delta-v budget *Hohmann transfer orbit *Bi-elliptic transfer ==References== Category:Orbits hu:Kozmikus sebességek#Szökési sebességek Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 250px|thumb|Animation of Orbit by eccentricity In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. This is an approximation that only holds true when the orbiting body is of considerably lesser mass than the central one, and eccentricity is close to zero. The maximum (instantaneous) orbital speed occurs at periapsis (perigee, perihelion, etc.), while the minimum speed for objects in closed orbits occurs at apoapsis (apogee, aphelion, etc.). Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). The empty focus (\mathbf{F2} = \left(f_x,f_y\right)) can be found by first determining the Eccentricity vector: :\mathbf{e} = \frac{\mathbf{r}}{|\mathbf{r}|} - \frac{\mathbf{v}\times \mathbf{h}}{\mu} Where \mathbf{h} is the specific angular momentum of the orbiting body: :\mathbf{h} = \mathbf{r} \times \mathbf{v} Then :\mathbf{F2} = -2a\mathbf{e} ====Using XY Coordinates==== This can be done in cartesian coordinates using the following procedure: The general equation of an ellipse under the assumptions above is: : \sqrt{ \left(f_x - x\right)^2 + \left(f_y - y\right)^2} + \sqrt{ x^2 + y^2 } = 2a \qquad\mid z=0 Given: :r_x, r_y \quad the initial position coordinates :v_x, v_y \quad the initial velocity coordinates and :\mu = Gm_1 \quad the gravitational parameter Then: :h = r_x v_y - r_y v_x \quad specific angular momentum :r = \sqrt{r_x^2 + r_y^2} \quad initial distance from F1 (at the origin) :a = \frac{\mu r}{2\mu - r \left(v_x^2 + v_y^2 \right)} \quad the semi-major axis length :e_x = \frac{r_x}{r} - \frac{h v_y}{\mu} \quad the Eccentricity vector coordinates :e_y = \frac{r_y}{r} + \frac{h v_x}{\mu} \quad Finally, the empty focus coordinates :f_x = - 2 a e_x \quad :f_y = - 2 a e_y \quad Now the result values fx, fy and a can be applied to the general ellipse equation above. == Orbital parameters == The state of an orbiting body at any given time is defined by the orbiting body's position and velocity with respect to the central body, which can be represented by the three-dimensional Cartesian coordinates (position of the orbiting body represented by x, y, and z) and the similar Cartesian components of the orbiting body's velocity. See radial hyperbolic trajectory * If the total energy is zero, (Ek = Ep): the orbit is a parabola with focus at the other body. For this case it is convenient to use the following assumptions which differ somewhat from the standard assumptions above: :# The central body's position is at the origin and is the primary focus (\mathbf{F1}) of the ellipse (alternatively, the center of mass may be used instead if the orbiting body has a significant mass) :# The central body's mass (m1) is known :# The orbiting body's initial position(\mathbf{r}) and velocity(\mathbf{v}) are known :# The ellipse lies within the XY-plane The fourth assumption can be made without loss of generality because any three points (or vectors) must lie within a common plane. Examples of elliptic orbits include: Hohmann transfer orbit, Molniya orbit, and tundra orbit. ==Velocity== Under standard assumptions, no other forces acting except two spherically symmetrical bodies m1 and m2, the orbital speed (v\,) of one body traveling along an elliptic orbit can be computed from the vis-viva equation as: :v = \sqrt{\mu\left({2\over{r}} - {1\over{a}}\right)} where: *\mu\, is the standard gravitational parameter, G(m1+m2), often expressed as GM when one body is much larger than the other. *r\, is the distance between the orbiting body and center of mass. *a\,\\! is the length of the semi-major axis.

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Find the value of the integral $\int_S(\nabla \times \mathbf{A}) \cdot d \mathbf{a}$ if the vector $\mathbf{A}=y \mathbf{i}+z \mathbf{j}+x \mathbf{k}$ and $S$ is the surface defined by the paraboloid $z=1-x^2-y^2$, where $z \geq 0$.

Let us note that the surface integral of this 2-form is the same as the surface integral of the vector field which has as components f_x, f_y and f_z. == Theorems involving surface integrals == Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem. == Dependence on parametrization == Let us notice that we defined the surface integral by using a parametrization of the surface S. For integrals of scalar fields, the answer to this question is simple; the value of the surface integral will be the same no matter what parametrization one uses. This formula defines the integral on the left (note the dot and the vector notation for the surface element). Then, the surface integral of f on S is given by :\iint_D \left[ f_{z} ( \mathbf{r} (s,t)) \frac{\partial(x,y)}{\partial(s,t)} + f_{x} ( \mathbf{r} (s,t)) \frac{\partial(y,z)}{\partial(s,t)} + f_{y} ( \mathbf{r} (s,t))\frac{\partial(z,x)}{\partial(s,t)} \right]\, \mathrm ds\, \mathrm dt where :{\partial \mathbf{r} \over \partial s}\times {\partial \mathbf{r} \over \partial t}=\left(\frac{\partial(y,z)}{\partial(s,t)}, \frac{\partial(z,x)}{\partial(s,t)}, \frac{\partial(x,y)}{\partial(s,t)}\right) is the surface element normal to S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, \Gamma = \oint_{\partial S} \mathbf{V}\cdot \mathrm{d}\mathbf{l} = \iint_S abla \times \mathbf{V} \cdot \mathrm{d}\mathbf{S}=\iint_S \boldsymbol{\omega} \cdot \mathrm{d}\mathbf{S} Here, the closed integration path is the boundary or perimeter of an open surface , whose infinitesimal element normal is oriented according to the right-hand rule. With the above notation, if \mathbf{F} is any smooth vector field on \R^3, thenRobert Scheichl, lecture notes for University of Bath mathematics course \oint_{\partial\Sigma} \mathbf{F}\, \cdot\, \mathrm{d}{\mathbf{\Gamma}} = \iint_{\Sigma} abla\times\mathbf{F}\, \cdot\, \mathrm{d}\mathbf{\Sigma}. We find the formula :\begin{align} \iint_S {\mathbf v}\cdot\mathrm d{\mathbf {s}} &= \iint_S \left({\mathbf v}\cdot {\mathbf n}\right)\,\mathrm ds\\\ &{}= \iint_T \left({\mathbf v}(\mathbf{r}(s, t)) \cdot {\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t} \over \left\|\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right\|}\right) \left\|\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right\| \mathrm ds\, \mathrm dt\\\ &{}=\iint_T {\mathbf v}(\mathbf{r}(s, t))\cdot \left(\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right) \mathrm ds\, \mathrm dt. \end{align} The cross product on the right-hand side of this expression is a (not necessarily unital) surface normal determined by the parametrisation. In other words, we have to integrate v with respect to the vector surface element \mathrm{d}\mathbf s = {\mathbf n} \mathrm{d}s, which is the vector normal to S at the given point, whose magnitude is \mathrm{d}s = \|\mathrm{d}{\mathbf s}\|. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal n to S at each point, which will give us a scalar field, and integrate the obtained field as above. Suppose now that it is desired to integrate only the normal component of the vector field over the surface, the result being a scalar, usually called the flux passing through the surface. Then, the surface integral is given by : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \left\|{\partial \mathbf{r} \over \partial s} \times {\partial \mathbf{r} \over \partial t}\right\| \mathrm ds\, \mathrm dt where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of , and is known as the surface element (which would, for example, yield a smaller value near the poles of a sphere. where the lines of longitude converge more dramatically, and latitudinal coordinates are more compactly spaced). A natural question is then whether the definition of the surface integral depends on the chosen parametrization. Such a surface is called non-orientable, and on this kind of surface, one cannot talk about integrating vector fields. == See also == * Divergence theorem * Stokes' theorem * Line integral * Volume element * Volume integral * Cartesian coordinate system * Volume and surface area elements in spherical coordinate systems * Volume and surface area elements in cylindrical coordinate systems * Holstein–Herring method ==References== == External links == * Surface Integral — from MathWorld * Surface Integral — Theory and exercises Category:Multivariable calculus Category:Area Category:Surfaces If a vector field \mathbf{F}(x,y,z) = (F_x(x, y, z), F_y(x, y, z), F_z(x, y, z)) is defined and has continuous first order partial derivatives in a region containing \Sigma, then \iint_\Sigma ( abla \times \mathbf{F}) \cdot \mathrm{d} \mathbf{\Sigma} = \oint_{\partial\Sigma} \mathbf{F} \cdot \mathrm{d}\mathbf{\Gamma}. thumb|right|An illustration of Stokes' theorem, with surface , its boundary and the normal vector . That is, A' below is also a vector potential of v; \mathbf{A'} (\mathbf{x}) = \frac{1}{4 \pi} \int_{\Omega} \frac{ abla_y \times \mathbf{v} (\mathbf{y})}{\left\|\mathbf{x} -\mathbf{y} \right\|} \, d^3\mathbf{y}. By analogy with Biot-Savart's law, the following \boldsymbol{A}(\textbf{x}) is also qualify as a vector potential for v. :\boldsymbol{A}(\textbf{x}) =\int_\Omega \frac{\boldsymbol{v}(\boldsymbol{y}) \times (\boldsymbol{x} - \boldsymbol{y})}{4 \pi |\boldsymbol{x} - \boldsymbol{y}|^3} d^3 \boldsymbol{y} Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law. The surface integral can also be expressed in the equivalent form : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \sqrt{g} \, \mathrm ds\, \mathrm dt where is the determinant of the first fundamental form of the surface mapping . If, however, the normals for these parametrizations point in opposite directions, the value of the surface integral obtained using one parametrization is the negative of the one obtained via the other parametrization. The circulation of a vector field around a closed curve is the line integral: \Gamma = \oint_{C}\mathbf{V} \cdot \mathrm d \mathbf{l}.

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A skier weighing $90 \mathrm{~kg}$ starts from rest down a hill inclined at $17^{\circ}$. He skis $100 \mathrm{~m}$ down the hill and then coasts for $70 \mathrm{~m}$ along level snow until he stops. Find the coefficient of kinetic friction between the skis and the snow.

Kinetic (or dynamic) friction occurs when the ski is moving over the snow. One type of friction acting on the skier is the kinetic friction between the skis and snow. The coefficient of kinetic friction, \mu_\mathrm{k}, is less than the coefficient of static friction for both ice and snow. The motion of a skier is determined by the physical principles of the conservation of energy and the frictional forces acting on the body. The second type of frictional force acting on a skier is drag. The kinetic friction can be reduced by applying wax to the bottom of the skis which reduces the coefficient of friction. The necessary speed required to keep the skier upright varies by the weight of the barefooter and can be approximated by the following formula: (W / 10) + 20, where W is the skier's weight in pounds and the result is in miles per hour. The Physics of Skiing. The force required for sliding on snow is the product of the coefficient of kinetic friction and the normal force: F_{k} = \mu_\mathrm{k} F_{n}\,. However, the heat generated by friction can be lost by conduction to a cold ski, thereby diminishing the production of the melt layer. The ability of a ski or other runner to slide over snow depends on both the properties of the snow and the ski to result in an optimum amount of lubrication from melting the snow by friction with the ski—too little and the ski interacts with solid snow crystals, too much and capillary attraction of meltwater retards the ski. ===Friction=== Before a ski can slide, it must overcome the maximum value static friction, F_{max} = \mu_\mathrm{s} F_{n}\,, for the ski/snow contact, where \mu_\mathrm{s} is the coefficient of static friction and F_{n}\, is the normal force of the ski on snow. *Moisture content: The percentage of mass that is liquid water and may create suction friction with the base of the ski as it slides. A skier with skis pointed perpendicular to the fall line, across the hill instead of down it, will accelerate more slowly. The physics of skiing refers to the analysis of the forces acting on a person while skiing. right|thumb|300x300px|The texture of this top layer dependent on the weather history. Kuzmin and Fuss suggest that the most favorable combination of ski base material properties to minimize ski sliding friction on snow include: increased hardness and lowered thermal conductivity of the base material to promote meltwater generation for lubrication, wear resistance in cold snow, and hydrophobicity to minimize capillary suction. The shape and construction material of a ski can also greatly impact the forces acting on a skier.D. A. Lind and S. P. Sanders. Typically, a sliding ski melts a thin and transitory film of lubricating layer of water, caused by the heat of friction between the ski and the snow in its passing. Additionally, the skier can use the same techniques to turn the ski away from the direction of movement, generating skidding forces between the skis and snow which further slow the descent. Wax is adjusted for hardness to minimize sliding friction as a function of snow properties, which include the effects of: *Age: Reflects the metamorphism of snow crystals that are sharp and well- defined, when new, but with aging become broken or truncated with wind action or rounded into ice granules with freeze-thaw, all of which affects a ski's coefficient of friction. The material of Mr. Snow is claimed to have very good sliding capacities, is predictable in all climates and does not harm the ski or sliding surface. In doing so, the snow resists passage of the stemmed ski, creating a force that retards downhill speed and sustains a turn in the opposite direction. Too little melting and sharp edges of snow crystals or too much suction impede the passage of the ski.

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Consider a comet moving in a parabolic orbit in the plane of Earth's orbit. If the distance of closest approach of the comet to the $\operatorname{Sun}$ is $\beta r_E$, where $r_E$ is the radius of Earth's (assumed) circular orbit and where $\beta<1$, show that the time the comet spends within the orbit of Earth is given by $$ \sqrt{2(1-\beta)} \cdot(1+2 \beta) / 3 \pi \times 1 \text { year } $$ If the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earth's orbit?

209P/LINEAR is a periodic comet with an orbital period of 5.1 years. 71P/Clark is a periodic comet in the Solar System with an orbital period of 5.5 years. The comet last came to perihelion (closest approach to the Sun) on February 1, 2022 and will next come to perihelion on December 11, 2028. Perihelion distance at recent epochs Epoch Perihelion (AU) 2028 1.310 2022 1.306 2015 1.349 2008 1.355 Comet Borrelly or Borrelly's Comet (official designation: 19P/Borrelly) is a periodic comet, which was visited by the spacecraft Deep Space 1 in 2001. Prediscovery images of the comet, dating back to December 2003, were found during 2009. 209P/LINEAR came to perihelion (closest approach to the Sun) on 6 May 2014. The 2014 Earth approach was the 9th closest known comet approach to Earth. 170P/Christensen is a periodic comet in the Solar System. Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 164P/Christensen is a periodic comet in the Solar System. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 164P/Christensen – Seiichi Yoshida @ aerith.net * Elements and Ephemeris for 164P/Christensen – Minor Planet Center Category:Periodic comets 0164 164P 20041221 The nucleus of the comet has a radius of 0.68 ± 0.04 kilometers, assuming a geometric albedo of 0.04, based on observations by Hubble Space Telescope, while observations by Keck indicate a radius of 1.305 km. ==See also== * List of numbered comets == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 71P/Clark – Seiichi Yoshida @ aerith.net Category:Periodic comets 0071 Category:Comets in 2011 Category:Comets in 2017 19730609 The radar imaging showed the comet nucleus is elongated and about 2.4 km by 3 km in size, later refined to 3.9 × 2.7 × 2.6 km. On 29 May 2014 the comet passed from Earth, but only brightened to about apparent magnitude 12. It came to perihelion in September 2014 at about apparent magnitude 18. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 170P on Seiichi Yoshida's comet list * Elements and Ephemeris for 170P/Christensen – Minor Planet Center Category:Periodic comets 0170 170P 20050617 All the trails from the comet from 1803 through 1924 were expected to intersect Earth's orbit during May 2014. The comet has extremely low activity for its size and is probably in the process of evolving into an extinct comet. == Observational history == The comet discovered on 3 February 2004 by Lincoln Near-Earth Asteroid Research (LINEAR) using a reflector. Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). The peak activity was expected to occur around 24 May 2014 7h UT when dust trails produced from past returns of the comet could pass from Earth. 2005 HC4 is the asteroid with the smallest known perihelion of any known object orbiting the Sun (except sungrazing comets). The close approach allowed the comet nucleus to be imaged by Arecibo, producing the most detailed radar image of a comet nucleus to that date. It was given the permanent number 209P on 12 December 2008 as it was the second observed appearance of the comet. Deep Space 1 returned images of the comet's nucleus from 3400 kilometers away. The comet also had very low water production, mol/s, from an active area measuring just 0.007 km². 209P/LINEAR was recovered on 31 December 2018 at magnitude 19.2 by Hidetaka Sato. ==Associated meteor showers== Preliminary results by Esko Lyytinen and Peter Jenniskens, later confirmed by other researchers, predicted 209P/LINEAR might a big meteor shower which would come from the constellation Camelopardalis on the night of 23/24 May 2014.

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An automobile drag racer drives a car with acceleration $a$ and instantaneous velocity $v$. The tires (of radius $r_0$ ) are not slipping. For what initial velocity in the rotating system will the hockey puck appear to be subsequently motionless in the fixed system?

For a swept angle the change in is a vector at right angles to and of magnitude , which in turn means that the magnitude of the acceleration is given by a_c = v \frac{d\theta}{dt} = v\omega = \frac{v^2}{r} Centripetal acceleration for some values of radius and magnitude of velocity colspan="2" rowspan="2" 1 m/s 3.6 km/h 2.2 mph 2 m/s 7.2 km/h 4.5 mph 5 m/s 18 km/h 11 mph 10 m/s 36 km/h 22 mph 20 m/s 72 km/h 45 mph 50 m/s 180 km/h 110 mph 100 m/s 360 km/h 220 mph Slow walk Bicycle City car Aerobatics 10 cm 3.9 in Laboratory centrifuge 10 m/s2 1.0 g 40 m/s2 4.1 g 250 m/s2 25 g 1.0 km/s2 100 g 4.0 km/s2 410 g 25 km/s2 2500 g 100 km/s2 10000 g 20 cm 7.9 in 5.0 m/s2 0.51 g 20 m/s2 2.0 g 130 m/s2 13 g 500 m/s2 51 g 2.0 km/s2 200 g 13 km/s2 1300 g 50 km/s2 5100 g 50 cm 1.6 ft 2.0 m/s2 0.20 g 8.0 m/s2 0.82 g 50 m/s2 5.1 g 200 m/s2 20 g 800 m/s2 82 g 5.0 km/s2 510 g 20 km/s2 2000 g 1 m 3.3 ft Playground carousel 1.0 m/s2 0.10 g 4.0 m/s2 0.41 g 25 m/s2 2.5 g 100 m/s2 10 g 400 m/s2 41 g 2.5 km/s2 250 g 10 km/s2 1000 g 2 m 6.6 ft 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 13 m/s2 1.3 g 50 m/s2 5.1 g 200 m/s2 20 g 1.3 km/s2 130 g 5.0 km/s2 510 g 5 m 16 ft 200 mm/s2 0.020 g 800 mm/s2 0.082 g 5.0 m/s2 0.51 g 20 m/s2 2.0 g 80 m/s2 8.2 g 500 m/s2 51 g 2.0 km/s2 200 g 10 m 33 ft Roller-coaster vertical loop 100 mm/s2 0.010 g 400 mm/s2 0.041 g 2.5 m/s2 0.25 g 10 m/s2 1.0 g 40 m/s2 4.1 g 250 m/s2 25 g 1.0 km/s2 100 g 20 m 66 ft 50 mm/s2 0.0051 g 200 mm/s2 0.020 g 1.3 m/s2 0.13 g 5.0 m/s2 0.51 g 20 m/s2 2 g 130 m/s2 13 g 500 m/s2 51 g 50 m 160 ft 20 mm/s2 0.0020 g 80 mm/s2 0.0082 g 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 8.0 m/s2 0.82 g 50 m/s2 5.1 g 200 m/s2 20 g 100 m 330 ft Freeway on-ramp 10 mm/s2 0.0010 g 40 mm/s2 0.0041 g 250 mm/s2 0.025 g 1.0 m/s2 0.10 g 4.0 m/s2 0.41 g 25 m/s2 2.5 g 100 m/s2 10 g 200 m 660 ft 5.0 mm/s2 0.00051 g 20 mm/s2 0.0020 g 130 m/s2 0.013 g 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 13 m/s2 1.3 g 50 m/s2 5.1 g 500 m 1600 ft 2.0 mm/s2 0.00020 g 8.0 mm/s2 0.00082 g 50 mm/s2 0.0051 g 200 mm/s2 0.020 g 800 mm/s2 0.082 g 5.0 m/s2 0.51 g 20 m/s2 2.0 g 1 km 3300 ft High-speed railway 1.0 mm/s2 0.00010 g 4.0 mm/s2 0.00041 g 25 mm/s2 0.0025 g 100 mm/s2 0.010 g 400 mm/s2 0.041 g 2.5 m/s2 0.25 g 10 m/s2 1.0 g ==Non-uniform== right|293 px|frameless In a non-uniform circular motion, an object is moving in a circular path with a varying speed. * The period of the motion is 2 seconds per turn. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion. ==Uniform circular motion== thumb|upright=0.82|Figure 1: Velocity and acceleration in uniform circular motion at angular rate ; the speed is constant, but the velocity is always tangent to the orbit; the acceleration has constant magnitude, but always points toward the center of rotation. thumb|upright=1.14|right|Figure 2: The velocity vectors at time and time are moved from the orbit on the left to new positions where their tails coincide, on the right. Rotation with no velocity, r_e \Omega e 0 and v = 0, means that \text{slip} = \infty. ==Lateral slip== The lateral slip of a tire is the angle between the direction it is moving and the direction it is pointing. This root sum of squares of separate radial and tangential accelerations is only correct for circular motion; for general motion within a plane with polar coordinates (r, \theta), the Coriolis term a_c = 2 \left(\frac{dr}{dt}\right)\left(\frac{d\theta}{dt}\right) should be added to a_t, whereas radial acceleration then becomes a_r = \frac{-v^2}{r} + \frac{d^2 r}{dt^2}. ==See also== * Angular momentum * Equations of motion for circular motion * * Fictitious force * Geostationary orbit * Geosynchronous orbit * Pendulum (mathematics) * Reactive centrifugal force * Reciprocating motion * * Sling (weapon) ==References== ==External links== * Physclips: Mechanics with animations and video clips from the University of New South Wales * Circular Motion – a chapter from an online textbook * Circular Motion Lecture – a video lecture on CM * – an online textbook with different analysis for circular motion Category:Rotation Category:Classical mechanics Category:Motion (physics) The speed of the object traveling the circle is: v = \frac{2 \pi r}{T} = \omega r The angle swept out in a time is: \theta = 2 \pi \frac{t}{T} = \omega t The angular acceleration, , of the particle is: \alpha = \frac{d\omega}{dt} In the case of uniform circular motion, will be zero. Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. In a uniform circular motion, the total acceleration of an object in a circular path is equal to the radial acceleration. The acceleration due to change in the direction is: a_c = \frac{v^2}{r} = \omega^2 r The centripetal and centrifugal force can also be found using acceleration: F_c = \dot{p} \mathrel\overset{\dot{m} = 0}{=} ma_c = \frac{mv^2}{r} The vector relationships are shown in Figure 1. To find the total acceleration of an object in a non uniform circular, find the vector sum of the tangential acceleration and the radial acceleration. \sqrt{a_r^2 + a_t^2} = a Radial acceleration is still equal to \frac{v^2}{r}. Locked brakes, r_e \Omega = 0, means that \text{slip} = -1 = -100\% and sliding without rotating. Therefore, the speed of travel around the orbit is v = r \frac{d\theta}{dt} = r\omega , where the angular rate of rotation is . Because the radius of the circle is constant, the radial component of the velocity is zero. In a non-uniform circular motion, the net acceleration (a) is along the direction of , which is directed inside the circle but does not pass through its center (see figure). The acceleration points radially inwards (centripetally) and is perpendicular to the velocity. With this notation, the velocity becomes: v = \dot{z} = \frac{d}{dt}\left(R e^{i\theta[t]}\right) = R \frac{d}{dt}\left(e^{i\theta[t]}\right) = R e^{i\theta(t)} \frac{d}{dt} \left(i \theta[t] \right) = iR\dot{\theta}(t) e^{i\theta(t)} = i\omega R e^{i\theta(t)} = i\omega z and the acceleration becomes: \begin{align} a &= \dot{v} = i\dot{\omega} z + i\omega\dot{z} = \left(i\dot{\omega} - \omega^2\right)z \\\ &= \left(i\dot{\omega} - \omega^2 \right) R e^{i\theta(t)} \\\ &= -\omega^2 R e^{i\theta(t)} + \dot{\omega} e^{i\frac{\pi}{2}} R e^{i\theta(t)} \, . \end{align} The first term is opposite in direction to the displacement vector and the second is perpendicular to it, just like the earlier results shown before. ====Velocity==== Figure 1 illustrates velocity and acceleration vectors for uniform motion at four different points in the orbit. Thus, is a constant, and the velocity vector also rotates with constant magnitude , at the same angular rate . ====Relativistic circular motion==== In this case, the three- acceleration vector is perpendicular to the three-velocity vector, \mathbf{u} \cdot \mathbf{a} = 0. and the square of proper acceleration, expressed as a scalar invariant, the same in all reference frames, \alpha^2 = \gamma^4 a^2 + \gamma^6 \left(\mathbf{u} \cdot \mathbf{a}\right)^2, becomes the expression for circular motion, \alpha^2 = \gamma^4 a^2. or, taking the positive square root and using the three-acceleration, we arrive at the proper acceleration for circular motion: \alpha = \gamma^2 \frac{v^2}{r}. ====Acceleration==== The left-hand circle in Figure 2 is the orbit showing the velocity vectors at two adjacent times. As , the acceleration vector becomes perpendicular to , which means it points toward the center of the orbit in the circle on the left. The only acceleration responsible for keeping an object moving in a circle is the radial acceleration. The rotor is rigid if R is independent of time. Consequently, the acceleration is: \begin{align} \mathbf{a}(t) &= R \left( \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) + \omega \frac{d \hat\mathbf{u}_\theta}{dt} \right) \\\ &= R \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) - \omega^2 R \hat\mathbf{u}_R(t) \,. \end{align} The centripetal acceleration is the radial component, which is directed radially inward: \mathbf{a}_R(t) = -\omega^2 R \hat\mathbf{u}_R(t) \, , while the tangential component changes the magnitude of the velocity: \mathbf{a}_\theta(t) = R \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) = \frac{d R \omega}{dt} \hat\mathbf{u}_\theta(t) = \frac{d \left|\mathbf{v}(t)\right|}{dt} \hat\mathbf{u}_\theta(t) \, . ====Using complex numbers==== Circular motion can be described using complex numbers. The equations of motion describe the movement of the center of mass of a body.

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A British warship fires a projectile due south near the Falkland Islands during World War I at latitude $50^{\circ} \mathrm{S}$. If the shells are fired at $37^{\circ}$ elevation with a speed of $800 \mathrm{~m} / \mathrm{s}$, by how much do the shells miss their target and in what direction? Ignore air resistance.

Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. The target range at the time of projectile impact can be estimated using Equation 1, which is illustrated in Figure 3. The Ballistic Trajectory Extended Range Munition (BTERM) was a failed program to develop a precision guided rocket-assisted 127 mm (5-inch) artillery shell for the U.S. Navy. The higher altitude readings were needed for firings of the coast defense mortars, which sent their shells on very high trajectories. In naval gunnery, when long-range guns became available, an enemy ship would move some distance after the shells were fired. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. The impact point of a projectile is a function of many variables: :* Air temperature :* Air density :* Wind :* Range :* Earth rotation :* Projectile, fuze, weapon characteristics :* Muzzle velocity :* Propellant temperature :* Drift :* Parallax between the guns and the rangefinders and radar systems :* Elevation difference between target and artillery piece The firing tables provide data for an artillery piece firing under standardized conditions and the corrections required to determine the point of impact under actual conditions. The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. The M110 155mm Projectile is an artillery shell used by the U.S. Army and U.S. Marine Corps. Adjustments were usually made by observing and plotting the fall (splashes) of the shells fired and reporting by how much they were left or right in azimuth or over or under in range.FM 4-15, Ch. 4 ===Factors influencing corrections=== Corrections could be made for the following factors: # Variations in muzzle velocity (including the results of variations in temperature of powder) # Variations in atmospheric density # Variations in atmospheric temperature # Height of site (taking account of the level of the tide) # Variations in weight of projectile # Travel of the target during the time of the projectile's flight # Wind # Rotation of the earth (for long range guns) # Drift The uncorrected firing data, to which such corrections were applied, were those derived, for instance, from using a plotting board to track the position of an observed target (e.g., a ship) and the range and azimuth to that target from the guns of a battery. ===Implementing corrections=== ====Meteorological data==== Several of the common corrections depended on meteorological data. The second solution is the useful one for determining the range of the projectile. (Equation 1) \begin{array}{lcr} R_{TP}&=& R_{T} + \frac{dR_{T}}{dt} \cdot t_{TOF'}\\\ &=& R_{T} + \frac{dR_{T}}{dt} \cdot \left( t_{TOF}+t_{Delay}\right) \end{array} where :* R_{TP}\, is the range to the target at the time of projectile impact. The projectile's advantages in terms of speed and rate of fire make ranging shots possible. The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. For example, a torpedo's time of flight is much longer than that of battleship's main gun projectile. The exact prediction of the target range at the time of projectile impact is difficult because it requires knowing the projectile time of flight, which is a function of the projected target position. Special devices, like the "deflection board" (for corrections in azimuth) or the "range correction board" (for corrections in range) were used to produce corrected firing data (described below).Coast Artillery Journal index at sill- www.army.mil The final stage (the red "3" in the diagram at right) had to do with using feedback from the battery's observers, who spotted the fall of the projectiles (over or under range, left or right in azimuth, or on target) and telephoned their data to the plotting room so that the aim of the guns could be corrected for future salvos."Seacoast Artillery Firing," Coast Artillery Journal, Vol. 63, No. 4, October, 1925, pp. 375–391 ==Fire control timing== thumb|left|This example shows the relationship of steps in the fire control process playing out over time. The projectile in this case would have had a time of flight of ~40 seconds (based on the 16 inch guns of the Iowa class). During World War I the Germans created an exceptionally large cannon, the Paris Gun, which could fire a shell more than 80 miles (130 km).

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Two double stars of the same mass as the sun rotate about their common center of mass. Their separation is 4 light years. What is their period of revolution?

The four stars are each about half the mass of the Sun and are approximately 500 million years old. The system is unusual in how closely the four stars are orbiting each other; one pair has an orbital separation of at most .04 astronomical units and an orbital period of about two days, the other pair has a separation of at most .26 astronomical units and a period of about 55 days, and the two pairs are separated by 5.8 AU and have an orbital period of less than nine years. Kepler-84 is a Sun-like star 4,700 light-years from the Sun. The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. For celestial objects in general, the orbital period is determined by a 360° revolution of one body around its primary, e.g. Earth around the Sun. Periods in astronomy are expressed in units of time, usually hours, days, or years. ==Small body orbiting a central body== thumb|upright=1.2|The semi-major axis (a) and semi-minor axis (b) of an ellipse According to Kepler's Third Law, the orbital period T of two point masses orbiting each other in a circular or elliptic orbit is: :T = 2\pi\sqrt{\frac{a^3}{GM}} where: * a is the orbit's semi-major axis * G is the gravitational constant, * M is the mass of the more massive body. Kepler-444 (or KOI-3158, KIC 6278762, 2MASS J19190052+4138043, BD+41°3306) is a triple star system, estimated to be 11.2 billion years old (more than 80% of the age of the universe), approximately away from Earth in the constellation Lyra. At that conference, the star was known as KOI-3158. ==Characteristics== The star, Kepler-444, is approximately 11.2 billion years old, whereas the Sun is only 4.6 billion years old. The star is believed to have 2 M dwarfs in orbit around it with > the fainter companion 1.8 arc-seconds from the main star. ==Stellar system== The Kepler-444 system consists of the planet hosting primary and a pair of M-dwarf stars. Gliese 623 is a dim double star 25.6 light years from Earth in the constellation Hercules. BD−22 5866 is a quadruple-star system located 166 light years from Earth. The age is that of Kepler-444 A, an orange main sequence star of spectral type K0. Another (which is a background star with a probability 0.5%) is a yellow star of mass 0.855 on projected separations of 0.18″ or 0.26″ (213.6 AU). ==Planetary system== Kepler-84 is orbited by five known planets, four small gas giants and a Super- Earth. The rotation period of a celestial object (e.g., star, gas giant, planet, moon, asteroid) may refer to its sidereal rotation period, i.e. the time that the object takes to complete a single revolution around its axis of rotation relative to the background stars, measured in sidereal time. This period differs from the sidereal period because both the orbital plane of the object and the plane of the ecliptic precess with respect to the fixed stars, so their intersection, the line of nodes, also precesses with respect to the fixed stars. Most are investigated by detailed complex astronomical theories using celestial mechanics using precise positional observations of celestial objects via astrometry. ===Synodic period=== One of the observable characteristics of two bodies which orbit a third body in different orbits, and thus have different orbital periods, is their synodic period, which is the time between conjunctions. Kepler-444 is the > densest star with detected solar-like oscillations. There are many periods related to the orbits of objects, each of which are often used in the various fields of astronomy and astrophysics, particularly they must not be confused with other revolving periods like rotational periods. For example, Hyperion, a moon of Saturn, exhibits this behaviour, and its rotation period is described as chaotic. ==Rotation period of selected objects== Celestial objects Rotation period with respect to distant stars, the sidereal rotation period (compared to Earth's mean Solar days) Rotation period with respect to distant stars, the sidereal rotation period (compared to Earth's mean Solar days) Synodic rotation period (mean Solar day) Apparent rotational period viewed from Earth Sun* 25.379995 days (Carrington rotation) 35 days (high latitude) 25d 9h 7m 11.6s 35d ~28 days (equatorial) Mercury 58.6462 days 58d 15h 30m 30s 176 days Venus −243.0226 daysThis rotation is negative because the pole which points north of the invariable plane rotates in the opposite direction to most other planets. −243d 0h 33m −116.75 days Earth 0.99726968 daysReference adds about 1 ms to Earth's stellar day given in mean solar time to account for the length of Earth's mean solar day in excess of 86400 SI seconds. 0d 23h 56m 4.0910s 1.00 days (24h 00m 00s) Moon 27.321661 days (equal to sidereal orbital period due to spin-orbit locking, a sidereal lunar month) 27d 7h 43m 11.5s 29.530588 days (equal to synodic orbital period, due to spin-orbit locking, a synodic lunar month) none (due to spin-orbit locking) Mars 1.02595675 days 1d 0h 37m 22.663s 1.02749125 days Ceres 0.37809 days 0d 9h 4m 27.0s 0.37818 days Jupiter 0.41354 days(average) 0.4135344 days (deep interiorRotation period of the deep interior is that of the planet's magnetic field.) 0.41007 days (equatorial) 0.4136994 days (high latitude) 0d 9h 55m 30s 0d 9h 55m 29.37s 0d 9h 50m 30s 0d 9h 55m 43.63s (9 h 55 m 33 s) (average) Saturn days (average, deep interiorFound through examination of Saturn's C Ring) 0.44401 days (deep interior) 0.4264 days (equatorial) 0.44335 days (high latitude) 0d 10h 39m 22.4s 0d 10h 13m 59s 0d 10h 38m 25.4s (10 h 32 m 36 s) Uranus −0.71833 days −0d 17h 14m 24s (−17 h 14 m 23 s) Neptune 0.67125 days 0d 16h 6m 36s (16 h 6 m 36 s) Pluto −6.38718 days (synchronous with Charon) –6d 9h 17m 32s (–6d 9h 17m 0s) Haumea 0.1631458 ±0.0000042 days 0d 3h 56m 43.80 ±0.36s 0.1631461 ±0.0000042 days Makemake 0.9511083 ±0.0000042 days 22h 49m 35.76 ±0.36s 0.9511164 ±0.0000042 days Eris ~1.08 days 25h ~54m ~1.08 days * See Solar rotation for more detail. == See also == * Apparent retrograde motion * List of slow rotators (minor planets) * List of fast rotators (minor planets) * Retrograde motion * Rotational speed * Synodic day ==References== ==External links== * Note, the rotation periods for Mercury and Earth in this work may be inaccurate. We use asteroseismology > to directly measure a precise age of 11.2+/-1.0 Gyr for the host star, > indicating that Kepler-444 formed when the Universe was less than 20% of its > current age and making it the oldest known system of terrestrial-size > planets. AM Canum Venaticorum 17.146 minutes Beta Lyrae AB 12.9075 days Alpha Centauri AB 79.91 years Proxima Centauri – Alpha Centauri AB 500,000 years or more ==See also== * Geosynchronous orbit derivation * Rotation period – time that it takes to complete one revolution around its axis of rotation * Satellite revisit period * Sidereal time * Sidereal year * Opposition (astronomy) * List of periodic comets ==Notes== ==Bibliography== * == External links == Category:Time in astronomy Period Category:Kinematic properties Period Category:Time in astronomy The age of Kepler-444 not only > suggests that thick-disk stars were among the hosts to the first Galactic > planets, but may also help to pinpoint the beginning of the era of planet > formation."

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To perform a rescue, a lunar landing craft needs to hover just above the surface of the moon, which has a gravitational acceleration of $g / 6$. The exhaust velocity is $2000 \mathrm{~m} / \mathrm{s}$, but fuel amounting to only 20 percent of the total mass may be used. How long can the landing craft hover?

The acceleration due to gravity on the surface of the Moon is approximately 1.625 m/s2, about 16.6% that on Earth's surface or 0.166 . The lunar GM = 4902.8001 km3/s2 from GRAIL analyses. Instead, after it arrested its velocity at an altitude of 3.4m it simply fell to the lunar surface. Since rocketry is used for descent and landing, the Moon's gravity necessitates the use of more fuel than is needed for asteroid landing. thumb|Maglev hover car A hover car is a personal vehicle that flies at a constant altitude of up to a few meters (yards) above the ground and used for personal transportation in the same way a modern automobile is employed. The lunar GM is 1/81.30057 of the Earth's GM. == Theory == For the lunar gravity field, it is conventional to use an equatorial radius of R = 1738.0 km. Orbital speed around the Moon can, depending on altitude, exceed 1500 m/s. A lunar lander or Moon lander is a spacecraft designed to land on the surface of the Moon. thumb| Project Horizon Lunar Landing-and-Return Vehicle. In comparison, the much lighter (292 kg) Surveyor 3 landed on the Moon in 1967 using nearly 700 kg of fuel. All lunar landers require rocket engines for descent. The relatively high gravity (higher than all known asteroids, but lower than all solar system planets) and lack of lunar atmosphere negates the use of aerobraking, so a lander must use propulsion to decelerate and achieve a soft landing. ==History== The Luna program was a series of robotic impactors, flybys, orbiters, and landers flown by the Soviet Union between 1958 and 1976. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. The idea is that engine exhaust and lunar regolith can cause problems if they were to be kicked back from the surface to the spacecraft, and thus the engines cut off just before touchdown. Over the entire surface, the variation in gravitational acceleration is about 0.0253 m/s2 (1.6% of the acceleration due to gravity). This was in lieu of a 12 million- pound thrust superbooster required for a direct-ascent lunar flight, which could not possibly be developed in time for the 1966 deployment target. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. For example, the 900-kg Curiosity rover was landed on Mars by a craft having a mass (at the time of Mars atmospheric entry) of 2400 kg, of which only 390 kg was fuel. The landing gear was designed to withstand landings with engine cut-out at up to of height, though it was intended for descent engine shutdown to commence when one of the probes touched the surface. The design requirements for these landers depend on factors imposed by the payload, flight rate, propulsive requirements, and configuration constraints.Lunar Lander Stage Requirements Based on the Civil Needs Data Base (PDF). During this period the rockets would transport some 220 tonnes of useful cargo to the Moon. Higher speeds can be attained if the skydiver pulls in their limbs (see also freeflying).

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In an elastic collision of two particles with masses $m_1$ and $m_2$, the initial velocities are $\mathbf{u}_1$ and $\mathbf{u}_2=\alpha \mathbf{u}_1$. If the initial kinetic energies of the two particles are equal, find the conditions on $u_1 / u_2$ such that $m_1$ is at rest after the collision and $\alpha$ is positive.

The conservation of the total momentum before and after the collision is expressed by: m_{1}u_{1}+m_{2}u_{2} \ =\ m_{1}v_{1} + m_{2}v_{2}. Consider particles 1 and 2 with masses m1, m2, and velocities u1, u2 before collision, v1, v2 after collision. In the center of momentum frame where the total momentum equals zero, \begin{align} p_1 &= - p_2 \\\ p_1^2 &= p_2^2 \\\ E &= \sqrt {m_1^2c^4 + p_1^2c^2} + \sqrt {m_2^2c^4 + p_2^2c^2} = E \\\ p_1 &= \pm \frac{\sqrt{E^4 - 2E^2m_1^2c^4 - 2E^2m_2^2c^4 + m_1^4c^8 - 2m_1^2m_2^2c^8 + m_2^4c^8}}{2cE} \\\ u_1 &= -v_1. \end{align} Here m_1, m_2 represent the rest masses of the two colliding bodies, u_1, u_2 represent their velocities before collision, v_1, v_2 their velocities after collision, p_1, p_2 their momenta, c is the speed of light in vacuum, and E denotes the total energy, the sum of rest masses and kinetic energies of the two bodies. Indeed, to derive the equations, one may first change the frame of reference so that one of the known velocities is zero, determine the unknown velocities in the new frame of reference, and convert back to the original frame of reference. ====Examples==== ;Before collision: :Ball 1: mass = 3 kg, velocity = 4 m/s :Ball 2: mass = 5 kg, velocity = −6 m/s ;After collision: :Ball 1: velocity = −8.5 m/s :Ball 2: velocity = 1.5 m/s Another situation: frame|center|Elastic collision of unequal masses. However, in the case of the proton, the evidence suggested three distinct concentrations of charge (quarks) and not one. ==Formula== The formula for the velocities after a one-dimensional collision is: \begin{align} v_a &= \frac{C_R m_b (u_b - u_a) + m_a u_a + m_b u_b} {m_a+m_b} \\\ v_b &= \frac{C_R m_a (u_a - u_b) + m_a u_a + m_b u_b} {m_a+m_b} \end{align} where *va is the final velocity of the first object after impact *vb is the final velocity of the second object after impact *ua is the initial velocity of the first object before impact *ub is the initial velocity of the second object before impact *ma is the mass of the first object *mb is the mass of the second object *CR is the coefficient of restitution; if it is 1 we have an elastic collision; if it is 0 we have a perfectly inelastic collision, see below. Assuming that the second particle is at rest before the collision, the angles of deflection of the two particles, \theta_1 and \theta_2, are related to the angle of deflection \theta in the system of the center of mass by \tan \theta_1=\frac{m_2 \sin \theta}{m_1+m_2 \cos \theta},\qquad \theta_2=\frac{{\pi}-{\theta}}{2}. In physics, an elastic collision is an encounter (collision) between two bodies in which the total kinetic energy of the two bodies remains the same. To see this, consider the center of mass at time t before collision and time t' after collision: \begin{align} \bar{x}(t) &= \frac{m_{1} x_{1}(t)+m_{2} x_{2}(t)}{m_{1}+m_{2}} \\\ \bar{x}(t') &= \frac{m_{1} x_{1}(t')+m_{2} x_{2}(t')}{m_{1}+m_{2}}. \end{align} Hence, the velocities of the center of mass before and after collision are: \begin{align} v_{ \bar{x} } &= \frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}} \\\ v_{ \bar{x} }' &= \frac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}. \end{align} The numerators of v_{ \bar{x} } and v_{ \bar{x} }' are the total momenta before and after collision. These equations may be solved directly to find v_1,v_2 when u_1,u_2 are known: \begin{array}{ccc} v_1 &=& \dfrac{m_1-m_2}{m_1+m_2} u_1 + \dfrac{2m_2}{m_1+m_2} u_2 \\\\[.5em] v_2 &=& \dfrac{2m_1}{m_1+m_2} u_1 + \dfrac{m_2-m_1}{m_1+m_2} u_2. \end{array} If both masses are the same, we have a trivial solution: \begin{align} v_{1} &= u_{2} \\\ v_{2} &= u_{1}. \end{align} This simply corresponds to the bodies exchanging their initial velocities to each other. In this frame most of the kinetic energy before the collision is that of the particle with the smaller mass. A useful special case of elastic collision is when the two bodies have equal mass, in which case they will simply exchange their momenta. In this example, momentum of the system is conserved because there is no friction between the sliding bodies and the surface. m_a u_a + m_b u_b = \left( m_a + m_b \right) v where v is the final velocity, which is hence given by v=\frac{m_a u_a + m_b u_b}{m_a + m_b} Another perfectly inelastic collision|frame|center The reduction of total kinetic energy is equal to the total kinetic energy before the collision in a center of momentum frame with respect to the system of two particles, because in such a frame the kinetic energy after the collision is zero. Assuming no friction, this gives the velocity updates: \begin{align} \Delta \vec{v_{a}} &= \frac{J_{n}}{m_{a}} \vec{n} \\\ \Delta \vec{v_{b}} &= -\frac{J_{n}}{m_{b}} \vec{n} \end{align} ==Perfectly inelastic collision== A completely inelastic collision between equal masses|frame|center A perfectly inelastic collision occurs when the maximum amount of kinetic energy of a system is lost. The final velocities can then be calculated from the two new component velocities and will depend on the point of collision. The reduction of kinetic energy E_r is hence: E_r = \frac{1}{2}\frac{m_a m_b}{m_a + m_b}|u_a - u_b|^2 With time reversed we have the situation of two objects pushed away from each other, e.g. shooting a projectile, or a rocket applying thrust (compare the derivation of the Tsiolkovsky rocket equation). ==Partially inelastic collisions== Partially inelastic collisions are the most common form of collisions in the real world. The magnitudes of the velocities of the particles after the collision are: \begin{align} v'_1 &= v_1\frac{\sqrt{m_1^2+m_2^2+2m_1m_2\cos \theta}}{m_1+m_2} \\\ v'_2 &= v_1\frac{2m_1}{m_1+m_2}\sin \frac{\theta}{2}. \end{align} ===Two-dimensional collision with two moving objects=== The final x and y velocities components of the first ball can be calculated as: \begin{align} v'_{1x} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\cos(\varphi)+v_{1}\sin(\theta_1-\varphi)\cos(\varphi + \tfrac{\pi}{2}) \\\\[0.8em] v'_{1y} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\sin(\varphi)+v_{1}\sin(\theta_1-\varphi)\sin(\varphi + \tfrac{\pi}{2}), \end{align} where and are the scalar sizes of the two original speeds of the objects, and are their masses, and are their movement angles, that is, v_{1x} = v_1\cos\theta_1,\; v_{1y}=v_1\sin\theta_1 (meaning moving directly down to the right is either a −45° angle, or a 315° angle), and lowercase phi () is the contact angle. Once v_1 is determined, v_2 can be found by symmetry. ====Center of mass frame==== With respect to the center of mass, both velocities are reversed by the collision: a heavy particle moves slowly toward the center of mass, and bounces back with the same low speed, and a light particle moves fast toward the center of mass, and bounces back with the same high speed. This is why a neutron moderator (a medium which slows down fast neutrons, thereby turning them into thermal neutrons capable of sustaining a chain reaction) is a material full of atoms with light nuclei which do not easily absorb neutrons: the lightest nuclei have about the same mass as a neutron. ====Derivation of solution==== To derive the above equations for v_1,v_2, rearrange the kinetic energy and momentum equations: \begin{align} m_1(v_1^2-u_1^2) &= m_2(u_2^2-v_2^2) \\\ m_1(v_1-u_1) &= m_2(u_2-v_2) \end{align} Dividing each side of the top equation by each side of the bottom equation, and using \tfrac{a^2-b^2}{(a-b)} = a+b, gives: v_1+u_1=u_2+v_2 \quad\Rightarrow\quad v_1-v_2 = u_2-u_1. In an elastic collision these magnitudes do not change. In such a collision, kinetic energy is lost by bonding the two bodies together. An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction. When considering energies, possible rotational energy before and/or after a collision may also play a role. ==Equations== ===One-dimensional Newtonian=== In an elastic collision, both momentum and kinetic energy are conserved.

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Astronaut Stumblebum wanders too far away from the space shuttle orbiter while repairing a broken communications satellite. Stumblebum realizes that the orbiter is moving away from him at $3 \mathrm{~m} / \mathrm{s}$. Stumblebum and his maneuvering unit have a mass of $100 \mathrm{~kg}$, including a pressurized tank of mass $10 \mathrm{~kg}$. The tank includes only $2 \mathrm{~kg}$ of gas that is used to propel him in space. The gas escapes with a constant velocity of $100 \mathrm{~m} / \mathrm{s}$. With what velocity will Stumblebum have to throw the empty tank away to reach the orbiter?

Escape velocity is the minimum speed an object without propulsion needs to have to move away indefinitely from the source of the gravity field. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes v = \sqrt\frac{2mg}{\rho A C_d} \tanh \left(t \sqrt{\frac{g \rho A C_d }{2m}}\right). Using the formula for terminal velocity V_t = \sqrt\frac{2mg}{\rho A C_d} the equation can be rewritten as v = V_t \tanh \left(t \frac{g}{V_t}\right). Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. The formula for terminal velocity (V)] appears on p. [52], equation (127). "Escape Velocity" is the fourth episode in the fourth season of the science fiction television series Battlestar Galactica. Substitution of equations (–) in equation () and solving for terminal velocity, V_t to yield the following expression In equation (), it is assumed that the object is denser than the fluid. So instead of m use the reduced mass m_r = m-\rho V in this and subsequent formulas. Dividing both sides by m gives \frac{\mathrm{d}v}{\mathrm{d}t} = g \left( 1 - \alpha^2 v^2 \right). The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0. Escape Velocity may also refer to: ==Books== * Escape Velocity (Doctor Who), a Doctor Who novel * Escape Velocity: Cyberculture at the End of the Century, a nonfiction book by Mark Dery * Escape Velocity, prequel to the Warlock series by Christopher Stasheff ==Video games== * Escape Velocity (video game) * Escape Velocity Override, its sequel * Escape Velocity Nova, the most recent title in the Escape Velocity franchise, along with an expandable card-driven board game based on it ==Music== * "Escape Velocity" (song), a 2010 song by The Chemical Brothers * Escape Velocity, an album by The Phenomenauts ==Film and television== * "Escape Velocity" (Battlestar Galactica), an episode of the TV show Battlestar Galactica * Escape Velocity (film), a 1998 Canadian thriller film ==See also== thumb|upright=1.3|A flight envelope diagram showing VS (Stall speed at 1G), VC (Corner/Maneuvering speed) and VD (Dive speed) thumb|upright=1.3|Vg diagram. Note the 1g stall speed, and the Maneuvering Speed (Corner Speed) for both positive and negative g. After a lengthy investigation, Velocity found and solved the cause of these stalls. An object with a large projected area relative to its mass, such as a parachute, has a lower terminal velocity than one with a small projected area relative to its mass, such as a dart. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. Note that this is a different concept than design maneuvering speed. If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below).

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Use the $D_0$ value of $\mathrm{H}_2(4.478 \mathrm{eV})$ and the $D_0$ value of $\mathrm{H}_2^{+}(2.651 \mathrm{eV})$ to calculate the first ionization energy of $\mathrm{H}_2$ (that is, the energy needed to remove an electron from $\mathrm{H}_2$ ).

The second way of calculating ionization energies is mainly used at the lowest level of approximation, where the ionization energy is provided by Koopmans' theorem, which involves the highest occupied molecular orbital or "HOMO" and the lowest unoccupied molecular orbital or "LUMO", and states that the ionization energy of an atom or molecule is equal to the negative value of energy of the orbital from which the electron is ejected. There are two main ways in which ionization energy is calculated. For hydrogen in the ground state Z=1 and n=1 so that the energy of the atom before ionization is simply E = - 13.6\ \mathrm{eV} After ionization, the energy is zero for a motionless electron infinitely far from the proton, so that the ionization energy is : I = E(\mathrm{H}^+) - E(\mathrm{H}) = +13.6\ \mathrm{eV}. The first ionization energy is quantitatively expressed as :X(g) + energy ⟶ X+(g) + e− where X is any atom or molecule, X+ is the resultant ion when the original atom was stripped of a single electron, and e− is the removed electron. In physics, ionization energy is usually expressed in electronvolts (eV) or joules (J). For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). The ionization energy will be the energy of photons hνi (h is the Planck constant) that caused a steep rise in the current: Ei = hνi. The 2s electrons then shield the 2p electron from the nucleus to some extent, and it is easier to remove the 2p electron from boron than to remove a 2s electron from beryllium, resulting in a lower ionization energy for B. In physics and chemistry, ionization energy (IE) (American English spelling), ionisation energy (British English spelling) is the minimum energy required to remove the most loosely bound electron of an isolated gaseous atom, positive ion, or molecule. To convert from "value of ionization energy" to the corresponding "value of molar ionization energy", the conversion is: *:1 eV = 96.48534 kJ/mol *:1 kJ/mol = 0.0103642688 eV == References == === WEL (Webelements) === As quoted at http://www.webelements.com/ from these sources: * J.E. Huheey, E.A. Keiter, and R.L. Keiter in Inorganic Chemistry : Principles of Structure and Reactivity, 4th edition, HarperCollins, New York, USA, 1993. As can be seen in the above graph for ionization energies, the sharp rise in IE values from (: 3.89 eV) to (: 5.21 eV) is followed by a small increase (with some fluctuations) as the f-block proceeds from to . The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion. Since the ionization cross section depends on the chemical nature of the sample and the energy of ionizing electrons a standard value of 70 eV is used. That 2p electron is much closer to the nucleus than the 3s electrons removed previously. thumb|350x350px|Ionization energies peak in noble gases at the end of each period in the periodic table of elements and, as a rule, dip when a new shell is starting to fill. The ionization energy is the lowest binding energy for a particular atom (although these are not all shown in the graph). ===Solid surfaces: work function=== Work function is the minimum amount of energy required to remove an electron from a solid surface, where the work function for a given surface is defined by the difference :W = -e\phi - E_{\rm F}, where is the charge of an electron, is the electrostatic potential in the vacuum nearby the surface, and is the Fermi level (electrochemical potential of electrons) inside the material. ==Note== ==See also== * Rydberg equation, a calculation that could determine the ionization energies of hydrogen and hydrogen-like elements. * Electron pairing energies: Half-filled subshells usually result in higher ionization energies. When the next ionization energy involves removing an electron from the same electron shell, the increase in ionization energy is primarily due to the increased net charge of the ion from which the electron is being removed. However, due to experimental limitations, the adiabatic ionization energy is often difficult to determine, whereas the vertical detachment energy is easily identifiable and measurable. ==Analogs of ionization energy to other systems== While the term ionization energy is largely used only for gas-phase atomic, cationic, or molecular species, there are a number of analogous quantities that consider the amount of energy required to remove an electron from other physical systems. ===Electron binding energy=== thumb|500px|Binding energies of specific atomic orbitals as a function of the atomic number. == Numerical values == For each atom, the column marked 1 is the first ionization energy to ionize the neutral atom, the column marked 2 is the second ionization energy to remove a second electron from the +1 ion, the column marked 3 is the third ionization energy to remove a third electron from the +2 ion, and so on. "use" and "WEL" give ionization energy in the unit kJ/mol; "CRC" gives atomic ionization energy in the unit eV. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 H hydrogen use 1312.0 WEL 1312.0 CRC 13.59844 2 He helium use 2372.3 5250.5 WEL 2372.3 5250.5 CRC 24.58738 54.41776 3 Li lithium use 520.2 7298.1 11815.0 WEL 520.2 7298.1 11815.0 CRC 5.39171 75.64009 122.45435 4 Be beryllium use 899.5 1757.1 14848.7 21006.6 WEL 899.5 1757.1 14848.7 21006.6 CRC 9.32269 18.21115 153.89620 217.71858 5 B boron use 800.6 2427.1 3659.7 25025.8 32826.7 WEL 800.6 2427.1 3659.7 25025.8 32826.7 CRC 8.29803 25.15484 37.93064 259.37521 340.22580 6 C carbon use 1086.5 2352.6 4620.5 6222.7 37831 47277.0 WEL 1086.5 2352.6 4620.5 6222.7 37831 47277.0 CRC 11.26030 24.38332 47.8878 64.4939 392.087 489.99334 7 N nitrogen use 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 WEL 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 CRC 14.53414 29.6013 47.44924 77.4735 97.8902 552.0718 667.046 8 O oxygen use 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 WEL 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 CRC 13.61806 35.11730 54.9355 77.41353 113.8990 138.1197 739.29 871.4101 9 F fluorine use 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 WEL 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 CRC 17.42282 34.97082 62.7084 87.1398 114.2428 157.1651 185.186 953.9112 1103.1176 10 Ne neon use 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 WEL 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 CRC 21.5646 40.96328 63.45 97.12 126.21 157.93 207.2759 239.0989 1195.8286 1362.1995 11 Na sodium use 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 159076 WEL 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 CRC 5.13908 47.2864 71.6200 98.91 138.40 172.18 208.50 264.25 299.864 1465.121 1648.702 12 Mg magnesium use 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 169988 189368 WEL 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 CRC 7.64624 15.03528 80.1437 109.2655 141.27 186.76 225.02 265.96 328.06 367.50 1761.805 1962.6650 13 Al aluminium use 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 42647 201266 222316 WEL 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 CRC 5.98577 18.82856 28.44765 119.992 153.825 190.49 241.76 284.66 330.13 398.75 442.00 2085.98 2304.1410 14 Si silicon use 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 45962 50502 235196 257923 WEL 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 CRC 8.15169 16.34585 33.49302 45.14181 166.767 205.27 246.5 303.54 351.12 401.37 476.36 523.42 2437.63 2673.182 15 P phosphorus use 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 46261 54110 59024 271791 296195 WEL 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 CRC 10.48669 19.7694 30.2027 51.4439 65.0251 220.421 263.57 309.60 372.13 424.4 479.46 560.8 611.74 2816.91 3069.842 16 S sulfur use 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 48710 54460 62930 68216 311048 337138 WEL 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 CRC 10.36001 23.3379 34.79 47.222 72.5945 88.0530 280.948 328.75 379.55 447.5 504.8 564.44 652.2 707.01 3223.78 3494.1892 17 Cl chlorine use 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 51068 57119 63363 72341 78095 352994 380760 WEL 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 CRC 12.96764 23.814 39.61 53.4652 67.8 97.03 114.1958 348.28 400.06 455.63 529.28 591.99 656.71 749.76 809.40 3658.521 3946.2960 18 Ar argon use 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 52002 59653 66199 72918 82473 88576 397605 427066 WEL 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 CRC 15.75962 27.62967 40.74 59.81 75.02 91.009 124.323 143.460 422.45 478.69 538.96 618.26 686.10 755.74 854.77 918.03 4120.8857 4426.2296 19 K potassium use 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 54490 60730 68950 75900 83080 93400 99710 444880 476063 WEL 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 CRC 4.34066 31.63 45.806 60.91 82.66 99.4 117.56 154.88 175.8174 503.8 564.7 629.4 714.6 786.6 861.1 968 1033.4 4610.8 4934.046 20 Ca calcium use 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 63410 70110 78890 86310 94000 104900 111711 494850 527762 WEL 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 CRC 6.11316 11.87172 50.9131 67.27 84.50 108.78 127.2 147.24 188.54 211.275 591.9 657.2 726.6 817.6 894.5 974 1087 1157.8 5128.8 5469.864 21 Sc scandium use 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 WEL 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 CRC 6.5615 12.79967 24.75666 73.4894 91.65 110.68 138.0 158.1 180.03 225.18 249.798 687.36 756.7 830.8 927.5 1009 1094 1213 1287.97 5674.8 6033.712 22 Ti titanium use 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 639294 WEL 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 CRC 6.8281 13.5755 27.4917 43.2672 99.30 119.53 140.8 170.4 192.1 215.92 265.07 291.500 787.84 863.1 941.9 1044 1131 1221 1346 1425.4 6249.0 6625.82 23 V vanadium use 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 661050 699144 WEL 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 CRC 6.7462 14.66 29.311 46.709 65.2817 128.13 150.6 173.4 205.8 230.5 255.7 308.1 336.277 896.0 976 1060 1168 1260 1355 1486 1569.6 6851.3 7246.12 24 Cr chromium use 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 166090 721870 761733 WEL 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 CRC 6.7665 16.4857 30.96 49.16 69.46 90.6349 160.18 184.7 209.3 244.4 270.8 298.0 354.8 384.168 1010.6 1097 1185 1299 1396 1496 1634 1721.4 7481.7 7894.81 25 Mn manganese use 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 172500 181380 785450 827067 WEL 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 CRC 7.43402 15.63999 33.668 51.2 72.4 95.6 119.203 194.5 221.8 248.3 286.0 314.4 343.6 403.0 435.163 1134.7 1224 1317 1437 1539 1644 1788 1879.9 8140.6 8571.94 26 Fe iron use 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 173600 188100 195200 851800 895161 WEL 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 CRC 7.9024 16.1878 30.652 54.8 75.0 99.1 124.98 151.06 233.6 262.1 290.2 330.8 361.0 392.2 457 489.256 1266 1358 1456 1582 1689 1799 1950 2023 8828 9277.69 27 Co cobalt use 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 178100 189300 204500 214100 920870 966023 WEL 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 CRC 7.8810 17.083 33.50 51.3 79.5 102.0 128.9 157.8 186.13 275.4 305 336 379 411 444 511.96 546.58 1397.2 1504.6 1603 1735 1846 1962 2119 2219.0 9544.1 10012.12 28 Ni nickel use 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 182700 194000 205600 221400 231490 992718 1039668 WEL 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 CRC 7.6398 18.16884 35.19 54.9 76.06 108 133 162 193 224.6 321.0 352 384 430 464 499 571.08 607.06 1541 1648 1756 1894 2011 2131 2295 2399.2 10288.8 10775.40 29 Cu copper use 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 184900 198800 210500 222700 239100 249660 1067358 1116105 WEL 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 CRC 7.72638 20.29240 36.841 57.38 79.8 103 139 166 199 232 265.3 369 401 435 484 520 557 633 670.588 1697 1804 1916 2060 2182 2308 2478 2587.5 11062.38 11567.617 30 Zn zinc use 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 WEL 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 CRC 9.3942 17.96440 39.723 59.4 82.6 108 134 174 203 238 274 310.8 419.7 454 490 542 579 619 698 738 1856 31 Ga gallium use 578.8 1979.3 2963 6180 WEL 578.8 1979.3 2963 6180 CRC 5.99930 20.5142 30.71 64 32 Ge germanium use 762 1537.5 3302.1 4411 9020 WEL 762 1537.5 3302.1 4411 9020 CRC 7.8994 15.93462 34.2241 45.7131 93.5 33 As arsenic use 947.0 1798 2735 4837 6043 12310 WEL 947.0 1798 2735 4837 6043 12310 CRC 9.7886 18.633 28.351 50.13 62.63 127.6 34 Se selenium use 941.0 2045 2973.7 4144 6590 7880 14990 WEL 941.0 2045 2973.7 4144 6590 7880 14990 CRC 9.75238 21.19 30.8204 42.9450 68.3 81.7 155.4 35 Br bromine use 1139.9 2103 3470 4560 5760 8550 9940 18600 WEL 1139.9 2103 3470 4560 5760 8550 9940 18600 CRC 11.81381 21.8 36 47.3 59.7 88.6 103.0 192.8 36 Kr krypton use 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 90400 96300 101400 111100 116290 282500 296200 311400 326200 WEL 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 CRC 13.99961 24.35985 36.950 52.5 64.7 78.5 111.0 125.802 230.85 268.2 308 350 391 447 492 541 592 641 786 833 884 937 998 1051 1151 1205.3 2928 3070 3227 3381 37 Rb rubidium use 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 WEL 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 CRC 4.17713 27.285 40 52.6 71.0 84.4 99.2 136 150 277.1 38 Sr strontium use 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 31270 WEL 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 CRC 5.6949 11.03013 42.89 57 71.6 90.8 106 122.3 162 177 324.1 39 Y yttrium use 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 19900 36090 WEL 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 CRC 6.2171 12.24 20.52 60.597 77.0 93.0 116 129 146.2 191 206 374.0 40 Zr zirconium use 640.1 1270 2218 3313 7752 9500 WEL 640.1 1270 2218 3313 7752 9500 CRC 6.63390 13.13 22.99 34.34 80.348 41 Nb niobium use 652.1 1380 2416 3700 4877 9847 12100 WEL 652.1 1380 2416 3700 4877 9847 12100 CRC 6.75885 14.32 25.04 38.3 50.55 102.057 125 42 Mo molybdenum use 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 93400 98420 104400 121900 127700 133800 139800 148100 154500 WEL 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 CRC 7.09243 16.16 27.13 46.4 54.49 68.8276 125.664 143.6 164.12 186.4 209.3 230.28 279.1 302.60 544.0 570 636 702 767 833 902 968 1020 1082 1263 1323 1387 1449 1535 1601 43 Tc technetium use 702 1470 2850 WEL 702 1470 2850 CRC 7.28 15.26 29.54 44 Ru ruthenium use 710.2 1620 2747 WEL 710.2 1620 2747 CRC 7.36050 16.76 28.47 45 Rh rhodium use 719.7 1740 2997 WEL 719.7 1740 2997 CRC 7.45890 18.08 31.06 46 Pd palladium use 804.4 1870 3177 WEL 804.4 1870 3177 CRC 8.3369 19.43 32.93 47 Ag silver use 731.0 2070 3361 WEL 731.0 2070 3361 CRC 7.5762 21.49 34.83 48 Cd cadmium use 867.8 1631.4 3616 WEL 867.8 1631.4 3616 CRC 8.9938 16.90832 37.48 49 In indium use 558.3 1820.7 2704 5210 WEL 558.3 1820.7 2704 5210 CRC 5.78636 18.8698 28.03 54 50 Sn tin use 708.6 1411.8 2943.0 3930.3 7456 WEL 708.6 1411.8 2943.0 3930.3 7456 CRC 7.3439 14.63225 30.50260 40.73502 72.28 51 Sb antimony use 834 1594.9 2440 4260 5400 10400 WEL 834 1594.9 2440 4260 5400 10400 CRC 8.6084 16.53051 25.3 44.2 56 108 52 Te tellurium use 869.3 1790 2698 3610 5668 6820 13200 WEL 869.3 1790 2698 3610 5668 6820 13200 CRC 9.0096 18.6 27.96 37.41 58.75 70.7 137 53 I iodine use 1008.4 1845.9 3180 WEL 1008.4 1845.9 3180 CRC 10.45126 19.1313 33 54 Xe xenon use 1170.4 2046.4 3099.4 WEL 1170.4 2046.4 3099.4 CRC 12.1298 21.20979 32.1230 55 Cs caesium use 375.7 2234.3 3400 WEL 375.7 2234.3 3400 CRC 3.89390 23.15745 56 Ba barium use 502.9 965.2 3600 WEL 502.9 965.2 3600 CRC 5.21170 10.00390 57 La lanthanum use 538.1 1067 1850.3 4819 5940 WEL 538.1 1067 1850.3 4819 5940 CRC 5.5769 11.060 19.1773 49.95 61.6 58 Ce cerium use 534.4 1050 1949 3547 6325 7490 WEL 534.4 1050 1949 3547 6325 7490 CRC 5.5387 10.85 20.198 36.758 65.55 77.6 59 Pr praseodymium use 527 1020 2086 3761 5551 WEL 527 1020 2086 3761 5551 CRC 5.473 10.55 21.624 38.98 57.53 60 Nd neodymium use 533.1 1040 2130 3900 WEL 533.1 1040 2130 3900 CRC 5.5250 10.73 22.1 40.41 61 Pm promethium use 540 1050 2150 3970 WEL 540 1050 2150 3970 CRC 5.582 10.90 22.3 41.1 62 Sm samarium use 544.5 1070 2260 3990 WEL 544.5 1070 2260 3990 CRC 5.6436 11.07 23.4 41.4 63 Eu europium use 547.1 1085 2404 4120 WEL 547.1 1085 2404 4120 CRC 5.6704 11.241 24.92 42.7 64 Gd gadolinium use 593.4 1170 1990 4250 WEL 593.4 1170 1990 4250 CRC 6.1501 12.09 20.63 44.0 65 Tb terbium use 565.8 1110 2114 3839 WEL 565.8 1110 2114 3839 CRC 5.8638 11.52 21.91 39.79 66 Dy dysprosium use 573.0 1130 2200 3990 WEL 573.0 1130 2200 3990 CRC 5.9389 11.67 22.8 41.47 67 Ho holmium use 581.0 1140 2204 4100 WEL 581.0 1140 2204 4100 CRC 6.0215 11.80 22.84 42.5 68 Er erbium use 589.3 1150 2194 4120 WEL 589.3 1150 2194 4120 CRC 6.1077 11.93 22.74 42.7 69 Tm thulium use 596.7 1160 2285 4120 WEL 596.7 1160 2285 4120 CRC 6.18431 12.05 23.68 42.7 70 Yb ytterbium use 603.4 1174.8 2417 4203 WEL 603.4 1174.8 2417 4203 CRC 6.25416 12.1761 25.05 43.56 71 Lu lutetium use 523.5 1340 2022.3 4370 6445 WEL 523.5 1340 2022.3 4370 6445 CRC 5.4259 13.9 20.9594 45.25 66.8 72 Hf hafnium use 658.5 1440 2250 3216 WEL 658.5 1440 2250 3216 CRC 6.82507 14.9 23.3 33.33 73 Ta tantalum use 761 1500 WEL 761 1500 CRC 7.5496 74 W tungsten use 770 1700 WEL 770 1700 CRC 7.8640 75 Re rhenium use 760 1260 2510 3640 WEL 760 1260 2510 3640 CRC 7.8335 76 Os osmium use 840 1600 WEL 840 1600 CRC 8.4382 77 Ir iridium use 880 1600 WEL 880 1600 CRC 8.9670 78 Pt platinum use 870 1791 WEL 870 1791 CRC 8.9587 18.563 28https://dept.astro.lsa.umich.edu/~cowley/ionen.htm archived 79 Au gold use 890.1 1980 WEL 890.1 1980 CRC 9.2255 20.5 30 80 Hg mercury use 1007.1 1810 3300 WEL 1007.1 1810 3300 CRC 10.43750 18.756 34.2 81 Tl thallium use 589.4 1971 2878 WEL 589.4 1971 2878 CRC 6.1082 20.428 29.83 82 Pb lead use 715.6 1450.5 3081.5 4083 6640 WEL 715.6 1450.5 3081.5 4083 6640 CRC 7.41666 15.0322 31.9373 42.32 68.8 83 Bi bismuth use 703 1610 2466 4370 5400 8520 WEL 703 1610 2466 4370 5400 8520 CRC 7.2856 16.69 25.56 45.3 56.0 88.3 84 Po polonium use 812.1 WEL 812.1 CRC 8.417 85 At astatine use 899.003 WEL 920 CRC talk Originally quoted as 9.31751(8) eV. 86 Rn radon use 1037 WEL 1037 CRC 10.74850 87 Fr francium use 380 WEL 380 CRC 4.0727 talk give 4.0712±0.00004 eV (392.811(4) kJ/mol) 88 Ra radium use 509.3 979.0 WEL 509.3 979.0 CRC 5.2784 10.14716 89 Ac actinium use 499 1170 WEL 499 1170 CRC 5.17 12.1 90 Th thorium use 587 1110 1930 2780 WEL 587 1110 1930 2780 CRC 6.3067 11.5 20.0 28.8 91 Pa protactinium use 568 WEL 568 CRC 5.89 92 U uranium use 597.6 1420 WEL 597.6 1420 CRC 6.19405 14.72 93 Np neptunium use 604.5 WEL 604.5 CRC 6.2657 94 Pu plutonium use 584.7 WEL 584.7 CRC 6.0262 95 Am americium use 578 WEL 578 CRC 5.9738 96 Cm curium use 581 WEL 581 CRC 5.9915 97 Bk berkelium use 601 WEL 601 CRC 6.1979 98 Cf californium use 608 WEL 608 CRC 6.2817 99 Es einsteinium use 619 WEL 619 CRC 6.42 100 Fm fermium use 627 WEL 627 CRC 6.50 101 Md mendelevium use 635 WEL 635 CRC 6.58 102 No nobelium use 642 WEL CRC 6.65 103 Lr lawrencium use 470 WEL CRC 4.9 104 Rf rutherfordium use 580 WEL CRC 6.0 == Notes == * Values from CRC are ionization energies given in the unit eV; other values are molar ionization energies given in the unit kJ/mol. The term ionization potential is an older and obsolete term for ionization energy, because the oldest method of measuring ionization energy was based on ionizing a sample and accelerating the electron removed using an electrostatic potential. == Determination of ionization energies == thumb|304x304px|Ionization energy measurement apparatus. |alt= The ionization energy of atoms, denoted Ei, is measured by finding the minimal energy of light quanta (photons) or electrons accelerated to a known energy that will kick out the least bound atomic electrons. This in turn makes its ionization energies increase by 18 kJ/mol−1. The kinetic energy of the bombarding electrons should have higher energy than the ionization energy of the sample molecule.

650000

2.9

15.0

11.58

15.425

E

Calculate the energy of one mole of UV photons of wavelength $300 \mathrm{~nm}$ and compare it with a typical single-bond energy of $400 \mathrm{~kJ} / \mathrm{mol}$.

To find the photon energy in electronvolts using the wavelength in micrometres, the equation is approximately :E\text{ (eV)} = \frac{1.2398}{\lambda\text{ (μm)}} since hc/e=1.239 \; 841 \; 984... \times 10^{-6} eVm where h is Planck's constant, c is the speed of light in m/sec, and e is the electron charge. Additionally, E = \frac{hc}{\lambda} where *E is photon energy *λ is the photon's wavelength *c is the speed of light in vacuum *h is the Planck constant The photon energy at 1 Hz is equal to 6.62607015 × 10−34 J That is equal to 4.135667697 × 10−15 eV === Electronvolt === Energy is often measured in electronvolts. The photon energy of near infrared radiation at 1 μm wavelength is approximately 1.2398 eV. ==Examples== An FM radio station transmitting at 100 MHz emits photons with an energy of about 4.1357 × 10−7 eV. These wavelengths correspond to photon energies of down to . The amount of energy is directly proportional to the photon's electromagnetic frequency and thus, equivalently, is inversely proportional to the wavelength. During photosynthesis, specific chlorophyll molecules absorb red-light photons at a wavelength of 700 nm in the photosystem I, corresponding to an energy of each photon of ≈ 2 eV ≈ 3 × 10−19 J ≈ 75 kBT, where kBT denotes the thermal energy. Photon energy is the energy carried by a single photon. Equivalently, the longer the photon's wavelength, the lower its energy. Very-high-energy gamma rays have photon energies of 100 GeV to over 1 PeV (1011 to 1015 electronvolts) or 16 nanojoules to 160 microjoules. Photon energy can be expressed using any unit of energy. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. The higher the photon's frequency, the higher its energy. Spectral irradiance of wavelengths in the solar spectrum. As one joule equals 6.24 × 1018 eV, the larger units may be more useful in denoting the energy of photons with higher frequency and higher energy, such as gamma rays, as opposed to lower energy photons as in the optical and radio frequency regions of the electromagnetic spectrum. ==Formulas== === Physics === Photon energy is directly proportional to frequency. The wavelengths in the solar spectrum range from approximately 0.3-2.0 μm. A minimum of 48 photons is needed for the synthesis of a single glucose molecule from CO2 and water (chemical potential difference 5 × 10−18 J) with a maximal energy conversion efficiency of 35%. ==See also== *Photon *Electromagnetic radiation *Electromagnetic spectrum *Planck constant *Planck–Einstein relation *Soft photon ==References== Category:Foundational quantum physics Category:Electromagnetic spectrum Category:Photons Ultra-high-energy gamma rays are gamma rays with photon energies higher than 100 TeV (0.1 PeV). UV-B lamps are lamps that emit a spectrum of ultraviolet light with wavelengths ranging from 290–320 nanometers. This minuscule amount of energy is approximately 8 × 10−13 times the electron's mass (via mass-energy equivalence). These tables list values of molar ionization energies, measured in kJ⋅mol−1. This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. Among the units commonly used to denote photon energy are the electronvolt (eV) and the joule (as well as its multiples, such as the microjoule).

0.54

399

2.0

6.0

0.118

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Calculate the magnitude of the spin angular momentum of a proton. Give a numerical answer.

The sum of the proton and neutron magnetic moments gives 0.879 µN, which is within 3% of the measured value 0.857 µN. The result is larger than μ by a factor equal to the ratio of the proton to electron mass, or about a factor of 1836. ==See also== * Nucleon magnetic moment ==References== ==External links== *. The nucleon magnetic moments are the intrinsic magnetic dipole moments of the proton and neutron, symbols μp and μn. The resulting value was not zero and had a sign opposite to that of the proton. In this calculation, the spins of the nucleons are aligned, but their magnetic moments offset because of the neutron's negative magnetic moment. == Nature of the nucleon magnetic moments == thumb|upright|A magnetic dipole moment can be created by either a current loop (top; Ampèrian) or by two magnetic monopoles (bottom; Gilbertian). The key question is how the nucleon's spin, whose magnitude is 1/2ħ, is carried by its constituent partons (quarks and gluons). The g-factor for the proton is 5.6, and the chargeless neutron, which should have no magnetic moment at all, has a g-factor of −3.8. The CODATA recommended value for the magnetic moment of the proton is or The best available measurement for the value of the magnetic moment of the neutron is Here, μN is the nuclear magneton, a standard unit for the magnetic moments of nuclear components, and μB is the Bohr magneton, both being physical constants. Nucleon spin structure describes the partonic structure of nucleon (proton and neutron) intrinsic angular momentum (spin). In SI units, these values are and A magnetic moment is a vector quantity, and the direction of the nucleon's magnetic moment is determined by its spin. The nuclear magneton is the spin magnetic moment of a Dirac particle, a charged, spin-1/2 elementary particle, with a proton's mass p, in which anomalous corrections are ignored. Since for the neutron the sign of γn is negative, the neutron's spin angular momentum precesses counterclockwise about the direction of the external magnetic field. ===Proton nuclear magnetic resonance=== Nuclear magnetic resonance employing the magnetic moments of protons is used for nuclear magnetic resonance (NMR) spectroscopy. The value of nuclear magneton System of units Value Unit SI J·T CGS Erg·G eV eV·T MHz/T (per h) MHz/T The nuclear magneton (symbol μ) is a physical constant of magnetic moment, defined in SI units by: :\mu_\text{N} = {{e \hbar} \over {2 m_\text{p}}} and in Gaussian CGS units by: :\mu_\text{N} = {{e \hbar} \over{2 m_\text{p}c}} where: :e is the elementary charge, :ħ is the reduced Planck constant, :m is the proton rest mass, and :c is the speed of light In SI units, its value is approximately: :μ = In Gaussian CGS units, its value can be given in convenient units as :μ = The nuclear magneton is the natural unit for expressing magnetic dipole moments of heavy particles such as nucleons and atomic nuclei. Baryon Magnetic moment of quark model Computed (\mu_\text{N}) Observed (\mu_\text{N}) p u − d 2.79 2.793 n d − u −1.86 −1.913 The results of this calculation are encouraging, but the masses of the up or down quarks were assumed to be the mass of a nucleon. Thus, in units of nuclear magneton, for the neutron and for the proton. The nuclear magneton is \mu_\text{N} = \frac{e \hbar}{2 m_\text{p}}, where is the elementary charge, and is the reduced Planck constant. Nucleon magnetic moments have been successfully computed from first principles, requiring significant computing resources. ==See also== * Neutron triple-axis spectrometry * LARMOR neutron microscope * Neutron electric dipole moment * Aharonov–Casher effect ==References== ==Bibliography== * S. W. Lovesey (1986). The magnetic moment of such a particle is parallel to its spin. The nuclear magnetic moment also includes contributions from the orbital motion of the charged protons. The calculation assumes that the quarks behave like pointlike Dirac particles, each having their own magnetic moment, as computed using an expression similar to the one above for the nuclear magneton: \ \mu_\text{q} = \frac{\ e_\text{q} \hbar\ }{2 m_\text{q}}\ , where the q-subscripted variables refer to quark magnetic moment, charge, or mass. For a nucleus of which the numbers of protons and of neutrons are both even in its ground state (i.e. lowest energy state), the nuclear spin and magnetic moment are both always zero. The magnetic moment is calculated through j, l and s of the unpaired nucleon, but nuclei are not in states of well defined l and s.

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The ${ }^7 \mathrm{Li}^1 \mathrm{H}$ ground electronic state has $D_0=2.4287 \mathrm{eV}, \nu_e / c=1405.65 \mathrm{~cm}^{-1}$, and $\nu_e x_e / c=23.20 \mathrm{~cm}^{-1}$, where $c$ is the speed of light. (These last two quantities are usually designated $\omega_e$ and $\omega_e x_e$ in the literature.) Calculate $D_e$ for ${ }^7 \mathrm{Li}^1 \mathrm{H}$.

The calculated abundance and ratio of 1H and 4He is in agreement with data from observations of young stars. ===The P-P II branch=== In stars, lithium-7 is made in a proton-proton chain reaction. thumb|Proton–proton II chain reaction :{| border="0" |- style="height:2em;" | ||+ || ||→ || ||+ || |- style="height:2em;" | ||+ || ||→ || ||+ || ||+ || ||/ || |- style="height:2em;" | ||+ || ||→ ||2 |} The P-P II branch is dominant at temperatures of 14 to . thumb|right|400px|Stable nuclides of the first few elements ==Observed abundance of lithium== Despite the low theoretical abundance of lithium, the actual observable amount is less than the calculated amount by a factor of 3–4. thumb|250px|7Li NMR spectrum of LiCl (1M) in D2O. However, they didn't use the Mössbauer effect but made magnetic resonance measurements of the nucleus of lithium-7, whose ground state possesses a spin of . E7, E07, E-7 or E7 may refer to: ==Science and engineering== * E7 liquid crystal mixture * E7, the Lie group in mathematics * E7 polytope, in geometry * E7 papillomavirus protein * E7 European long distance path ==Transport== * EMD E7, a diesel locomotive * European route E07, an international road * Peugeot E7, a hackney cab * PRR E7, a steam locomotive * Carbon Motors E7,a police car * E7 series, a Japanese high-speed train * Nihonkai-Tōhoku Expressway and Akita Expressway (between Kawabe JCT and Kosaka JCT), route E7 in Japan * Cheras–Kajang Expressway, route E7 in Malaysia ==Other uses== * Boeing E-7, either: ** Boeing E-7 ARIA, the original designation assigned by the United States Air Force under the Mission Designation System to the EC-18B Advanced Range Instrumentation Aircraft. The SD7 is a model of 6-axle diesel locomotive built by General Motors Electro-Motive Division between May 1951 and November 1953. The molecular formula C7H7I (molar mass: 218.03 g/mol) may refer to: * Benzyl iodide * Iodotoluene Other isotopes including 2H, 3H, 3He, 6Li, 7Li, and 7Be are much rarer; the estimated abundance of primordial lithium is 10−10 relative to hydrogen. Though it transmutes into two atoms of helium due to collision with a proton at temperatures above 2.4 million degrees Celsius (most stars easily attain this temperature in their interiors), lithium is more abundant than current computations would predict in later-generation stars. The discrepancy is highlighted in a so-called "Schramm plot", named in honor of astrophysicist David Schramm, which depicts these primordial abundances as a function of cosmic baryon content from standard BBN predictions. ==Origin of lithium== Minutes after the Big Bang, the universe was made almost entirely of hydrogen and helium, with trace amounts of lithium and beryllium, and negligibly small abundances of all heavier elements. ===Lithium synthesis in the Big Bang=== Big Bang nucleosynthesis produced both lithium-7 and beryllium-7, and indeed the latter dominates the primordial synthesis of mass 7 nuclides. In summary, accurate measurements of the primordial lithium abundance is the current focus of progress, and it could be possible that the final answer does not lie in astrophysical solutions. The ground state is split into four equally spaced magnetic energy levels when measured in a magnetic field in accordance with its allowed magnetic quantum number. They test the framework of Tsallis non-extensive statistics.Their result suggest that 1.069 === Nuclear physics solutions === When one considers the possibility that the measured primordial lithium abundance is correct and based on the Standard Model of particle physics and the standard cosmology, the lithium problem implies errors in the BBN light element predictions. Namely, the most widely accepted models of the Big Bang suggest that three times as much primordial lithium, in particular lithium-7, should exist. EMD ended production in November 1953 and began producing the SD7's successor, the SD9, in January 1954. == Original buyers == Owner Quantity Numbers Notes Electro-Motive Division 2 990 to Southern Pacific 5308 then 2715 to 1415 ne 1518 Electro-Motive Division 2 991 to Baltimore and Ohio 760 Baltimore and Ohio Railroad 4 761–764 These units were built with the 567BC engine. Since a series of coefficients are tested in those experiments, only the value of maximal sensitivity is given (for precise data, see the individual articles): Author Year SME constraints SME constraints SME constraints Description Author Year Proton Neutron Electron Description Prestage et al. 1985 10−27 Comparing the nuclear spin-flip transition of (stored in a penning trap) with a hydrogen maser transition. EMD SD7 Original Owners. Retrieved on August 27, 2006 * Diesel Era Volume 6 Number 6 November/December 1995, "EMD's SD7" by Paul K. Withers pp 5-20; 47-50. == External links == * Locomotive Truck EMD FlexiCoil C SD07 Category:C-C locomotives Category:Diesel-electric locomotives of the United States Category:Railway locomotives introduced in 1952 Category:Freight locomotives Category:Standard gauge locomotives of the United States Spectroscopic observations of stars in NGC 6397, a metal-poor globular cluster, are consistent with an inverse relation between lithium abundance and age, but a theoretical mechanism for diffusion has not been formalized. EMD produced its first examples of the SD7 in May 1951, using the 567B engine. Furthermore, more observations on lithium depletion remain important since present lithium levels might not reflect the initial abundance in the star. This was the first model in EMD's SD (Special Duty) series of locomotives, a lengthened B-B GP7 with a C-C truck arrangement. == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition.

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The positron has charge $+e$ and mass equal to the electron mass. Calculate in electronvolts the ground-state energy of positronium-an "atom" that consists of a positron and an electron.

While the electron has a negative electric charge, the positron has a positive electric charge, and is produced naturally in certain types of radioactive decay. Thus, for positronium, its reduced mass only differs from the electron by a factor of 2. So finally, the energy levels of positronium are given by E_n = -\frac{1}{2} \frac{m_\mathrm{e} q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2} = \frac{-6.8~\mathrm{eV}}{n^2}. Positronium can also be considered by a particular form of the two-body Dirac equation; Two particles with a Coulomb interaction can be exactly separated in the (relativistic) center-of-momentum frame and the resulting ground-state energy has been obtained very accurately using finite element methods of Janine Shertzer. The electron is commonly symbolized by , and the positron is symbolized by . However, because of the reduced mass, the frequencies of the spectral lines are less than half of those for the corresponding hydrogen lines. ==States== The mass of positronium is 1.022 MeV, which is twice the electron mass minus the binding energy of a few eV. The antiparticle of the electron is called the positron; it is identical to the electron, except that it carries electrical charge of the opposite sign. Approximately: * ~60% of positrons will directly annihilate with an electron without forming positronium. In this approximation, the energy levels are different because of a different effective mass, μ, in the energy equation (see electron energy levels for a derivation): E_n = -\frac{\mu q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2}, where: * is the charge magnitude of the electron (same as the positron), * is the Planck constant, * is the electric constant (otherwise known as the permittivity of free space), * is the reduced mass: \mu = \frac{m_\mathrm{e} m_\mathrm{p}}{m_\mathrm{e} + m_\mathrm{p}} = \frac{m_\mathrm{e}^2}{2m_\mathrm{e}} = \frac{m_\mathrm{e}}{2}, where and are, respectively, the mass of the electron and the positron (which are the same by definition as antiparticles). The lowest energy level of positronium () is . Positronium (Ps) is a system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom, specifically an onium. In 2012, Cassidy et al. were able to produce the excited molecular positronium L = 1 angular momentum state. ==See also== *Hydrogen molecule *Hydrogen molecular ion *Positronium *Protonium *Exotic atom ==References== ==External links== *Molecules of Positronium Observed in the Laboratory for the First Time, press release, University of California, Riverside, September 12, 2007. Upon slowing down in the silica, the positrons captured ordinary electrons to form positronium atoms. The electron's mass is approximately 1/1836 that of the proton. If the electron and positron have negligible momentum, a positronium atom can form before annihilation results in two or three gamma ray photons totalling 1.022 MeV. thumb|350px|Naturally occurring electron-positron annihilation as a result of beta plus decay Electron–positron annihilation occurs when an electron () and a positron (, the electron's antiparticle) collide. The lowest energy orbital state of positronium is 1S, and like with hydrogen, it has a hyperfine structure arising from the relative orientations of the spins of the electron and the positron. The resulting weakly bound electron and positron spiral inwards and eventually annihilate, with an estimated lifetime of years. == See also == * Breit equation * Antiprotonic helium * Di-positronium * Quantum electrodynamics * Protonium * Two-body Dirac equations == References == == External links == * The annihilation of positronium - The Feynman Lectures on Physics * The Search for Positronium * Obituary of Martin Deutsch, discoverer of Positronium Category:Molecular physics Category:Quantum electrodynamics Category:Spintronics Category:Onia Category:Antimatter Category:Substances discovered in the 1950s The electron (symbol e) is on the left. The opposite is also true: the antiparticle of the positron is the electron. For example, the antiparticle of the electron is the positron (also known as an antielectron). Positron paths in a cloud- chamber trace the same helical path as an electron but rotate in the opposite direction with respect to the magnetic field direction due to their having the same magnitude of charge-to-mass ratio but with opposite charge and, therefore, opposite signed charge-to-mass ratios.

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What is the value of the angular-momentum quantum number $l$ for a $t$ orbital?

Here is the total orbital angular momentum quantum number. An atomic electron's angular momentum, L, is related to its quantum number ℓ by the following equation: \mathbf{L}^2\Psi = \hbar^2 \ell(\ell + 1) \Psi, where ħ is the reduced Planck's constant, L2 is the orbital angular momentum operator and \Psi is the wavefunction of the electron. When referring to angular momentum, it is better to simply use the quantum number ℓ. The associated quantum number is the main total angular momentum quantum number j. Each orbital is characterized by its number , where takes integer values from to , and its angular momentum number , where takes integer values from to . It is also known as the orbital angular momentum quantum number, orbital quantum number, subsidiary quantum number, or second quantum number, and is symbolized as (pronounced ell). == Derivation == Connected with the energy states of the atom's electrons are four quantum numbers: n, ℓ, mℓ, and ms. The quantum number ℓ is always a non-negative integer: 0, 1, 2, 3, etc. L has no real meaning except in its use as the angular momentum operator. It can take the following range of values, jumping only in integer steps: \vert \ell - s\vert \le j \le \ell + s where ℓ is the azimuthal quantum number (parameterizing the orbital angular momentum) and s is the spin quantum number (parameterizing the spin). In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin). In general, the values of range from to , where is the spin quantum number, associated with the particle's intrinsic spin angular momentum: :. Furthermore, the eigenvectors of j, s, mj and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of ℓ, s, mℓ and ms. == List of angular momentum quantum numbers == * Intrinsic (or spin) angular momentum quantum number, or simply spin quantum number * orbital angular momentum quantum number (the subject of this article) * magnetic quantum number, related to the orbital momentum quantum number * total angular momentum quantum number == History == The azimuthal quantum number was carried over from the Bohr model of the atom, and was posited by Arnold Sommerfeld. This number gives the information about the direction of spinning of the electron present in any orbital. Shape of orbital is also given by azimuthal quantum number. ====Magnetic quantum number==== The magnetic quantum number describes the specific orbital (or "cloud") within that subshell, and yields the projection of the orbital angular momentum along a specified axis: : The values of range from to , with integer intervals. Simultaneous measurement of electron energy and orbital angular momentum is allowed because the Hamiltonian commutes with the angular momentum operator related to L_z. In solutions of the Schrödinger-Pauli equation, angular momentum is quantized according to this number, so that magnitude of the spin angular momentum is \| \bold{s} \| = \hbar\sqrt{s(s+1)} = \frac{\sqrt{3}}{2}\ \hbar ~. The value of is the component of spin angular momentum, in units of the reduced Planck constant , parallel to a given direction (conventionally labelled the –axis). Depending on the value of n, there is an angular momentum quantum number ℓ and the following series. The value of ranges from 0 to , so the first p orbital () appears in the second electron shell (), the first d orbital () appears in the third shell (), and so on: : A quantum number beginning in = 3, = 0, describes an electron in the s orbital of the third electron shell of an atom. If s is the particle's spin angular momentum and ℓ its orbital angular momentum vector, the total angular momentum j is \mathbf j = \mathbf s + \boldsymbol {\ell} ~. These were identified as, respectively, the electron "shell" number , the "orbital" number , and the "orbital angular momentum" number . The azimuthal quantum number can also denote the number of angular nodes present in an orbital. Each of the different angular momentum states can take 2(2ℓ + 1) electrons.

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How many states belong to the carbon configurations $1 s^2 2 s^2 2 p^2$?

thumb|upright=1.3|Carbon dioxide pressure-temperature phase diagram Supercritical carbon dioxide (s) is a fluid state of carbon dioxide where it is held at or above its critical temperature and critical pressure. secondary Carbon 150x150px Structural formula of propane (secondary carbon is highlighted red) A secondary carbon is a carbon atom bound to two other carbon atoms. Franckeite, chemical formula Pb5Sn3Sb2S14, belongs to a family of complex sulfide minerals. thumb|Schematic of a binary star system with one planet on an S-type orbit and one on a P-type orbit. quaternary carbon 150x150px Structural formula of neopentane (quaternary carbon is highlighted red) A quaternary carbon is a carbon atom bound to four other carbon atoms. For this reason, quaternary carbon atoms are found only in hydrocarbons having at least five carbon atoms. Quaternary carbon atoms can occur in branched alkanes, but not in linear alkanes. primary carbon secondary carbon tertiary carbon quaternary carbon General structure (R = Organyl group) frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px Partial Structural formula frameless=1.0|96x96px frameless=1.0|102x102px frameless=1.0|98x98px frameless=1.0|105x105px == Synthesis == The formation of chiral quaternary carbon centers has been a synthetic challenge. In unbranched alkanes, the inner carbon atoms are always secondary carbon atoms (see figure). primary carbon secondary carbon tertiary carbon quaternary carbon General structure (R = Organyl group) frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px Partial Structural formula frameless=1.0|96x96px frameless=1.0|102x102px frameless=1.0|98x98px frameless=1.0|105x105px == References == Category:Chemical nomenclature Category:Organic chemistry For this reason, secondary carbon atoms are found in all hydrocarbons having at least three carbon atoms. The limits of stability for S-type and P-type orbits within binary as well as trinary stellar systems have been established as a function of the orbital characteristics of the stars, for both prograde and retrograde motions of stars and planets. ==See also== *Astrobiology *Circumstellar habitable zone *Habitability of yellow dwarf systems *Planetary habitability *Circumbinary planet ==References== Binary star systems Category:Binary stars Red atoms are oxygens. thumb|upright=1.3|TEM images of amorphous HBS Two-dimensional silica (2D silica) is a layered polymorph of silicon dioxide. Planets that orbit just one star in a binary pair are said to have "S-type" orbits, whereas those that orbit around both stars have "P-type" or "circumbinary" orbits. Typical estimates often suggest that 50% or more of all star systems are binary systems. Habitability of binary star systems is determined by many factors from a variety of sources. The planets have semi-major axes that lie between 1.09 and 1.46 times this critical radius. Carbon dioxide usually behaves as a gas in air at standard temperature and pressure (STP), or as a solid called dry ice when cooled and/or pressurised sufficiently. This would have implications for bulk thermal and nuclear generation of electricity, because the supercritical properties of carbon dioxide at above 500 °C and 20 MPa enable thermal efficiencies approaching 45 percent. thumb|upright=1.3|Top and side views of graphene (left) and HBS structures (right). The minimum stable star-to-circumbinary- planet separation is about 2–4 times the binary star separation, or orbital period about 3–8 times the binary period. Volume 2001, Issue 40 , Pages 2482–2486 Heck reaction, Enyne cyclization, cycloaddition reactions, Quasdorf, K.W.; Overman, L. E. Nature Volume 2014, Volume 516, Pages 181 C–H activation, Allylic substitution, Pauson–Khand reaction, Ishizaki, M.; Niimi, Y.; Hoshino, O.; Hara, H.; Takahashi, T. Tetrahedron Volume 2001, Issue 61, Pages 4053–4065 etc. to construct asymmetric quaternary carbon atoms. == References == Category:Chemical nomenclature Category:Organic chemistry For example, Kepler-47c is a gas giant in the circumbinary habitable zone of the Kepler-47 system. It was shown to be a member of the auxetics materials family with a negative Poisson's ratio. ==References== Category:Two-dimensional nanomaterials Category:Silicon dioxide Category:Silica polymorphs

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Calculate the energy needed to compress three carbon-carbon single bonds and stretch three carbon-carbon double bonds to the benzene bond length $1.397 Å$. Assume a harmonicoscillator potential-energy function for bond stretching and compression. Typical carboncarbon single- and double-bond lengths are 1.53 and $1.335 Å$; typical stretching force constants for carbon-carbon single and double bonds are 500 and $950 \mathrm{~N} / \mathrm{m}$.

Bond energy (BE) is the average of all bond-dissociation energies of a single type of bond in a given molecule.Madhusha (2017), Difference Between Bond Energy and Bond Dissociation Energy, Pediaa, Difference Between Bond Energy and Bond Dissociation Energy The bond-dissociation energies of several different bonds of the same type can vary even within a single molecule. IUPAC defines bond energy as the average value of the gas-phase bond-dissociation energy (usually at a temperature of 298.15 K) for all bonds of the same type within the same chemical species. The molecule has eight bond lengths ranging between 0.137 and 0.146 nm. right|thumb|illustrative example of C-C length molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C angle molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C-C torsion molecular energy dependence, numerical accuracy is not guaranteed A Molecular spring is a device or part of a biological system based on molecular mechanics and is associated with molecular vibration. Applying a harmonic approximation to the potential minimum (at V(r_m) = -\varepsilon), the exponent n and the energy parameter \varepsilon can be related to the harmonic spring constant: k = 2 \varepsilon \left(\frac{n}{r_0}\right)^2, from which n can be calculated if k is known. In chemistry, bond energy (BE), also called the mean bond enthalpyClark, J (2013), BOND ENTHALPY (BOND ENERGY), Chemguide, BOND ENTHALPY (BOND ENERGY) or average bond enthalpy is a measure of bond strength in a chemical bond. Most authors prefer to use the BDE values at 298.15 K.Luo, Yu-Ran and Jin-Pei Cheng "Bond Dissociation Energies". The first reduction requires around 1.0 V (Fc/), indicating that C70 is an electron acceptor. === Solution === Saturated solubility of C70 (S, mg/mL) Solvent S (mg/mL) 1,2-dichlorobenzene 36.2 carbon disulfide 9.875 xylene 3.985 toluene 1.406 benzene 1.3 carbon tetrachloride 0.121 n-hexane 0.013 cyclohexane 0.08 pentane 0.002 octane 0.042 decane 0.053 dodecane 0.098 heptane 0.047 isopropanol 0.0021 mesitylene 1.472 dichloromethane 0.080 Fullerenes are sparingly soluble in many aromatic solvents such as toluene and others like carbon disulfide, but not in water. C70 fullerene is the fullerene molecule consisting of 70 carbon atoms. This step yields a solution containing up to 70% of C60 and 15% of C70, as well as other fullerenes. For example, the carbon–hydrogen bond energy in methane BE(C–H) is the enthalpy change (∆H) of breaking one molecule of methane into a carbon atom and four hydrogen radicals, divided by four. Graph of the Lennard-Jones potential function: Intermolecular potential energy as a function of the distance of a pair of particles. In addition, the force needed to draw molecular string to its maximum length could be impractically high - comparable to the tensile strength of particular polymer molecule (~100GPa for some carbon compounds) == See also == *Ultra high molecular weight polyethylene *Carbon nanotube *Carbon nanotube springs ==References == * Stretching molecular springs:elasticity of titin filaments in vertebrate striated muscle, W.A. Linke, Institute of Physiology II, University of Heidelberg, Heidelberg, Germany Category:Nanotechnology Category:Molecular physics Valence bond (VB) computer programs for modern valence bond calculations:- * CRUNCH, by Gordon A. Gallup and his group. The bond energy for H2O is the average of energy required to break each of the two O–H bonds in sequence: : \begin{array}{lcl} \mathrm{H-O-H} & \rightarrow & \mathrm{H\cdot + \cdot O-H} & , D_1 \\\ \mathrm{\cdot O-H} & \rightarrow & \mathrm{\cdot O\cdot + \cdot H} & , D_2 \\\ \mathrm{H-O-H} & \rightarrow & \mathrm{H\cdot + \cdot O\cdot + \cdot H} & , D =(D_1 + D_2)/2 \\\ \end{array} Although the two bonds are the equivalent in the original symmetric molecule, the bond-dissociation energy of an oxygen–hydrogen bond varies slightly depending on whether or not there is another hydrogen atom bonded to the oxygen atom. Each carbon atom in the structure is bonded covalently with 3 others. thumb|left|The structure of C70 molecule. A related fullerene molecule, named buckminsterfullerene (C60 fullerene), consists of 60 carbon atoms. The amount of energy storable in molecular spring is limited by the value of deformation the molecule can withstand until it undergoes chemical change. The resulting structural unit [-C≡(-CH2-)3≡C-] is a rigid cage, consisting of two carbon atoms joined by three methylene bridges; therefore the joined units are constrained to lie on a straight line. The bond dissociation energy (enthalpy) is also referred to as bond disruption energy, bond energy, bond strength, or binding energy (abbreviation: BDE, BE, or D). * Two center Lennard-Jones potential The two center Lennard-Jones potential consists of two identical Lennard-Jones interaction sites (same \varepsilon, \sigma, m) that are bonded as a rigid body. For the LJTS potential with r_\mathrm{end} = 2.5\,\sigma , the potential energy shift is approximately 1/60 of the dispersion energy at the potential well: V_\mathrm{LJ}(r_\mathrm{end} = 2.5\,\sigma) = -0.0163\,\varepsilon .

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When a particle of mass $9.1 \times 10^{-28} \mathrm{~g}$ in a certain one-dimensional box goes from the $n=5$ level to the $n=2$ level, it emits a photon of frequency $6.0 \times 10^{14} \mathrm{~s}^{-1}$. Find the length of the box.

In particle physics, the radiation length is a characteristic of a material, related to the energy loss of high energy particles electromagnetically interacting with it. The radiation length for a given material consisting of a single type of nucleus can be approximated by the following expression: (http://pdg.lbl.gov/) X_0 = 716.4 \text{ g cm}^{-2} \frac{A}{Z (Z+1) \ln{\frac{287}{\sqrt{Z}}}} = 1433 \text{ g cm}^{-2} \frac{A}{Z (Z+1) (11.319 - \ln{Z})}, where is the atomic number and is mass number of the nucleus. This length is useful for renormalizing a non-isotropic scattering problem into an isotropic one in order to use classical diffusion laws (Fick law and Brownian motion). The four-frequency of a massless particle, such as a photon, is a four-vector defined by :N^a = \left( u, u \hat{\mathbf{n}} \right) where u is the photon's frequency and \hat{\mathbf{n}} is a unit vector in the direction of the photon's motion. The characteristic amount of matter traversed for these related interactions is called the radiation length , usually measured in g·cm−2. The parameter a_s of dimension length is defined as the scattering length. The transport length in a strongly diffusing medium (noted l*) is the length over which the direction of propagation of the photon is randomized. The scattering length in quantum mechanics describes low-energy scattering. To relate the scattering length to physical observables that can be measured in a scattering experiment we need to compute the cross section \sigma. It is also the appropriate length scale for describing high-energy electromagnetic cascades. Comprehensive tables for radiation lengths and other properties of materials are available from the Particle Data Group. ==See also== * Mean free path * Attenuation length * Attenuation coefficient * Attenuation * Range (particle radiation) * Stopping power (particle radiation) * Electron energy loss spectroscopy ==References== Category:Experimental particle physics The 10.5 cm leFH 18M ( "light field howitzer") was a German light howitzer used in the Second World War. The formal way to solve this problem is to do a partial wave expansion (somewhat analogous to the multipole expansion in classical electrodynamics), where one expands in the angular momentum components of the outgoing wave. It is defined as the mean length (in cm) into the material at which the energy of an electron is reduced by the factor 1/e. ==Definition== In materials of high atomic number (e.g. tungsten, uranium, plutonium) the electrons of energies >~10 MeV predominantly lose energy by , and high-energy photons by pair production. The transport length might be measured by transmission experiments and backscattering experiments. For , a good approximation is \frac{1}{X_0} = 4 \left( \frac{\hbar}{m_\mathrm{e} c} \right)^2 Z(Z+1) \alpha^3 n_\mathrm{a} \log\left(\frac{183}{Z^{1/3}}\right), where * is the number density of the nucleus, *\hbar denotes the reduced Planck constant, * is the electron rest mass, * is the speed of light, * is the fine-structure constant. thumb|230px|The three leading Slepian sequences for T=1000 and 2WT=6. For our potential we have therefore a=r_0, in other words the scattering length for a hard sphere is just the radius. The concept of the scattering length can also be extended to potentials that decay slower than 1/r^3 as r\to \infty. It is both the mean distance over which a high- energy electron loses all but of its energy by , and of the mean free path for pair production by a high-energy photon. An observer moving with four-velocity V^b will observe a frequency :\frac{1}{c}\eta\left(N^a, V^b\right) Where \eta is the Minkowski inner-product (+−−−) Closely related to the four-frequency is the four-wavevector defined by :K^a = \left(\frac{\omega}{c}, \mathbf{k}\right) where \omega = 2 \pi u, c is the speed of light and \mathbf{k} = \frac{2 \pi}{\lambda}\hat{\mathbf{n}} and \lambda is the wavelength of the photon. Sterling Publishing Company, Inc., 2002, p.144 ==History== The 10.5 cm leFH 18M superseded the 10.5 cm leFH 18 as the standard German divisional field howitzer used during the Second World War.

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Use the normalized Numerov-method harmonic-oscillator wave functions found by going from -5 to 5 in steps of 0.1 to estimate the probability of being in the classically forbidden region for the $v=0$ state.

\\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. {\pi R^2} + \frac{\arcsin\\!\left(\frac{x}{R}\right)}{\pi}\\! for -R\leq x \leq R| mean =0\,| median =0\,| mode =0\,| variance =\frac{R^2}{4}\\!| skewness =0\,| kurtosis =-1\,| entropy =\ln (\pi R) - \frac12 \,| mgf =2\,\frac{I_1(R\,t)}{R\,t}| char =2\,\frac{J_1(R\,t)}{R\,t}| }} The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): :f(x)={2 \over \pi R^2}\sqrt{R^2-x^2\,}\, for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. Phys. 2009, 39, 337–356 The normal distribution is recovered as q → 1\. If we denote by u_{0\ell} the wave function subject to the given potential with total energy E=0 and azimuthal quantum number \ell, the Sturm Oscillation Theorem implies that N_\ell equals the number of nodes of u_{0\ell}. The entropy is calculated as H_{N}(n)=\int_{-1}^{+1} f_{X}(x;n)\ln (f_{X}(x;n))dx The first 5 moments (n=-1 to 3), such that R=1 are \ -\ln(2/\pi) ; n=-1 \ -\ln(2) ;n=0 \ -1/2+\ln(\pi) ;n=1 \ 5/3-\ln(3) ;n=2 \ -7/4-\ln(1/3\pi) ; n=3 == N-sphere Wigner distribution with odd symmetry applied == The marginal PDF distribution with odd symmetry is f{_X}(x;n) ={(1-x^2)^{(n-1)/2)}\Gamma (1+n/2) \over \ {\sqrt{\pi}} \Gamma((n+1)/2)}\sgn(x)\, ; such that R=1 Hence, the CF is expressed in terms of Struve functions CF(t;n) ={ \Gamma(n/2+1) H_{n/2}(t)/(t/2)^{(n/2)} }\, \urcorner (n>=-1); "The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by" Z= { \rho c \pi a^2 [R_1 (2ka)-i X_1 (2 ka)], } R_1 ={1-{2 J_1(x) \over 2x} , } X_1 ={{2 H_1(x) \over x} , } == Example (Normalized Received Signal Strength): quadrature terms == The normalized received signal strength is defined as |R| ={{1 \over N} | }\sum_{k=1}^N \exp [i x_n t]| and using standard quadrature terms x ={{1 \over N} }\sum_{k=1}^N \cos ( x_n t) y ={{1 \over N} }\sum_{k=1}^N \sin ( x_n t) Hence, for an even distribution we expand the NRSS, such that x = 1 and y = 0, obtaining {\sqrt{x^2+y^2}}=x+{3 \over 2}y^2-{3 \over 2}xy^2+{1 \over 2}x^2y^2 + O(y^3) +O(y^3)(x-1) +O(y^3)(x-1)^2 +O(x-1)^3 The expanded form of the Characteristic function of the received signal strength becomes E[x] = {1\over N }CF(t;n) E[y^2] ={1\over 2 N}(1 - CF(2t;n)) E[x^2] ={1\over 2N}(1 + CF(2t;n)) E[xy^2] = {t^2 \over 3N^2} CF(t;n)^3+({N-1 \over 2N^2})(1-t CF(2t;n))CF(t;n) E[x^2y^2] = {1\over 8N^3} (1-CF(4t;n))+({N-1 \over 4N^3})(1-CF(2t;n)^2) +({N-1 \over 3N^3})t^2CF(t;n)^4 +({(N-1)(N-2)\over N^3})CF(t;n)^2(1-CF(2t;n)) == See also == * Wigner surmise * The Wigner semicircle distribution is the limit of the Kesten–McKay distributions, as the parameter d tends to infinity. thumb|right|Figure 1 Part of a semi-Markovian discrete system in one dimension with directional jumping time probability density functions (JT-PDFs), including "death" terms (the JT-PDFs from state i in state I). Let V:\mathbb{R}^3\to\mathbb{R}:\mathbf{r}\mapsto V(r) be a spherically symmetric potential, such that it is piecewise continuous in r, V(r)=O(1/r^a) for r\to0 and V(r)=O(1/r^b) for r\to+\infty, where a\in(2,+\infty) and b\in(-\infty,2). If :\int_0^{+\infty}r|V(r)|dr<+\infty, then the number of bound states N_\ell with azimuthal quantum number \ell for a particle of mass m obeying the corresponding Schrödinger equation, is bounded from above by :N_\ell<\frac{1}{2\ell+1}\frac{2m}{\hbar^2}\int_0^{+\infty}r|V(r)|dr. Initial attempts at approximating Chernoff's distribution via solving the heat equation, however, did not achieve satisfactory precision due to the nature of the boundary conditions. It takes the form :N_\ell < \frac{1}{2\ell+1} \frac{2m}{\hbar^2} \int_0^\infty r |V(r)|\, dr This limit is the best possible upper bound in such a way that for a given \ell, one can always construct a potential V_\ell for which N_\ell is arbitrarily close to this upper bound. \right] for q < 1 | pdf ={\sqrt{\beta} \over C_q} e_q({-\beta x^2}) | cdf = | mean =0\text{ for }q<2, otherwise undefined| median =0| mode =0| variance = { 1 \over {\beta (5-3q)}} \text{ for } q < {5 \over 3} \infty \text{ for } {5 \over 3} \le q < 2 \text{Undefined for }2 \le q <3| skewness = 0 \text{ for } q < {3 \over 2} | kurtosis = 6{q-1 \over 7-5q} \text{ for } q < {7 \over 5} | entropy =| mgf =| cf =| }} The q-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. In quantum mechanics, Bargmann's limit, named for Valentine Bargmann, provides an upper bound on the number N_\ell of bound states with azimuthal quantum number \ell in a system with central potential V. Groeneboom (1989) shows that : f_Z (z) \sim \frac{1}{2} \frac{4^{4/3} |z|}{\operatorname{Ai}' (\tilde{a}_1)} \exp \left( - \frac{2}{3} |z|^3 + 2^{1/3} \tilde{a}_1 |z| \right) \text{ as }z \rightarrow \infty where \tilde{a}_1 \approx -2.3381 is the largest zero of the Airy function Ai and where \operatorname{Ai}' (\tilde{a}_1 ) \approx 0.7022. The choice of the q-Gaussian form may arise if the system is non-extensive, or if there is lack of a connection to small samples sizes. ==Characterization== ===Probability density function=== The standard q-Gaussian has the probability density function : f(x) = {\sqrt{\beta} \over C_q} e_q(-\beta x^2) where :e_q(x) = [1+(1-q)x]_+^{1 \over 1-q} is the q-exponential and the normalization factor C_q is given by :C_q = {{2 \sqrt{\pi} \Gamma\left({1 \over 1-q}\right)} \over {(3-q) \sqrt{1-q} \Gamma\left({3-q \over 2(1-q)}\right)}} \text{ for } -\infty < q < 1 : C_q = \sqrt{\pi} \text{ for } q = 1 \, :C_q = { {\sqrt{\pi} \Gamma\left({3-q \over 2(q-1)}\right)} \over {\sqrt{q-1} \Gamma\left({1 \over q-1}\right)}} \text{ for }1 < q < 3 . The parabolic Wigner distribution is also considered the monopole moment of the hydrogen like atomic orbitals. == Wigner n-sphere distribution == The normalized N-sphere probability density function supported on the interval [−1, 1] of radius 1 centered at (0, 0): f_n(x;n)={(1-x^2)^{(n-1)/2}\Gamma (1+n/2) \over \sqrt{\pi} \Gamma((n+1)/2)}\, (n>= -1) , for −1 ≤ x ≤ 1, and f(x) = 0 if |x| > 1\. Just as the cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is normal, so also, the free cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is Wigner's semicircle distribution. == Related distributions == === Wigner (spherical) parabolic distribution === The parabolic probability distribution supported on the interval [−R, R] of radius R centered at (0, 0): f(x)={3 \over \ 4 R^3}{(R^2-x^2)}\, for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. Example. If : V(a,c) = \underset{s \in \mathbf{R}}{\operatorname{argmax}} \ (W(s) - c(s-a)^2), then V(0, c) has density : f_c(t) = \frac{1}{2} g_c(t) g_c(-t) where gc has Fourier transform given by : \hat{g}_c (s) = \frac{(2/c)^{1/3}}{\operatorname{Ai} (i (2c^2)^{-1/3} s)}, \ \ \ s \in \mathbf{R} and where Ai is the Airy function. The solution is based on the path representation of the Green's function, calculated when including all the path probability density functions of all lengths: Here, : \bar{\Psi}_{i}(s) =\sum_j \bar{\Psi}_{ij}(s) and : \bar{\Psi}_{ij}(s)=\frac{1-\bar{\psi}_{ij}(s)}{s}. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). For the corresponding wave function with total energy E=0 and azimuthal quantum number \ell, denoted by \phi_{0\ell}, the radial Schrödinger equation becomes :\frac{d^{2}}{d r^{2}} \phi_{0\ell}(r)-\frac{\ell(\ell+1)}{r^{2}} \phi_{0\ell}(r)=-W(r) \phi_{0\ell}(r), with W=2m|V|/\hbar^2. The following formula will generate deviates from a q-Gaussian with specified parameter q and \beta = {1 \over {3-q}} :Z = \sqrt{-2 \text{ ln}_{q'}(U_1)} \text{ cos}(2 \pi U_2) where \text{ ln}_q is the q-logarithm and q' = { {1+q} \over {3-q}} These deviates can be transformed to generate deviates from an arbitrary q-Gaussian by : Z' = \mu + {Z \over \sqrt{\beta (3-q)}} == Applications == === Physics === It has been shown that the momentum distribution of cold atoms in dissipative optical lattices is a q-Gaussian.

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Calculate the de Broglie wavelength of an electron moving at 1/137th the speed of light. (At this speed, the relativistic correction to the mass is negligible.)

Thus the uncertainty in position must be greater than half of the reduced Compton wavelength . ==Relationship to other constants== Typical atomic lengths, wave numbers, and areas in physics can be related to the reduced Compton wavelength for the electron {2\pi}\simeq 386~\textrm{fm})}} and the electromagnetic fine-structure constant The Bohr radius is related to the Compton wavelength by: a_0 = \frac{1}{\alpha}\left(\frac{\lambda_\text{e}}{2\pi}\right) = \frac{\bar{\lambda}_\text{e}}{\alpha} \simeq 137\times\bar{\lambda}_\text{e}\simeq 5.29\times 10^4~\textrm{fm} The classical electron radius is about 3 times larger than the proton radius, and is written: r_\text{e} = \alpha\left(\frac{\lambda_\text{e}}{2\pi}\right) = \alpha\bar{\lambda}_\text{e} \simeq\frac{\bar{\lambda}_\text{e}}{137}\simeq 2.82~\textrm{fm} The Rydberg constant, having dimensions of linear wavenumber, is written: \frac{1}{R_\infty}=\frac{2\lambda_\text{e}}{\alpha^2} \simeq 91.1~\textrm{nm} \frac{1}{2\pi R_\infty} = \frac{2}{\alpha^2}\left(\frac{\lambda_\text{e}}{2\pi}\right) = 2 \frac{\bar{\lambda}_\text{e}}{\alpha^2} \simeq 14.5~\textrm{nm} This yields the sequence: r_{\text{e}} = \alpha \bar{\lambda}_{\text{e}} = \alpha^2 a_0 = \alpha^3 \frac{1}{4\pi R_\infty}. The standard Compton wavelength of a particle is given by \lambda = \frac{h}{m c}, while its frequency is given by f = \frac{m c^2}{h}, where is the Planck constant, is the particle's proper mass, and is the speed of light. The CODATA 2018 value for the Compton wavelength of the electron is .CODATA 2018 value for Compton wavelength for the electron from NIST. The Compton wavelength for this particle is the wavelength of a photon of the same energy. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. For photons of frequency , energy is given by E = h f = \frac{h c}{\lambda} = m c^2, which yields the Compton wavelength formula if solved for . ==Limitation on measurement== The Compton wavelength expresses a fundamental limitation on measuring the position of a particle, taking into account quantum mechanics and special relativity. The reduced Compton wavelength appears in the relativistic Klein–Gordon equation for a free particle: \mathbf{ abla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi = \left(\frac{m c}{\hbar} \right)^2 \psi. The Fermi space observatory detected a gamma-ray with an energy of at least 94 billion electron volts. The Compton wavelength is a quantum mechanical property of a particle, defined as the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). In this case, the Compton wavelength is equal to the square root of the quantum metric, a metric describing the quantum space: \sqrt{g_{kk}}=\lambda_c ==See also== * de Broglie wavelength * Planck–Einstein relation ==References== ==External links== * Length Scales in Physics: the Compton Wavelength Category:Atomic physics Category:Foundational quantum physics de:Compton-Effekt#Compton-Wellenlänge The Planck mass and length are defined by: m_{\rm P} = \sqrt{\hbar c/G} l_{\rm P} = \sqrt{\hbar G /c^3}. ==Geometrical interpretation== A geometrical origin of the Compton wavelength has been demonstrated using semiclassical equations describing the motion of a wavepacket. thumb|left|upright=1.0|Spectrum of WR 137 showing the prominent emission lines of ionised Carbon and Helium WR 137 is a variable Wolf-Rayet star located around 6,000 light years away from Earth in the constellation of Cygnus. This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. Relativistic electron beams are streams of electrons moving at relativistic speeds. Species Mass (kg) \lambda_{\mathrm{th}} (m) Electron Photon 0 H2 O2 ==References== * Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28. Equations that pertain to the wavelengths of photons interacting with mass use the non-reduced Compton wavelength. The Planck mass is the order of mass for which the Compton wavelength and the Schwarzschild radius r_{\rm S} = 2 G M /c^2 are the same, when their value is close to the Planck length (l_{\rm P}). Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source. ==Massive particles== For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. Other particles have different Compton wavelengths. == Reduced Compton wavelength == The reduced Compton wavelength (barred lambda, denoted below by \bar\lambda) is defined as the Compton wavelength divided by : : \bar\lambda = \frac{\lambda}{2 \pi} = \frac{\hbar}{m c}, where is the reduced Planck constant. ==Role in equations for massive particles== The inverse reduced Compton wavelength is a natural representation for mass on the quantum scale, and as such, it appears in many of the fundamental equations of quantum mechanics. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light.

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12

D

Calculate the angle that the spin vector $S$ makes with the $z$ axis for an electron with spin function $\alpha$.

The component of the spin along a specified axis is given by the spin magnetic quantum number, conventionally written . * The spin value of an electron, proton, neutron is . The direct observation of the electron's intrinsic angular momentum was achieved in the Stern–Gerlach experiment. === Stern–Gerlach experiment === The theory of spatial quantization of the spin moment of the momentum of electrons of atoms situated in the magnetic field needed to be proved experimentally. The component of nuclear spin parallel to the –axis can have (2 + 1) values , –1, ..., . Given an arbitrary direction (usually determined by an external magnetic field) the spin -projection is given by :s_z = m_s \, \hbar where is the secondary spin quantum number, ranging from − to + in steps of one. The direction of spin is described by spin quantum number. The electron spin magnetic moment is given by the formula: \ \boldsymbol{\mu}_s = -\frac{e}{\ 2m\ }\ g\ \mathbf{s}\ where : is the charge of the electron : is the Landé g-factor and by the equation: \ \mu_z = \pm \frac{1}{2}\ g\ \mu_\mathsf{B}\ where \ \mu_\mathsf{B}\ is the Bohr magneton. In nuclear magnetic resonance spectroscopy and magnetic resonance imaging, the Ernst angle is the flip angle (a.k.a. "tip" or "nutation" angle) for excitation of a particular spin that gives the maximal signal intensity in the least amount of time when signal averaging over many transients. In solutions of the Schrödinger-Pauli equation, angular momentum is quantized according to this number, so that magnitude of the spin angular momentum is \| \bold{s} \| = \hbar\sqrt{s(s+1)} = \frac{\sqrt{3}}{2}\ \hbar ~. The name "spin" comes from a physical spinning of the electron about an axis, as proposed by Uhlenbeck and Goudsmit. The value of is the component of spin angular momentum, in units of the reduced Planck constant , parallel to a given direction (conventionally labelled the –axis). This number gives the information about the direction of spinning of the electron present in any orbital. Nuclear-spin quantum numbers are conventionally written for spin, and or for the -axis component. * The magnitude spin quantum number of an electron cannot be changed. Here is the total orbital angular momentum quantum number. In the electron, the two different spin orientations are sometimes called "spin-up" or "spin-down". In physics, the spin quantum number is a quantum number (designated ) that describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. Spin engineering describes the control and manipulation of quantum spin systems to develop devices and materials. In the figure below, x and z name the directions of the (inhomogenous) magnetic field, with the x-z-plane being orthogonal to the particle beam. First of all, spin satisfies the fundamental commutation relation: \ [S_i, S_j ] = i\ \hbar\ \epsilon_{ijk}\ S_k\ , \ \left[S_i, S^2 \right] = 0\ where \ \epsilon_{ijk}\ is the (antisymmetric) Levi-Civita symbol. thumb|upright=1.35|Stern–Gerlach experiment: Silver atoms travelling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. When atoms have even numbers of electrons the spin of each electron in each orbital has opposing orientation to that of its immediate neighbor(s).

54.7

0.2553

0.9992093669

92

48

A

The AM1 valence electronic energies of the atoms $\mathrm{H}$ and $\mathrm{O}$ are $-11.396 \mathrm{eV}$ and $-316.100 \mathrm{eV}$, respectively. For $\mathrm{H}_2 \mathrm{O}$ at its AM1-calculated equilibrium geometry, the AM1 valence electronic energy (core-core repulsion omitted) is $-493.358 \mathrm{eV}$ and the AM1 core-core repulsion energy is $144.796 \mathrm{eV}$. For $\mathrm{H}(g)$ and $\mathrm{O}(g), \Delta H_{f, 298}^{\circ}$ values are 52.102 and $59.559 \mathrm{kcal} / \mathrm{mol}$, respectively. Find the AM1 prediction of $\Delta H_{f, 298}^{\circ}$ of $\mathrm{H}_2 \mathrm{O}(g)$.

==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. Indeed, AM1* is an extension of AM1 molecular orbital theory and uses AM1 parameters and theory unchanged for the elements H, C, N, O and F. ==Earth bulk continental crust and upper continental crust== *C1 — Crust: CRC Handbook *C2 — Crust: Kaye and Laby *C3 — Crust: Greenwood *C4 — Crust: Ahrens (Taylor) *C5 — Crust: Ahrens (Wänke) *C6 — Crust: Ahrens (Weaver) *U1 — Upper crust: Ahrens (Taylor) *U2 — Upper crust: Ahrens (Shaw) Element C1 C2 C3 C4 C5 C6 U1 U2 01 H hydrogen 1.40×10−3 1.52 _0_ ×10−3 02 He helium 8×10−9 03 Li lithium 2.0×10−5 2.0×10−5 1.8×10−5 1.3×10−5 1.37×10−5 2. _0_ ×10−5 2.2×10−5 04 Be beryllium 2.8×10−6 2.0×10−6 2×10−6 1.5 _00_ ×10−6 3. _000_ ×10−6 05 B boron 1.0×10−5 7.0×10−6 9×10−6 1. _0000_ ×10−5 1.5 _000_ ×10−5 06 C carbon 2.00×10−4 1.8 _0_ ×10−4 3.76×10−3 07 N nitrogen 1.9×10−5 2.0×10−5 1.9×10−5 08 O oxygen 4.61×10−1 3.7×10−1 4.55 _000_ ×10−1 09 F fluorine 5.85×10−4 4.6×10−4 5.44×10−4 5.25×10−4 10 Ne neon 5×10−9 11 Na sodium 2.36×10−2 2.3×10−2 2.27 _00_ ×10−2 2.3 _000_ ×10−2 2.44 _00_ ×10−2 3.1 _000_ ×10−2 2.89×10−2 2.57×10−2 12 Mg magnesium 2.33×10−2 2.8×10−2 2.764 _0_ ×10−2 3.20×10−2 2.37×10−2 1.69×10−2 1.33×10−2 1.35×10−2 13 Al aluminium 8.23×10−2 8.0×10−2 8.3 _000_ ×10−2 8.41 _00_ ×10−2 8.305 _0_ ×10−2 8.52 _00_ ×10−2 8.04 _00_ ×10−2 7.74 _00_ ×10−2 14 Si silicon 2.82×10−1 2.7×10−1 2.72 _000_ ×10−1 2.677×10−1 2.81×10−1 2.95×10−1 3.08×10−1 3.04×10−1 15 P phosphorus 1.05×10−3 1.0×10−3 1.12 _0_ ×10−3 7.63×10−4 8.3 _0_ ×10−4 16 S sulfur 3.50×10−4 3.0×10−4 3.4 _0_ ×10−4 8.81×10−4 17 Cl chlorine 1.45×10−4 1.9×10−4 1.26×10−4 1.9 _00_ ×10−3 18 Ar argon 3.5×10−6 19 K potassium 2.09×10−2 1.7×10−2 1.84 _00_ ×10−2 9.1 _00_ ×10−3 1.76 _00_ ×10−2 1.7 _000_ ×10−2 2.8 _000_ ×10−2 2.57 _00_ ×10−2 20 Ca calcium 4.15×10−2 5.1×10−2 4.66 _00_ ×10−2 5.29 _00_ ×10−2 4.92 _00_ ×10−2 3.4 _000_ ×10−2 3. _0000_ ×10−2 2.95 _00_ ×10−2 21 Sc scandium 2.2×10−5 2.2×10−5 2.5×10−5 3. _0_ ×10−5 2.14×10−5 1.1×10−5 7×10−6 22 Ti titanium 5.65×10−3 8.6×10−3 6.32 _0_ ×10−3 5.4 _00_ ×10−3 5.25 _0_ ×10−3 3.6 _00_ ×10−3 3. _000_ ×10−3 3.12 _0_ ×10−3 23 V vanadium 1.20×10−4 1.7×10−4 1.36×10−4 2.3 _0_ ×10−4 1.34×10−4 6. _0_ ×10−5 5.3×10−5 24 Cr chromium 1.02×10−4 9.6×10−5 1.22×10−4 1.85×10−4 1.46×10−4 5.6×10−5 3.5×10−5 3.5×10−5 25 Mn manganese 9.50×10−4 1.0×10−3 1.06 _0_ ×10−3 1.4 _00_ ×10−3 8.47×10−4 1. _000_ ×10−3 6. _00_ ×10−4 5.27×10−4 26 Fe iron 5.63×10−2 5.8×10−2 6.2 _000_ ×10−2 7.07×10−2 4.92×10−2 3.8×10−2 3.50×10−2 3.09×10−2 27 Co cobalt 2.5×10−5 2.8×10−5 2.9×10−5 2.9×10−5 2.54×10−5 1. _0_ ×10−5 1.2×10−5 28 Ni nickel 8.4×10−5 7.2×10−5 9.9×10−5 1.05×10−4 6.95×10−5 3.5×10−5 2×10−5 1.9×10−5 29 Cu copper 6.0×10−5 5.8×10−5 6.8×10−5 7.5×10−5 4.7×10−5 2.5×10−5 1.4×10−5 30 Zn zinc 7.0×10−5 8.2×10−5 7.6×10−5 8. _0_ ×10−5 7.6×10−5 7.1×10−5 5.2×10−5 31 Ga gallium 1.9×10−5 1.7×10−5 1.9×10−5 1.8×10−5 1.86×10−5 1.7×10−5 1.4×10−5 32 Ge germanium 1.5×10−6 1.3×10−6 1.5×10−6 1.6×10−6 1.32×10−6 1.6×10−6 33 As arsenic 1.8×10−6 2.0×10−6 1.8×10−6 1.0×10−6 2.03×10−6 1.5×10−6 34 Se selenium 5×10−8 5×10−8 5×10−8 5×10−8 1.53×10−7 5×10−8 35 Br bromine 2.4×10−6 4.0×10−6 2.5×10−6 6.95×10−6 36 Kr krypton 1×10−10 37 Rb rubidium 9.0×10−5 7.0×10−5 7.8×10−5 3.2×10−5 7.90×10−5 6.1×10−5 1.12×10−4 1.1 _0_ ×10−4 38 Sr strontium 3.70×10−4 4.5×10−4 3.84×10−4 2.6 _0_ ×10−4 2.93×10−4 5.03×10−4 3.5 _0_ ×10−4 3.16×10−4 39 Y yttrium 3.3×10−5 3.5×10−7 3.1×10−5 2. _0_ ×10−5 1.4×10−5 2.2×10−5 2.1×10−5 40 Zr zirconium 1.65×10−4 1.4×10−4 1.62×10−4 1. _00_ ×10−4 2.1 _0_ ×10−4 1.9 _0_ ×10−4 2.4 _0_ ×10−4 41 Nb niobium 2.0×10−5 2.0×10−5 2. _0_ ×10−5 1.1 _000_ ×10−5 1.3 _000_ ×10−5 2.5 _000_ ×10−5 2.6 _000_ ×10−5 42 Mo molybdenum 1.2×10−6 1.2×10−6 1.2×10−6 1. _000_ ×10−6 1.5 _00_ ×10−6 43 Tc technetium 44 Ru ruthenium 1×10−9 1×10−10 45 Rh rhodium 1×10−9 1×10−10 46 Pd palladium 1.5×10−8 3×10−9 1.5×10−8 1.0×10−9 5×10−10 47 Ag silver 7.5×10−8 8×10−8 8×10−8 8. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 48 Cd cadmium 1.5×10−7 1.8×10−7 1.6×10−7 9.8×10−8 1. _00_ ×10−7 9.8×10−8 49 In indium 2.5×10−7 2×10−7 2.4×10−7 5. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 50 Sn tin 2.3×10−6 1.5×10−6 2.1×10−6 2.5 _00_ ×10−6 5.5 _00_ ×10−6 51 Sb antimony 2×10−7 2×10−7 2×10−7 2. _00_ ×10−7 2.03×10−7 2. _00_ ×10−7 52 Te tellurium 1×10−9 1×10−9 2.03×10−9 53 I iodine 4.5×10−7 5×10−7 4.6×10−7 1.54 _0_ ×10−6 54 Xe xenon 3×10−11 55 Cs caesium 3×10−6 1.6×10−6 2.6×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 3.7 _00_ ×10−6 56 Ba barium 4.25×10−4 3.8×10−4 3.9 _0_ ×10−4 2.5 _0000_ ×10−4 5.42 _000_ ×10−4 7.07 _000_ ×10−4 5.5 _0000_ ×10−4 1.07 _0000_ ×10−3 57 La lanthanum 3.9×10−5 5.0×10−5 3.5×10−5 1.6 _000_ ×10−5 2.9 _000_ ×10−5 2.8 _000_ ×10−5 3. _0000_ ×10−5 3.2 _00_ ×10−6 58 Ce cerium 6.65×10−5 8.3×10−5 6.6×10−5 3.3 _000_ ×10−5 5.42 _00_ ×10−5 5.7 _000_ ×10−5 6.4 _000_ ×10−5 6.5 _000_ ×10−5 59 Pr praseodymium 9.2×10−6 1.3×10−5 9.1×10−6 3.9 _00_ ×10−6 7.1 _00_ ×10−6 60 Nd neodymium 4.15×10−5 4.4×10−5 4. _0_ ×10−5 1.6 _000_ ×10−5 2.54 _00_ ×10−5 2.3 _000_ ×10−5 2.6 _000_ ×10−5 2.6 _000_ ×10−5 61 Pm promethium 62 Sm samarium 7.05×10−6 7.7×10−6 7.0×10−6 3.5 _00_ ×10−6 5.59 _0_ ×10−6 4.1 _00_ ×10−6 4.5 _00_ ×10−6 4.5 _00_ ×10−6 63 Eu europium 2.0×10−6 2.2×10−6 2.1×10−6 1.1 _00_ ×10−6 1.407×10−6 1.09 _0_ ×10−6 8.8 _0_ ×10−7 9.4 _0_ ×10−7 64 Gd gadolinium 6.2×10−6 6.3×10−6 6.1×10−6 3.3 _00_ ×10−6 8.14 _0_ ×10−6 3.8 _00_ ×10−6 2.8 _00_ ×10−6 65 Tb terbium 1.2×10−6 1.0×10−6 1.2×10−6 6. _00_ ×10−7 1.02 _0_ ×10−6 5.3 _0_ ×10−7 6.4 _0_ ×10−7 4.8 _0_ ×10−7 66 Dy dysprosium 5.2×10−6 8.5×10−6 3.7 _00_ ×10−6 6.102×10−6 3.5 _00_ ×10−6 67 Ho holmium 1.3×10−6 1.6×10−6 1.3×10−6 7.8 _0_ ×10−7 1.86 _0_ ×10−6 8. _00_ ×10−7 6.2 _0_ ×10−7 68 Er erbium 3.5×10−6 3.6×10−6 3.5×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 2.3 _00_ ×10−6 69 Tm thulium 5.2×10−7 5.2×10−7 5×10−7 3.2 _0_ ×10−7 2.4 _0_ ×10−7 3.3 _0_ ×10−7 70 Yb ytterbium 3.2×10−6 3.4×10−6 3.1×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 1.53 _0_ ×10−6 2.2 _00_ ×10−6 1.5 _00_ ×10−6 71 Lu lutetium 8×10−7 8×10−7 3. _00_ ×10−7 5.76×10−7 2.3 _0_ ×10−7 3.2 _0_ ×10−7 2.3 _0_ ×10−7 72 Hf hafnium 3.0×10−6 4×10−6 2.8×10−6 3. _000_ ×10−6 3.46 _0_ ×10−6 4.7 _00_ ×10−6 5.8 _00_ ×10−6 5.8 _00_ ×10−6 73 Ta tantalum 2.0×10−6 2.4×10−6 1.7×10−6 1. _000_ ×10−6 2.203×10−6 2.2 _00_ ×10−6 74 W tungsten 1.25×10−6 1.0×10−6 1.2×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 2. _000_ ×10−6 75 Re rhenium 7×10−10 4×10−10 7×10−10 5×10−10 1.02×10−9 5×10−10 76 Os osmium 1.5×10−9 2×10−10 5×10−9 1.02×10−9 77 Ir iridium 1×10−9 2×10−10 1×10−9 1×10−10 1.02×10−9 2×10−11 78 Pt platinum 5×10−9 1×10−8 79 Au gold 4×10−9 2×10−9 4×10−9 3.0×10−9 4.07×10−9 1.8×10−9 80 Hg mercury 8.5×10−8 2×10−8 8×10−8 81 Tl thallium 8.5×10−7 4.7×10−7 7×10−7 3.6 _0_ ×10−7 7.5 _0_ ×10−7 5.2 _0_ ×10−7 82 Pb lead 1.4×10−5 1.0×10−5 1.3×10−5 8. _000_ ×10−6 1.5 _000_ ×10−5 2. _0000_ ×10−5 1.7 _000_ ×10−5 83 Bi bismuth 8.5×10−9 4×10−9 8×10−9 6. _0_ ×10−8 1.27×10−7 84 Po polonium 2×10−16 85 At astatine 86 Rn radon 4×10−19 87 Fr francium 88 Ra radium 9×10−13 89 Ac actinium 5.5×10−16 90 Th thorium 9.6×10−6 5.8×10−6 8.1×10−6 3.5 _00_ ×10−6 5.7 _00_ ×10−6 1.07 _00_ ×10−5 1. _0000_ ×10−5 91 Pa protactinium 1.4×10−12 92 U uranium 2.7×10−6 1.6×10−6 2.3×10−6 9.1 _0_ ×10−7 1.2 _00_ ×10−6 1.3 _00_ ×10−6 2.8 _00_ ×10−6 2.5 _00_ ×10−6 93 Np neptunium 94 Pu plutonium ==Urban soils== The established abundances of chemical elements in urban soils can be considered a geochemical (ecological and geochemical) characteristic, the accumulated impact of technogenic and natural processes at the beginning of the 21st century. AM1* is a semiempirical molecular orbital technique in computational chemistry. But, other elements have been parameterized using an additional set of d-orbitals in the basis set and with two-center core–core parameters, rather than the Gaussian functions used to modify the core–core potential in AM1. Element S1 Y1 Y2 01 H hydrogen 2.8×104 2.8×104* 2.79×104 02 He helium 2.7×103 2.7×103* 2.72×103 03 Li lithium 4.0×10−7 5.7×10−5 5.71×10−5 (9.2%) 04 Be beryllium 4.0×10−7 7.0×10−7 7.30×10−7 (9.5%) 05 B boron 1.1×10−5 2.1×10−5 2.12×10−5 (10%) 06 C carbon 1.0×101 1.0×101* 1.01×101 07 N nitrogen 3.1×100 3.1×100* 3.13×100 08 O oxygen 2.4×101 2.4×101* 2.38×101 (10%) 09 F fluorine about 1.0×10−3 8.5×10−4 8.43×10−4 (15%) 10 Ne neon 3.0×100 3.0×100* 3.44×100 (14%) 11 Na sodium 6.0×10−2 5.7×10−2 5.74×10−2 (7.1%) 12 Mg magnesium 1.0×100 1.1×100 1.074×100 (3.8%) 13 Al aluminium 8.3×10−2 8.5×10−2 8.49×10−2 (3.6%) 14 Si silicon 1.0×100 1.0×100 1.0×100 (4.4%) 15 P phosphorus 8.0×10−3 1.0×10−2 1.04×10−2 (10%) 16 S sulfur 4.5×10−1 5.2×10−1 5.15×10−1 (13%) 17 Cl chlorine about 9.0×10−3 5.2×10−3 5.24×10−3 (15%) 18 Ar argon 1.0×10−1* 1.0×10−1* 1.01×10−1 (6%) 19 K potassium 3.7×10−3 3.8×10−3 3.77×10−3 (7.7%) 20 Ca calcium 6.4×10−2 6.1×10−2 6.11×10−2 (7.1%) 21 Sc scandium 3.5×10−5 3.4×10−5 3.42×10−5 (8.6%) 22 Ti titanium 2.7×10−3 2.4×10−3 2.40×10−3 (5.0%) 23 V vanadium 2.8×10−4 2.9×10−4 2.93×10−4 (5.1%) 24 Cr chromium 1.3×10−2 1.3×10−2 1.35×10−2 (7.6%) 25 Mn manganese 6.9×10−3 9.5×10−3 9.55×10−3 (9.6%) 26 Fe iron 9.0×10−1 9.0×10−1 9.00×10−1 (2.7%) 27 Co cobalt 2.3×10−3 2.3×10−3 2.25×10−3 (6.6%) 28 Ni nickel 5.0×10−2 5.0×10−2 4.93×10−2 (5.1%) 29 Cu copper 4.5×10−4 5.2×10−4 5.22×10−4 (11%) 30 Zn zinc 1.1×10−3 1.3×10−3 1.26×10−3 (4.4%) 31 Ga gallium 2.1×10−5 3.8×10−5 3.78×10−5 (6.9%) 32 Ge germanium 7.2×10−5 1.2×10−4 1.19×10−4 (9.6%) 33 As arsenic 6.6×10−6 6.56×10−6 (12%) 34 Se selenium 6.3×10−5 6.21×10−5 (6.4%) 35 Br bromine 1.2×10−5 1.18×10−5 (19%) 36 Kr krypton 4.8×10−5 4.50×10−5 (18%) 37 Rb rubidium 1.1×10−5 7.0×10−6 7.09×10−6 (6.6%) 38 Sr strontium 2.2×10−5 2.4×10−5 2.35×10−5 (8.1%) 39 Y yttrium 4.9×10−6 4.6×10−6 4.64×10−6 (6.0%) 40 Zr zirconium 1.12×10−5 1.14×10−5 1.14×10−5 (6.4%) 41 Nb niobium 7.0×10−7 7.0×10−7 6.98×10−7 (1.4%) 42 Mo molybdenum 2.3×10−6 2.6×10−6 2.55×10−6 (5.5%) 43 Tc technetium 44 Ru ruthenium 1.9×10−6 1.9×10−6 1.86×10−6 (5.4%) 45 Rh rhodium 4.0×10−7 3.4×10−7 3.44×10−7 (8%) 46 Pd palladium 1.4×10−6 1.4×10−6 1.39×10−6 (6.6%) 47 Ag silver about 2.0×10−7 4.9×10−7 4.86×10−7 (2.9%) 48 Cd cadmium 2.0×10−6 1.6×10−6 1.61×10−6 (6.5%) 49 In indium about 1.3×10−6 1.9×10−7 1.84×10−7 (6.4%) 50 Sn tin about 3.0×10−6 3.9×10−6 3.82×10−6 (9.4%) 51 Sb antimony about 3.0×10−7 3.1×10−7 3.09×10−7 (18%) 52 Te tellurium 4.9×10−6 4.81×10−6 (10%) 53 I iodine 9.0×10−7 9.00×10−7 (21%) 54 Xe xenon 4.8×10−6 4.70×10−6 (20%) 55 Cs caesium 3.7×10−7 3.72×10−7 (5.6%) 56 Ba barium 3.8×10−6 4.5×10−6 4.49×10−6 (6.3%) 57 La lanthanum 5.0×10−7 4.4×10−7 4.46×10−7 (2.0%) 58 Ce cerium 1.0×10−6 1.1×10−6 1.136×10−6 (1.7%) 59 Pr praseodymium 1.4×10−7 1.7×10−7 1.669×10−7 (2.4%) 60 Nd neodymium 9.0×10−7 8.3×10−7 8.279×10−7 (1.3%) 61 Pm promethium 62 Sm samarium 3.0×10−7 2.6×10−7 2.582×10−7 (1.3%) 63 Eu europium 9.0×10−8 9.7×10−8 9.73×10−8 (1.6%) 64 Gd gadolinium 3.7×10−7 3.3×10−7 3.30×10−7 (1.4%) 65 Tb terbium about 2.0×10−8 6.0×10−8 6.03×10−8 (2.2%) 66 Dy dysprosium 3.5×10−7 4.0×10−7 3.942×10−7 (1.4%) 67 Ho holmium about 5.0×10−8 8.9×10−8 8.89×10−8 (2.4%) 68 Er erbium 2.4×10−7 2.5×10−7 2.508×10−7 (1.3%) 69 Tm thulium about 3.0×10−8 3.8×10−8 3.78×10−8 (2.3%) 70 Yb ytterbium 3.4×10−7 2.5×10−7 2.479×10−7 (1.6%) 71 Lu lutetium about 1.5×10−7 3.7×10−8 3.67×10−8 (1.3%) 72 Hf hafnium 2.1×10−7 1.5×10−7 1.54×10−7 (1.9%) 73 Ta tantalum 3.8×10−8 2.07×10−8 (1.8%) 74 W tungsten about 3.6×10−7 1.3×10−7 1.33×10−7 (5.1%) 75 Re rhenium 5.0×10−8 5.17×10−8 (9.4%) 76 Os osmium 8.0×10−7 6.7×10−7 6.75×10−7 (6.3%) 77 Ir iridium 6.0×10−7 6.6×10−7 6.61×10−7 (6.1%) 78 Pt platinum about 1.8×10−6 1.34×10−6 1.34×10−6 (7.4%) 79 Au gold about 3.0×10−7 1.9×10−7 1.87×10−7 (15%) 80 Hg mercury 3.4×10−7 3.40×10−7 (12%) 81 Tl thallium about 2.0×10−7 1.9×10−7 1.84×10−7 (9.4%) 82 Pb lead 2.0×10−6 3.1×10−6 3.15×10−6 (7.8%) 83 Bi bismuth 1.4×10−7 1.44×10−7 (8.2%) 84 Po polonium 85 At astatine 86 Rn radon 87 Fr francium 88 Ra radium 89 Ac actinium 90 Th thorium 5.0×10−8 4.5×10−8 3.35×10−8 (5.7%) 91 Pa protactinium 92 U uranium 1.8×10−8 9.00×10−9 (8.4%) 93 Np neptunium 94 Pu plutonium ==See also== *Chemical elements data references ==Notes== Due to the estimate nature of these values, no single recommendations are given. ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). AM1* parameters are now available for H, C, N, O, F, Al, Si, P, S, Cl, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Br, Zr, Mo, Pd, Ag, I and Au. Additionally, for transition metal-hydrogen interactions, a distance dependent term is used to calculate core-core potentials rather than the constant term. :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? AM1* is implemented in VAMP 10.0 Clark T, Alex A, Beck B, Chandrasekhar J, Gedeck P, Horn AHC, Hutter M, Martin B, Rauhut G, Sauer W, Schindler T, Steinke T (2005) Computer- Chemie-Centrum. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? (The average density of sea water in the surface is 1.025 kg/L) Element W1 W2 01 H hydrogen 1.08×10−1 1.1×10−1 02 He helium 7×10−12 7.2×10−12 03 Li lithium 1.8×10−7 1.7×10−7 04 Be beryllium 5.6×10−12 6×10−13 05 B boron 4.44×10−6 4.4×10−6 06 C carbon 2.8×10−5 2.8×10−5 07 N nitrogen 5×10−7 1.6×10−5 08 O oxygen 8.57×10−1 8.8×10−1 09 F fluorine 1.3×10−6 1.3×10−6 10 Ne neon 1.2×10−10 1.2×10−10 11 Na sodium 1.08×10−2 1.1×10−2 12 Mg magnesium 1.29×10−3 1.3×10−3 13 Al aluminium 2×10−9 1×10−9 14 Si silicon 2.2×10−6 2.9×10−6 15 P phosphorus 6×10−8 8.8×10−8 16 S sulfur 9.05×10−4 9.0×10−4 17 Cl chlorine 1.94×10−2 1.9×10−2 18 Ar argon 4.5×10−7 4.5×10−7 19 K potassium 3.99×10−4 3.9×10−4 20 Ca calcium 4.12×10−4 4.1×10−4 21 Sc scandium 6×10−13 < 4×10−12 22 Ti titanium 1×10−9 1×10−9 23 V vanadium 2.5×10−9 1.9×10−9 24 Cr chromium 3×10−10 2×10−10 25 Mn manganese 2×10−10 1.9×10−9 26 Fe iron 2×10−9 3.4×10−9 27 Co cobalt 2×10−11 3.9×10−10 28 Ni nickel 5.6×10−10 6.6×10−9 29 Cu copper 2.5×10−10 2.3×10−8 30 Zn zinc 4.9×10−9 1.1×10−8 31 Ga gallium 3×10−11 3×10−11 32 Ge germanium 5×10−11 6×10−11 33 As arsenic 3.7×10−9 2.6×10−9 34 Se selenium 2×10−10 9.0×10−11 35 Br bromine 6.73×10−5 6.7×10−5 36 Kr krypton 2.1×10−10 2.1×10−10 37 Rb rubidium 1.2×10−7 1.2×10−7 38 Sr strontium 7.9×10−6 8.1×10−6 39 Y yttrium 1.3×10−11 1.3×10−12 40 Zr zirconium 3×10−11 2.6×10−11 41 Nb niobium 1×10−11 1.5×10−11 42 Mo molybdenum 1×10−8 1.0×10−8 43 Tc technetium 44 Ru ruthenium 7×10−13 45 Rh rhodium 46 Pd palladium 47 Ag silver 4×10−11 2.8×10−10 48 Cd cadmium 1.1×10−10 1.1×10−10 49 In indium 2×10−8 50 Sn tin 4×10−12 8.1×10−10 51 Sb antimony 2.4×10−10 3.3×10−10 52 Te tellurium 53 I iodine 6×10−8 6.4×10−8 54 Xe xenon 5×10−11 4.7×10−11 55 Cs caesium 3×10−10 3.0×10−10 56 Ba barium 1.3×10−8 2.1×10−8 57 La lanthanum 3.4×10−12 3.4×10−12 58 Ce cerium 1.2×10−12 1.2×10−12 59 Pr praseodymium 6.4×10−13 6.4×10−13 60 Nd neodymium 2.8×10−12 2.8×10−12 61 Pm promethium 62 Sm samarium 4.5×10−13 4.5×10−13 63 Eu europium 1.3×10−13 1.3×10−13 64 Gd gadolinium 7×10−13 7.0×10−13 65 Tb terbium 1.4×10−13 1.4×10−12 66 Dy dysprosium 9.1×10−13 9.1×10−13 67 Ho holmium 2.2×10−13 2.2×10−13 68 Er erbium 8.7×10−13 8.7×10−12 69 Tm thulium 1.7×10−13 1.7×10−13 70 Yb ytterbium 8.2×10−13 8.2×10−13 71 Lu lutetium 1.5×10−13 1.5×10−13 72 Hf hafnium 7×10−12 < 8×10−12 73 Ta tantalum 2×10−12 < 2.5×10−12 74 W tungsten 1×10−10 < 1×10−12 75 Re rhenium 4×10−12 76 Os osmium 77 Ir iridium 78 Pt platinum 79 Au gold 4×10−12 1.1×10−11 80 Hg mercury 3×10−11 1.5×10−10 81 Tl thallium 1.9×10−11 82 Pb lead 3×10−11 3×10−11 83 Bi bismuth 2×10−11 2×10−11 84 Po polonium 1.5×10−20 85 At astatine 86 Rn radon 6×10−22 87 Fr francium 88 Ra radium 8.9×10−17 89 Ac actinium 90 Th thorium 1×10−12 1.5×10−12 91 Pa protactinium 5×10−17 92 U uranium 3.2×10−9 3.3×10−9 93 Np neptunium 94 Pu plutonium ==Sun and solar system== *S1 — Sun: Kaye & Laby *Y1 — Solar system: Kaye & Laby *Y2 — Solar system: Ahrens, with uncertainty s (%) Atom mole fraction relative to silicon = 1. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Element Atomic number Abundance in urban soils Ag 47 0.37 Al 13 38200 As 33 15.9 B 5 45 Ba 56 853.12 Be 4 3.3 Bi 83 1.12 C 6 45100 Ca 20 53800 Cd 48 0.9 Cl 17 285 Co 27 14.1 Cr 24 80 Cs 55 5.0 Cu 29 39 Fe 26 22300 Ga 31 16.2 Ge 32 1.8 H 1 15000 Hg 80 0.88 K 19 13400 La 57 34 Li 3 49.5 Mg 12 7900 Mn 25 729 Mo 42 2.4 N 7 10000 Na 11 5800 Nb 41 15.7 Ni 28 33 O 8 490000 P 15 1200 Pb 82 54.5 Rb 37 58 S 16 1200 Sb 51 1.0 Sc 21 9.4 Si 14 289000 Sn 50 6.8 Sr 38 458 Ta 73 1.5 Ti 22 4758 Tl 81 1.1 V 23 104.9 W 74 2.9 Y 39 23.4 Yb 70 2.4 Zn 30 158 Zr 40 255.6 ==Sea water== *W1 — CRC Handbook *W2 — Kaye & Laby Mass per volume fraction, in kg/L.

-2

3.00

1.2

-59.24

2598960

D

Given that $D_e=4.75 \mathrm{eV}$ and $R_e=0.741 Å$ for the ground electronic state of $\mathrm{H}_2$, find $U\left(R_e\right)$ for this state.

The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . The International Conference on Electronic Properties of Two-Dimensional Systems (EP2DS), is a biannual event, where over hundred scientists meet for the presentation of new developments on the special field of two-dimensional electron systems in semiconductors. * After dividing by the number of electrons, the standard potential E° is related to the standard Gibbs free energy of formation ΔGf° by: E = \frac{\sum \Delta G_\text{left}-\sum \Delta G_\text{right}}{F} where F is the Faraday constant. E2D may refer to: * Estradiol decanoate * trans-4,5-Epoxy-(E)-2-decenal * E2-D the fourth variant of the Northrop Grumman E-2 Hawkeye reconnaissance plane. ER2, ER-2, ER II etc. may refer to: * Elizabeth II's royal cypher E II R (sometimes written as ER II) for Elizabeth II Regina (Elizabeth II, Queen) *"ER2" (Kanjani Eight song), a single by Japanese boy band Kanjani Eight *ER2 electric trainset, an electric passenger railcar built in Latvia and Russia from 1962 to 1984 *NASA ER-2, "Earth Resources 2", an American very high- altitude civilian atmospheric research fixed-wing aircraft based on the Lockheed U-2 reconnaissance aircraft In the weakened potential at the surface, new electronic states can be formed, so called surface states. ==Origin at condensed matter interfaces== thumbnail|350px|Figure 1. Te (aq) + 2 + 2 (s) + 4 1.02 2 . Since the potential is periodic deep inside the crystal, the electronic wave functions must be Bloch waves here. This means that the US Government's use of E85 is effectively doubled as of August 8, 2005 with the signing into law of the Energy Policy Act of 2005. Surface states are electronic states found at the surface of materials. E85 is an abbreviation for an ethanol fuel blend of between 51% and 83% denatured ethanol fuel and gasoline or other hydrocarbon (HC) by volume. ==Availability== All data August 2014 from the Department of Energy, e85prices.com, and E85refueling.com.http://www.e85refueling.com Links go to each state's list of stations; see notes below for caveats. It can be shown that the energies of these states all lie within the band gap. The investigation tries to understand electronic states in ideal crystals of finite size based on the mathematical theory of periodic differential equations. This theory provides some fundamental new understandings of those electronic states, including surface states. The Shockley states are then found as solutions to the one- dimensional single electron Schrödinger equation : \begin{align} \left[-\frac{\hbar^2}{2m}\frac{d^2}{dz^2}+V(z)\right]\Psi(z) &=& E\Psi(z), \end{align} with the periodic potential : \begin{align} V(z)=\left\\{ \begin{array}{cc} P\delta(z+la),& \textrm{for}\quad z<0 \\\ V_0,&\textrm{for} \quad z>0 \end{array}\right., \end{align} where l is an integer, and P is the normalization factor. For example: : \+ Cu(s) ( = +0.520 V) Cu + 2 Cu(s) ( = +0.337 V) Cu + ( = +0.159 V) :Calculating the potential using Gibbs free energy ( = 2 – ) gives the potential for as 0.154 V, not the experimental value of 0.159 V. : __TOC__ ==Table of standard electrode potentials== Legend: (s) - solid; (l) - liquid; (g) - gas; (aq) - aqueous (default for all charged species); (Hg) - amalgam; bold - water electrolysis equations. Shockley states are thus states that arise due to the change in the electron potential associated solely with the crystal termination. :The Nernst equation will then give potentials at concentrations, pressures, and temperatures other than standard. All of the reactions should be divided by the stoichiometric coefficient for the electron to get the corresponding corrected reaction equation. For example, the equation Fe + 2 Fe(s) (–0.44 V) means that it requires 2 × 0.44 eV = 0.88 eV of energy to be absorbed (hence the minus sign) in order to create one neutral atom of Fe(s) from one Fe ion and two electrons, or 0.44 eV per electron, which is 0.44 J/C of electrons, which is 0.44 V. The energy levels of such states are expected to significantly shift from the bulk values. The nearly free electron approximation can be used to derive the basic properties of surface states for narrow gap semiconductors.

-8

0

'-5.0'

0.14

-31.95

E

For $\mathrm{NaCl}, R_e=2.36 Å$. The ionization energy of $\mathrm{Na}$ is $5.14 \mathrm{eV}$, and the electron affinity of $\mathrm{Cl}$ is $3.61 \mathrm{eV}$. Use the simple model of $\mathrm{NaCl}$ as a pair of spherical ions in contact to estimate $D_e$. [One debye (D) is $3.33564 \times 10^{-30} \mathrm{C} \mathrm{m}$.]

Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. right|thumb|The Madelung constant being calculated for the NaCl ion labeled 0 in the expanding spheres method. For example, the inter-ionic distance in RbI is 356 pm, giving 142 pm for the ionic radius of Rb+. If the distances are normalized to the nearest neighbor distance , the potential may be written :V_i = \frac{e}{4 \pi \varepsilon_0 r_0 } \sum_{j} \frac{z_j r_0}{r_{ij}} = \frac{e}{4 \pi \varepsilon_0 r_0 } M_i with being the (dimensionless) Madelung constant of the th ion :M_i = \sum_{j} \frac{z_j}{r_{ij}/r_0}. Comparison between observed and calculated ion separations (in pm) MX Observed Soft-sphere model LiCl 257.0 257.2 LiBr 275.1 274.4 NaCl 282.0 281.9 NaBr 298.7 298.2 In the soft-sphere model, k has a value between 1 and 2. The Debye–Hückel limiting law enables one to determine the activity coefficient of an ion in a dilute solution of known ionic strength. In this way values for the radii of 8 ions were determined. Pauling used effective nuclear charge to proportion the distance between ions into anionic and a cationic radii.Pauling, L. (1960). For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). One approach to improving the calculated accuracy is to model ions as "soft spheres" that overlap in the crystal. In the next step, D&H; assume that there is a certain radius a_i, beyond which no ions in the atmosphere may approach the (charge) center of the singled out ion. The electrostatic energy of the ion at site then is the product of its charge with the potential acting at its site :E_{el,i} = z_ieV_i = \frac{e^2}{4 \pi \varepsilon_0 r_0 } z_i M_i. D&H; say that this approximation holds at large distances between ions, which is the same as saying that the concentration is low. _Effective_ ionic radii in pm of elements as a function of ionic charge and spin (ls = low spin, hs = high spin). All compounds crystallize in the NaCl structure. thumb|300 px|Relative radii of atoms and ions. For ions on lower-symmetry sites significant deviations of their electron density from a spherical shape may occur. These tables list values of molar ionization energies, measured in kJ⋅mol−1. The iodide ions nearly touch (but don't quite), indicating that Landé's assumption is fairly good. To be consistent with Pauling's radii, Shannon has used a value of rion(O2−) = 140 pm; data using that value are referred to as "effective" ionic radii. For table salt in 0.01 M solution at 25 °C, a typical value of (\kappa a)^2 is 0.0005636, while a typical value of Z_0 is 7.017, highlighting the fact that, in low concentrations, (\kappa a)^2 is a target for a zero order of magnitude approximation such as perturbation analysis. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy. == Formal expression == The Madelung constant allows for the calculation of the electric potential of all ions of the lattice felt by the ion at position :V_i = \frac{e}{4 \pi \varepsilon_0 } \sum_{j eq i} \frac{z_j}{r_{ij}}\,\\! where r_{ij} = |r_i-r_j| is the distance between the th and the th ion. Note also how the fluorite structure being intermediate between the caesium chloride and sphalerite structures is reflected in the Madelung constants. == Formula == A fast converging formula for the Madelung constant of NaCl is :12 \, \pi \sum_{m, n \geq 1, \, \mathrm{odd}} \operatorname{sech}^2\left(\frac{\pi}{2}(m^2+n^2)^{1/2}\right) == Generalization == It is assumed for the calculation of Madelung constants that an ion's charge density may be approximated by a point charge.

4.56

3.0

0.1353

30

3.61

A

Find the number of CSFs in a full CI calculation of $\mathrm{CH}_2 \mathrm{SiHF}$ using a 6-31G** basis set.

The molecular formula C6H8S (molar mass: 112.19 g/mol, exact mass: 112.0347 u) may refer to: * 2,3-Dihydrothiepine * 2,7-Dihydrothiepine * 2,5-Dimethylthiophene Category:Molecular formulas Caesium fluoride or cesium fluoride is an inorganic compound with the formula CsF and it is a hygroscopic white salt. Caltech Intermediate Form (CIF) is a file format for describing integrated circuits. The molecular formula C6H6O2S (molar mass: 142.18 g/mol, exact mass: 142.0089 u) may refer to: * 3,4-Ethylenedioxythiophene (EDOT) * Phenylsulfinic acid * 3-Thiophene acetic acid * Thiophene-2-acetic acid All numbers in CIF are integers that refer to centimicrons of distance, unless subroutine scaling is specified (described later). CsF reaches a vapor pressure of 1 kilopascal at 825 °C, 10 kPa at 999 °C, and 100 kPa at 1249 °C. The molecular formula C8H6S (molar mass: 134.20 g/mol, exact mass: 134.0190 u) may refer to: *Benzo[c]thiophene *Benzothiophene Category:Molecular formulas The molecular formula C3H2F6O (molar mass: 168.038 g/mol, exact mass: 168.0010 u) may refer to: * Desflurane * Hexafluoro-2-propanol (HFIP) CsF is an alternative to tetra-n-butylammonium fluoride (TBAF) and TAS-fluoride (TASF). ===As a base=== As with other soluble fluorides, CsF is moderately basic, because HF is a weak acid. Extensions to CIF can be done with the numeric statements `0` through `9`. CsF gives higher yields in Knoevenagel condensation reactions than KF or NaF. ===Formation of Cs-F bonds=== Caesium fluoride serves as a source of fluoride in organofluorine chemistry. CsF chains with a thickness as small as one or two atoms can be grown inside carbon nanotubes. ==Structure== Caesium fluoride has the halite structure, which means that the Cs+ and F− pack in a cubic closest packed array as do Na+ and Cl− in sodium chloride. ==Applications in organic synthesis== Being highly dissociated, CsF is a more reactive source of fluoride than related salts. The reaction is shown below: :Cs2CO3 \+ 2 HF → 2 CsF + H2O + CO2 CsF is more soluble than sodium fluoride or potassium fluoride in organic solvents. FIGURE B.5 Typical user extensions to CIF. The final statement in a CIF file is the `END` statement (or the letter `E`). The reaction is shown below: :CsOH + HF → CsF + H2O Using the same reaction, another way to create caesium fluoride is to treat caesium carbonate (Cs2CO3) with hydrofluoric acid and again, the resulting salt can then be purified by recrystallization. Solutions of caesium fluoride in THF or DMF attack a wide variety of organosilicon compounds to produce an organosilicon fluoride and a carbanion, which can then react with electrophiles, for example: :500px ==Precautions== Like other soluble fluorides, CsF is moderately toxic.MSDS Listing for cesium fluoride . www.hazard.com . Similarly to potassium fluoride, CsF reacts with hexafluoroacetone to form a stable perfluoroalkoxide salt. FIGURE B.1 CIF layer names for MOS processes. CIF provides a limited set of graphics primitives that are useful for describing the two-dimensional shapes on the different layers of a chip. Note that the magnitude of this rotation vector has no meaning. thumb|333px|right|FIGURE B.2 A sample CIF "wire" statement. Caesium also has the highest electropositivity of all known elements and fluorine has the highest electronegativity of all known elements. ==Synthesis and properties== Caesium fluoride can be prepared by the reaction of caesium hydroxide (CsOH) with hydrofluoric acid (HF) and the resulting salt can then be purified by recrystallization.

1.86

1855

0.686

62.8318530718

93.4

A

Calculate the ratio of the electrical and gravitational forces between a proton and an electron.

However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. A resolution came in 2019, when two different studies, using different techniques involving the Lamb shift of the electron in hydrogen, and electron–proton scattering, found the radius of the proton to be 0.833 fm, with an uncertainty of ±0.010 fm, and 0.831 fm. Together with the mass-to-charge ratio, the electron mass was determined with reasonable precision. Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. Walter Kaufmann's measurement of the electron charge-to- mass ratio for different velocities of the electron. In January 2013, an updated value for the charge radius of a proton——was published. Since the ratio doesn't vary for resting electrons, the data points should be on a single horizontal line (see Fig. 6). Historically the proton charge radius was measured by two independent methods, which converged to a value of about 0.877 femtometres (1 fm = 10−15 m). The proton radius puzzle is an unanswered problem in physics relating to the size of the proton. The result is again ~5% smaller than the previously-accepted proton radius. Their measurement of the root-mean-square charge radius of a proton is ", which differs by 5.0 standard deviations from the CODATA value of ". Because protons are not fundamental particles, they possess a measurable size; the root mean square charge radius of a proton is about 0.84–0.87 fm ( = ). The radius of the proton is linked to the form factor and momentum-transfer cross section. However, in such an association with an electron, the character of the bound proton is not changed, and it remains a proton. Physical parameter 1H 16O relative atomic mass of the XZ+ ion relative atomic mass of the Z electrons correction for the binding energy relative atomic mass of the neutral atom The principle can be shown by the determination of the electron relative atomic mass by Farnham et al. at the University of Washington (1995). The internationally accepted value of a proton's charge radius is . It is likely future experiments will be able to both explain and settle the proton radius puzzle. === 2022 analysis === A re-analysis of experimental data, published in February 2022, found a result consistent with the smaller value of approximately 0.84 fm. ==Footnotes== ==References== Category:2010 in science Category:2019 in science Category:Proton For his final analysis, Bucherer recalculated the measured values of five runs with Lorentz's and Abraham's formulas respectively, in order to obtain the charge-to-mass ratio as if the electrons were at rest. The proton radius was a puzzle as of 2017. Another recent paper has pointed out how a simple, yet theory motivated change to previous fits will also give the smaller radius. === 2019 measurements === In September 2019, Bezginov et al. reported the remeasurement of the proton's charge radius for electronic hydrogen and found a result consistent with Pohl's value for muonic hydrogen. This value is based on measurements involving a proton and an electron (namely, electron scattering measurements and complex calculation involving scattering cross section based on Rosenbluth equation for momentum-transfer cross section), and studies of the atomic energy levels of hydrogen and deuterium. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton-to- electron mass ratio).

0

14.44

479.0

2

22

D

A one-particle, one-dimensional system has the state function $$ \Psi=(\sin a t)\left(2 / \pi c^2\right)^{1 / 4} e^{-x^2 / c^2}+(\cos a t)\left(32 / \pi c^6\right)^{1 / 4} x e^{-x^2 / c^2} $$ where $a$ is a constant and $c=2.000 Å$. If the particle's position is measured at $t=0$, estimate the probability that the result will lie between $2.000 Å$ and $2.001 Å$.

Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. A quantum particle is in a bound state if it is never found “too far away from any finite region R\subseteq X”, i.e. using a wavefunction representation, \begin{align} 0 &= \lim_{R\to\infty}{\mathbb{P}(\text{particle measured inside }X\setminus R)} \\\ &= \lim_{R\to\infty}{\int_{X\setminus R}|\psi(x)|^2\,d\mu(x)} \end{align} Consequently, \int_X{|\psi(x)|^{2}\,d\mu(x)} is finite. If a state has energy E < \max{\left(\lim_{x\to\infty}{V(x)}, \lim_{x\to-\infty}{V(x)}\right)}, then the wavefunction satisfies, for some X > 0 :\frac{\psi^{\prime\prime}}{\psi}=\frac{2m}{\hbar^2}(V(x)-E) > 0\text{ for }x > X so that is exponentially suppressed at large . Therefore, it became necessary to formulate clearly the difference between the state of something that is uncertain in the way just described, such as an electron in a probability cloud, and the state of something having a definite value. As stated above, when the wavefunction collapses because the position of an electron has been determined, the electron's state becomes an "eigenstate of position", meaning that its position has a known value, an eigenvalue of the eigenstate of position. 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. *If the state evolution of "moves this wave package constantly to the right", e.g. if [t-1,t+1] \in \mathrm{Supp}(\rho(t)) for all t \geq 0, then is not bound state with respect to position. Likewise, it is impossible to determine the exact location of that particle once its momentum has been measured at a particular instant. thumb|CEP concept and hit probability. 0.2% outside the outmost circle. In probability theory, the spectral expansion solution method is a technique for computing the stationary probability distribution of a continuous-time Markov chain whose state space is a semi-infinite lattice strip. In quantum physics, a bound state is a quantum state of a particle subject to a potential such that the particle has a tendency to remain localized in one or more regions of space. thumb|Graph of f(x) = e^x (blue) with its linear approximation P_1(x) = 1 + x (red) at a = 0. To obtain g(0), the following limit is applied, g(0) = \lim_{a \to 0} \frac{1}{a} \int\limits_{- \infty}^{\infty} dx\ g(x) \mathrm{rect}(\frac{x}{a}) and this can be written in terms of the Dirac delta function as, g(0) = \int\limits_{- \infty}^{\infty} dx\ g(x) \delta (x).The Fourier transform of the Dirac delta function \delta (t) is \delta (f) = \int_{-\infty}^\infty \delta (t) \cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \frac{1}{a} \int_{-\infty}^\infty \mathrm{rect}(\frac{t}{a})\cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \mathrm{sinc}{(a f)}. where the sinc function here is the normalized sinc function. Because the first zero of the sinc function is at f = 1 / a and a goes to infinity, the Fourier transform of \delta (t) is \delta (f) = 1, means that the frequency spectrum of the Dirac delta function is infinitely broad. The approximation error is the gap between the curves, and it increases for x values further from 0. In order for the first bound state to exist at all, D\gtrsim 0.8. Schrödinger's wave equation gives wavefunction solutions, the squares of which are probabilities of where the electron might be, just as Heisenberg's probability distribution does. The characteristic function is \varphi(k) = \frac{\sin(k/2)}{k/2}, and its moment-generating function is M(k) = \frac{\sinh(k/2)}{k/2}, where \sinh(t) is the hyperbolic sine function. ==Rational approximation== The pulse function may also be expressed as a limit of a rational function: \Pi(t) = \lim_{n\rightarrow \infty, n\in \mathbb(Z)} \frac{1}{(2t)^{2n}+1}. ===Demonstration of validity=== First, we consider the case where |t|<\frac{1}{2}. Its periodic version is called a rectangular wave. ==History== The rect function has been introduced by Woodward in as an ideal cutout operator, together with the sinc function as an ideal interpolation operator, and their counter operations which are sampling (comb operator) and replicating (rep operator), respectively. ==Relation to the boxcar function== The rectangular function is a special case of the more general boxcar function: \operatorname{rect}\left(\frac{t-X}{Y} \right) = H(t - (X - Y/2)) - H(t - (X + Y/2)) = H(t - X + Y/2) - H(t - X - Y/2) where H(x) is the Heaviside step function; the function is centered at X and has duration Y, from X-Y/2 to X+Y/2. ==Fourier transform of the rectangular function== thumb|400px|right|Plot of normalized \mathrm{sinc}(x) function (i.e. \mathrm{sinc}(\pi x)) with its spectral frequency components. If v e 0, the relative error is : \eta = \frac{\epsilon}{|v|} = \left| \frac{v-v_\text{approx}}{v} \right| = \left| 1 - \frac{v_\text{approx}}{v} \right|, and the percent error (an expression of the relative error) is :\delta = 100\%\times\eta = 100\%\times\left| \frac{v-v_\text{approx}}{v} \right|. The approximation error in a data value is the discrepancy between an exact value and some approximation to it.

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A one-particle, one-dimensional system has the state function $$ \Psi=(\sin a t)\left(2 / \pi c^2\right)^{1 / 4} e^{-x^2 / c^2}+(\cos a t)\left(32 / \pi c^6\right)^{1 / 4} x e^{-x^2 / c^2} $$ where $a$ is a constant and $c=2.000 Å$. If the particle's position is measured at $t=0$, estimate the probability that the result will lie between $2.000 Å$ and $2.001 Å$.

Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. A quantum particle is in a bound state if it is never found “too far away from any finite region R\subseteq X”, i.e. using a wavefunction representation, \begin{align} 0 &= \lim_{R\to\infty}{\mathbb{P}(\text{particle measured inside }X\setminus R)} \\\ &= \lim_{R\to\infty}{\int_{X\setminus R}|\psi(x)|^2\,d\mu(x)} \end{align} Consequently, \int_X{|\psi(x)|^{2}\,d\mu(x)} is finite. Initial attempts at approximating Chernoff's distribution via solving the heat equation, however, did not achieve satisfactory precision due to the nature of the boundary conditions. Groeneboom (1989) shows that : f_Z (z) \sim \frac{1}{2} \frac{4^{4/3} |z|}{\operatorname{Ai}' (\tilde{a}_1)} \exp \left( - \frac{2}{3} |z|^3 + 2^{1/3} \tilde{a}_1 |z| \right) \text{ as }z \rightarrow \infty where \tilde{a}_1 \approx -2.3381 is the largest zero of the Airy function Ai and where \operatorname{Ai}' (\tilde{a}_1 ) \approx 0.7022. If a state has energy E < \max{\left(\lim_{x\to\infty}{V(x)}, \lim_{x\to-\infty}{V(x)}\right)}, then the wavefunction satisfies, for some X > 0 :\frac{\psi^{\prime\prime}}{\psi}=\frac{2m}{\hbar^2}(V(x)-E) > 0\text{ for }x > X so that is exponentially suppressed at large . 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. Therefore, it became necessary to formulate clearly the difference between the state of something that is uncertain in the way just described, such as an electron in a probability cloud, and the state of something having a definite value. *If the state evolution of "moves this wave package constantly to the right", e.g. if [t-1,t+1] \in \mathrm{Supp}(\rho(t)) for all t \geq 0, then is not bound state with respect to position. However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. As stated above, when the wavefunction collapses because the position of an electron has been determined, the electron's state becomes an "eigenstate of position", meaning that its position has a known value, an eigenvalue of the eigenstate of position. In quantum physics, a bound state is a quantum state of a particle subject to a potential such that the particle has a tendency to remain localized in one or more regions of space. thumb|CEP concept and hit probability. 0.2% outside the outmost circle. If : V(a,c) = \underset{s \in \mathbf{R}}{\operatorname{argmax}} \ (W(s) - c(s-a)^2), then V(0, c) has density : f_c(t) = \frac{1}{2} g_c(t) g_c(-t) where gc has Fourier transform given by : \hat{g}_c (s) = \frac{(2/c)^{1/3}}{\operatorname{Ai} (i (2c^2)^{-1/3} s)}, \ \ \ s \in \mathbf{R} and where Ai is the Airy function. Likewise, it is impossible to determine the exact location of that particle once its momentum has been measured at a particular instant. To obtain g(0), the following limit is applied, g(0) = \lim_{a \to 0} \frac{1}{a} \int\limits_{- \infty}^{\infty} dx\ g(x) \mathrm{rect}(\frac{x}{a}) and this can be written in terms of the Dirac delta function as, g(0) = \int\limits_{- \infty}^{\infty} dx\ g(x) \delta (x).The Fourier transform of the Dirac delta function \delta (t) is \delta (f) = \int_{-\infty}^\infty \delta (t) \cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \frac{1}{a} \int_{-\infty}^\infty \mathrm{rect}(\frac{t}{a})\cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \mathrm{sinc}{(a f)}. where the sinc function here is the normalized sinc function. In order for the first bound state to exist at all, D\gtrsim 0.8. thumb|Graph of f(x) = e^x (blue) with its linear approximation P_1(x) = 1 + x (red) at a = 0. Because the first zero of the sinc function is at f = 1 / a and a goes to infinity, the Fourier transform of \delta (t) is \delta (f) = 1, means that the frequency spectrum of the Dirac delta function is infinitely broad. The characteristic function is \varphi(k) = \frac{\sin(k/2)}{k/2}, and its moment-generating function is M(k) = \frac{\sinh(k/2)}{k/2}, where \sinh(t) is the hyperbolic sine function. ==Rational approximation== The pulse function may also be expressed as a limit of a rational function: \Pi(t) = \lim_{n\rightarrow \infty, n\in \mathbb(Z)} \frac{1}{(2t)^{2n}+1}. ===Demonstration of validity=== First, we consider the case where |t|<\frac{1}{2}. Schrödinger's wave equation gives wavefunction solutions, the squares of which are probabilities of where the electron might be, just as Heisenberg's probability distribution does. The approximation error is the gap between the curves, and it increases for x values further from 0. Its periodic version is called a rectangular wave. ==History== The rect function has been introduced by Woodward in as an ideal cutout operator, together with the sinc function as an ideal interpolation operator, and their counter operations which are sampling (comb operator) and replicating (rep operator), respectively. ==Relation to the boxcar function== The rectangular function is a special case of the more general boxcar function: \operatorname{rect}\left(\frac{t-X}{Y} \right) = H(t - (X - Y/2)) - H(t - (X + Y/2)) = H(t - X + Y/2) - H(t - X - Y/2) where H(x) is the Heaviside step function; the function is centered at X and has duration Y, from X-Y/2 to X+Y/2. ==Fourier transform of the rectangular function== thumb|400px|right|Plot of normalized \mathrm{sinc}(x) function (i.e. \mathrm{sinc}(\pi x)) with its spectral frequency components.

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The $J=2$ to 3 rotational transition in a certain diatomic molecule occurs at $126.4 \mathrm{GHz}$, where $1 \mathrm{GHz} \equiv 10^9 \mathrm{~Hz}$. Find the frequency of the $J=5$ to 6 absorption in this molecule.

As 1 GHz = 109 Hz, the numerical conversion can be expressed as :\tilde u / \text{cm}^{-1} \approx \frac{ u / \text{GHz}}{30}. ===Effect of vibration on rotation=== The population of vibrationally excited states follows a Boltzmann distribution, so low- frequency vibrational states are appreciably populated even at room temperatures. For the so-called R branch of the spectrum, J' = J + 1 so that there is simultaneous excitation of both vibration and rotation. However, since only integer values of J are allowed, the maximum line intensity is observed for a neighboring integer J. :J = \sqrt{\frac{kT}{2hcB}} - \frac{1}{2} The diagram at the right shows an intensity pattern roughly corresponding to the spectrum above it. ====Centrifugal distortion==== When a molecule rotates, the centrifugal force pulls the atoms apart. The energy difference between successive J terms in any of these triplets is about 2 cm−1 (60 GHz), with the single exception of J = 1←0 difference which is about 4 cm−1. For a linear molecule, analysis of the rotational spectrum provides values for the rotational constantThis article uses the molecular spectroscopist's convention of expressing the rotational constant B in cm−1. Selection rules dictate that during emission or absorption the rotational quantum number has to change by unity; i.e., \Delta J = J^{\prime} - J^{\prime\prime} = \pm 1 . Constants of Diatomic Molecules, by K. P. Huber and Gerhard Herzberg (Van nostrand Reinhold company, New York, 1979, ), is a classic comprehensive multidisciplinary reference text contains a critical compilation of available data for all diatomic molecules and ions known at the time of publication - over 900 diatomic species in all - including electronic energies, vibrational and rotational constants, and observed transitions. Under the rigid rotor model, the rotational energy levels, F(J), of the molecule can be expressed as, : F\left( J \right) = B J \left( J+1 \right) \qquad J = 0,1,2,... where B is the rotational constant of the molecule and is related to the moment of inertia of the molecule. J is the quantum number of the lower rotational state. The Schumann–Runge bands are a set of absorption bands of molecular oxygen that occur at wavelengths between 176 and 192.6 nanometres. Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase. This spectrum is also interesting because it shows clear evidence of Coriolis coupling in the asymmetric structure of the band. ===Linear molecules=== right|thumb|300px|Energy levels and line positions calculated in the rigid rotor approximation The rigid rotor is a good starting point from which to construct a model of a rotating molecule. The particular pattern of energy levels (and, hence, of transitions in the rotational spectrum) for a molecule is determined by its symmetry. The number of molecules in an excited state with quantum number J, relative to the number of molecules in the ground state, NJ/N0 is given by the Boltzmann distribution as :\frac{N_J}{N_0} = e^{-\frac{E_J}{kT}} = e^{-\frac {BhcJ(J+1)}{kT}}, where k is the Boltzmann constant and T the absolute temperature. The Rydberg–Klein–Rees method is a procedure used in the analysis of rotational-vibrational spectra of diatomic molecules to obtain a potential energy curve from the experimentally-known line positions. To account for this a centrifugal distortion correction term is added to the rotational energy levels of the diatomic molecule. In this approximation, the vibration-rotation wavenumbers of transitions are :\tilde u = \tilde u_\text{vib} + BJ(J + 1) - B'J'(J' + 1), where B and B' are rotational constants for the upper and lower vibrational state respectively, while J and J' are the rotational quantum numbers of the upper and lower levels. Adjacent J^{\prime\prime}{\leftarrow}J^{\prime} transitions are separated by 2B in the observed spectrum. This gives the transition wavenumbers as : \tilde u_{J^{\prime}\leftrightarrow J^{\prime\prime},K} = F\left( J^{\prime},K \right) - F\left( J^{\prime\prime},K \right) = 2 B \left( J^{\prime\prime} + 1 \right) \qquad J^{\prime\prime} = 0,1,2,... which is the same as in the case of a linear molecule. In a linear molecule the moment of inertia about an axis perpendicular to the molecular axis is unique, that is, I_B = I_C, I_A=0 , so : B = {h \over{8\pi^2cI_B}}= {h \over{8\pi^2cI_C}} For a diatomic molecule : I=\frac{m_1m_2}{m_1 +m_2}d^2 where m1 and m2 are the masses of the atoms and d is the distance between them. The first study of the microwave spectrum of a molecule () was performed by Cleeton & Williams in 1934. J band may refer to: * J band (infrared), an atmospheric transmission window centred on 1.25 μm * J band (JRC), radio frequency bands from 139.5 to 140.5 and 148 to 149 MHz * J band (NATO), a radio frequency band from 10 to 20 GHz

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Assume that the charge of the proton is distributed uniformly throughout the volume of a sphere of radius $10^{-13} \mathrm{~cm}$. Use perturbation theory to estimate the shift in the ground-state hydrogen-atom energy due to the finite proton size. The potential energy experienced by the electron when it has penetrated the nucleus and is at distance $r$ from the nuclear center is $-e Q / 4 \pi \varepsilon_0 r$, where $Q$ is the amount of proton charge within the sphere of radius $r$. The evaluation of the integral is simplified by noting that the exponential factor in $\psi$ is essentially equal to 1 within the nucleus.

Another recent paper has pointed out how a simple, yet theory motivated change to previous fits will also give the smaller radius. === 2019 measurements === In September 2019, Bezginov et al. reported the remeasurement of the proton's charge radius for electronic hydrogen and found a result consistent with Pohl's value for muonic hydrogen. Measurements of hydrogen's energy levels are now so precise that the accuracy of the proton radius is the limiting factor when comparing experimental results to theoretical calculations. His personal assumption is that past measurements have misgauged the Rydberg constant and that the current official proton size is inaccurate. ===Quantum chromodynamic calculation=== In a paper by Belushkin et al. (2007), including different constraints and perturbative quantum chromodynamics, a smaller proton radius than the then-accepted 0.877 femtometres was predicted. ===Proton radius extrapolation=== Papers from 2016 suggested that the problem was with the extrapolations that had typically been used to extract the proton radius from the electron scattering data though these explanation would require that there was also a problem with the atomic Lamb shift measurements. ===Data analysis method=== In one of the attempts to resolve the puzzle without new physics, Alarcón et al. (2018) of Jefferson Lab have proposed that a different technique to fit the experimental scattering data, in a theoretically as well as analytically justified manner, produces a proton charge radius from the existing electron scattering data that is consistent with the muonic hydrogen measurement. The result is again ~5% smaller than the previously-accepted proton radius. By measuring the energy required to excite hydrogen atoms from the 2S to the 2P state, the Rydberg constant could be calculated, and from this the proton radius inferred. For hydrogen, whose nucleus consists only of one proton, this indirectly measures the proton charge radius. The form of the potential, in terms of the distance r from the center of nucleus, is: V(r) = -\frac{V_0}{1+\exp({r-R\over a})} where V0 (having dimension of energy) represents the potential well depth, a is a length representing the "surface thickness" of the nucleus, and R = r_0 A^{1/3} is the nuclear radius where and A is the mass number. Consistent with the spectroscopy method, this produces a proton radius of about . ===2010 experiment=== In 2010, Pohl et al. published the results of an experiment relying on muonic hydrogen as opposed to normal hydrogen. Effectively, this approach attributes the cause of the proton radius puzzle to a failure to use a theoretically motivated function for the extraction of the proton charge radius from the experimental data. Any one of these constants can be written in terms of any of the others using the fine-structure constant \alpha : :r_{\mathrm{e}} = \alpha \frac{\lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0. ==Hydrogen atom and similar systems== The Bohr radius including the effect of reduced mass in the hydrogen atom is given by : \ a_0^* \ = \frac{m_\text{e}}{\mu}a_0 , where \mu = m_\text{e} m_\text{p} / (m_\text{e} + m_\text{p}) is the reduced mass of the electron–proton system (with m_\text{p} being the mass of proton). He also noted that in the asymptotic limit (far away from the nucleus), his approximate form coincides with the exact hydrogen-like wave function in the presence of a nuclear charge of Z-s and in the state with a principal quantum number n equal to his effective quantum number n*. this opinion is not yet universally held. ==Problem== Prior to 2010, the proton charge radius was measured using one of two methods: one relying on spectroscopy, and one relying on nuclear scattering. ===Spectroscopy method=== The spectroscopy method uses the energy levels of electrons orbiting the nucleus. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the "reduced" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). It is likely future experiments will be able to both explain and settle the proton radius puzzle. === 2022 analysis === A re-analysis of experimental data, published in February 2022, found a result consistent with the smaller value of approximately 0.84 fm. ==Footnotes== ==References== Category:2010 in science Category:2019 in science Category:Proton Historically the proton charge radius was measured by two independent methods, which converged to a value of about 0.877 femtometres (1 fm = 10−15 m). A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. Since it takes nearly as much energy to excite the hydrogen atom's electron from n = 1 to n = 3 (12.1 eV, via the Rydberg formula) as it does to ionize the hydrogen atom (13.6 eV), ionization is far more probable than excitation to the n = 3 level. thumb|right|300px|Woods–Saxon potential for , relative to V0 with a and R=4.6 fm The Woods–Saxon potential is a mean field potential for the nucleons (protons and neutrons) inside the atomic nucleus, which is used to describe approximately the forces applied on each nucleon, in the nuclear shell model for the structure of the nucleus. thumb|300px|Comparison of a wavefunction in the Coulomb potential of the nucleus (blue) to the one in the pseudopotential (red). This method produces a proton radius of about , with approximately 1% relative uncertainty. ===Nuclear scattering=== The nuclear method is similar to Rutherford's scattering experiments that established the existence of the nucleus. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon.

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An electron in a three-dimensional rectangular box with dimensions of $5.00 Å, 3.00 Å$, and $6.00 Å$ makes a radiative transition from the lowest-lying excited state to the ground state. Calculate the frequency of the photon emitted.

We may derive the two- photon Rabi frequency by returning to the equations \begin{align} i \dot{c}_1(t) &= \frac{\Omega_{1i} c_2}{2} e^{i\Delta t}\\\ i \dot{c}_i(t) &= \frac{\Omega^*_{1i} c_1}{2} e^{-i\Delta t} \end{align} which now describe excitation between the ground and intermediate states. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. The Rabi frequency is the frequency at which the probability amplitudes of two atomic energy levels fluctuate in an oscillating electromagnetic field. Photoexcitation is the production of an excited state of a quantum system by photon absorption. Transition frequency may refer to: *A measure of the high-frequency operating characteristics of a transistor, usually symbolized as *A characteristic of spectral lines *The frequency of the radiation associated with a transition between hyperfine structure energy states of an atom *Turnover frequency in enzymology In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. Also note that as the incident light frequency shifts further from the transition frequency, the amplitude of the Rabi oscillation decreases, as is illustrated by the dashed envelope in the above plot. == Two-Photon Rabi Frequency == Coherent Rabi oscillations may also be driven by two-photon transitions. Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. After excitation the atom may return to the ground state or a lower excited state, by emitting a photon with a characteristic energy. Approximately half the time, this cascade will include the n = 3 to n = 2 transition and the atom will emit H-alpha light. The excited state originates from the interaction between a photon and the quantum system. A two-photon transition is not the same as excitation from the ground to intermediate state, and then out of the intermediate state to the excited state. In this case we consider a system with three atomic energy levels, |1\rangle , |i\rangle , and |2\rangle , where |i\rangle represents a so-called intermediate state with corresponding frequency \omega_i , and an electromagnetic field with two frequency components: \hat{V}(t) = e\mathbf{r} \cdot \mathbf{E}_{L1} \cos(\omega_{L1} t) + e\mathbf{r} \cdot \mathbf{E}_{L2} \cos(\omega_{L2} t) Now, \omega_i may be much greater than both \omega_1 and \omega_2 , or \omega_2 > \omega_i > \omega_1 , as illustrated in the figure on the right. thumb|Two photon excitation schema. \omega_i >> \omega_2 > \omega_1 is shown on the left, while \omega_2 > \omega_i > \omega_1 is shown on the right. In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. By giving the atom additional energy (for example, by absorption of a photon of an appropriate energy), the electron moves into an excited state (one with one or more quantum numbers greater than the minimum possible). Emission of photons from atoms in various excited states leads to an electromagnetic spectrum showing a series of characteristic emission lines (including, in the case of the hydrogen atom, the Lyman, Balmer, Paschen and Brackett series). The Rabi frequency is a semiclassical concept since it treats the atom as an object with quantized energy levels and the electromagnetic field as a continuous wave. A plot as a function of detuning and ramping the time from 0 to t = \frac{\pi}{\Omega} gives: Animation of optical resonance, frequency domain We see that for \delta = 0 the population will oscillate between the two states at the Rabi frequency. == Generalized Rabi frequency == The quantity \sqrt{\Omega^2 + \delta^2} is commonly referred to as the "generalized Rabi frequency." The energy released is equal to the difference in energy levels between the electron energy states. The absorption of the photon takes place in accordance with Planck's quantum theory. The next rule follows from the Frank-Condon Principle, which states that the absorption of a photon by an electron and the subsequent jump in energy levels is near-instantaneous.

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Do $\mathrm{HF} / 6-31 \mathrm{G}^*$ geometry optimizations on one conformers of $\mathrm{HCOOH}$ with $\mathrm{OCOH}$ dihedral angle of $0^{\circ}$. Calculate the dipole moment.

Order-6 hexagonal tiling honeycomb Order-6 hexagonal tiling honeycomb 320px Perspective projection view from center of Poincaré disk model 320px Perspective projection view from center of Poincaré disk model Type Hyperbolic regular honeycomb Paracompact uniform honeycomb Schläfli symbol {6,3,6} {6,3[3]} Coxeter diagram ↔ ↔ Cells {6,3} 40px Faces hexagon {6} Edge figure hexagon {6} Vertex figure {3,6} or {3[3]} 40px 40px Dual Self-dual Coxeter group \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Regular, quasiregular In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. Order-6 dodecahedral honeycomb Order-6 dodecahedral honeycomb 320px Perspective projection view within Poincaré disk model 320px Perspective projection view within Poincaré disk model Type Hyperbolic regular honeycomb Paracompact uniform honeycomb Schläfli symbol {5,3,6} {5,3[3]} Coxeter diagram ↔ Cells {5,3} 40px Faces pentagon {5} Edge figure hexagon {6} Vertex figure 80px 80px triangular tiling Dual Order-5 hexagonal tiling honeycomb Coxeter group \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Regular, quasiregular The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. Order-5 hexagonal tiling honeycomb Order-5 hexagonal tiling honeycomb 320px Perspective projection view from center of Poincaré disk model 320px Perspective projection view from center of Poincaré disk model Type Hyperbolic regular honeycomb Paracompact uniform honeycomb Schläfli symbol {6,3,5} Coxeter-Dynkin diagrams 80px ↔ Cells {6,3} 40px Faces hexagon {6} Edge figure pentagon {5} Vertex figure icosahedron Dual Order-6 dodecahedral honeycomb Coxeter group \overline{HV}_3, [5,3,6] Properties Regular In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). alt=|thumb|300x300px|The flow structure of the Lamb-Chaplygin dipole The Lamb–Chaplygin dipole model is a mathematical description for a particular inviscid and steady dipolar vortex flow. This dipole is the two-dimensional analogue of Hill's spherical vortex. The Lamb–Chaplygin model follows from demanding the following characteristics: * The dipole has a circular atmosphere/separatrix with radius R: \psi\left(r = R\right) = 0. It is a quasiregular honeycomb. === Cantic order-5 hexagonal tiling honeycomb === Cantic order-5 hexagonal tiling honeycomb Cantic order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol h2{6,3,5} Coxeter diagram ↔ Cells h2{6,3} 40px t{3,5} 40px r{5,3} 40px Faces triangle {3} pentagon {5} hexagon {6} Vertex figure 80px triangular prism Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex- transitive The cantic order-5 hexagonal tiling honeycomb, h2{6,3,5}, ↔ , has trihexagonal tiling, truncated icosahedron, and icosidodecahedron facets, with a triangular prism vertex figure. === Runcic order-5 hexagonal tiling honeycomb === Runcic order-5 hexagonal tiling honeycomb Runcic order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol h3{6,3,5} Coxeter diagram ↔ Cells {3[3]} 40px rr{5,3} 40px {5,3} 40px {}x{3} 40px Faces triangle {3} square {4} pentagon {5} Vertex figure 80px triangular cupola Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive The runcic order-5 hexagonal tiling honeycomb, h3{6,3,5}, ↔ , has triangular tiling, rhombicosidodecahedron, dodecahedron, and triangular prism facets, with a triangular cupola vertex figure. === Runcicantic order-5 hexagonal tiling honeycomb === Runcicantic order-5 hexagonal tiling honeycomb Runcicantic order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol h2,3{6,3,5} Coxeter diagram ↔ Cells h2{6,3} 40px tr{5,3} 40px t{5,3} 40px {}x{3} 40px Faces triangle {3} square {4} hexagon {6} decagon {10} Vertex figure 80px rectangular pyramid Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive The runcicantic order-5 hexagonal tiling honeycomb, h2,3{6,3,5}, ↔ , has trihexagonal tiling, truncated icosidodecahedron, truncated dodecahedron, and triangular prism facets, with a rectangular pyramid vertex figure. == See also == * Convex uniform honeycombs in hyperbolic space * Regular tessellations of hyperbolic 3-space * Paracompact uniform honeycombs == References == *Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . All vertices are on the ideal surface. : 180px === Truncated order-5 hexagonal tiling honeycomb === Truncated order-5 hexagonal tiling honeycomb Truncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t{6,3,5} or t0,1{6,3,5} Coxeter diagram Cells {3,5} 40px t{6,3} 40px Faces triangle {3} dodecagon {12} Vertex figure 80px pentagonal pyramid Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The truncated order-5 hexagonal tiling honeycomb, t0,1{6,3,5}, has icosahedron and truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure. 480px === Bitruncated order-5 hexagonal tiling honeycomb === Bitruncated order-5 hexagonal tiling honeycomb Bitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol 2t{6,3,5} or t1,2{6,3,5} Coxeter diagram ↔ Cells t{3,6} 40px t{3,5} 40px Faces pentagon {5} hexagon {6} Vertex figure 80px digonal disphenoid Coxeter groups {\overline{HV}}_3, [5,3,6] {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, has hexagonal tiling and truncated icosahedron facets, with a digonal disphenoid vertex figure. 480px === Cantellated order-5 hexagonal tiling honeycomb === Cantellated order-5 hexagonal tiling honeycomb Cantellated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol rr{6,3,5} or t0,2{6,3,5} Coxeter diagram Cells r{3,5} 40px rr{6,3} 40px {}x{5} 40px Faces triangle {3} square {4} pentagon {5} hexagon {6} Vertex figure 80px wedge Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The cantellated order-5 hexagonal tiling honeycomb, t0,2{6,3,5}, has icosidodecahedron, rhombitrihexagonal tiling, and pentagonal prism facets, with a wedge vertex figure. 480px === Cantitruncated order-5 hexagonal tiling honeycomb === Cantitruncated order-5 hexagonal tiling honeycomb Cantitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol tr{6,3,5} or t0,1,2{6,3,5} Coxeter diagram Cells t{3,5} 40px tr{6,3} 40px {}x{5} 40px Faces square {4} pentagon {5} hexagon {6} dodecagon {12} Vertex figure 80px mirrored sphenoid Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex- transitive The cantitruncated order-5 hexagonal tiling honeycomb, t0,1,2{6,3,5}, has truncated icosahedron, truncated trihexagonal tiling, and pentagonal prism facets, with a mirrored sphenoid vertex figure. 480px === Runcinated order-5 hexagonal tiling honeycomb === Runcinated order-5 hexagonal tiling honeycomb Runcinated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,3{6,3,5} Coxeter diagram Cells {6,3} 40px {5,3} 40px {}x{6} 40px {}x{5} 40px Faces square {4} pentagon {5} hexagon {6} Vertex figure 80px irregular triangular antiprism Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The runcinated order-5 hexagonal tiling honeycomb, t0,3{6,3,5}, has dodecahedron, hexagonal tiling, pentagonal prism, and hexagonal prism facets, with an irregular triangular antiprism vertex figure. 480px === Runcitruncated order-5 hexagonal tiling honeycomb === Runcitruncated order-5 hexagonal tiling honeycomb Runcitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,3{6,3,5} Coxeter diagram Cells t{6,3} 40px rr{5,3} 40px {}x{5} 40px {}x{12} 40px Faces triangle {3} square {4} pentagon {5} dodecagon {12} Vertex figure 80px isosceles-trapezoidal pyramid Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The runcitruncated order-5 hexagonal tiling honeycomb, t0,1,3{6,3,5}, has truncated hexagonal tiling, rhombicosidodecahedron, pentagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure. 480px === Runcicantellated order-5 hexagonal tiling honeycomb === The runcicantellated order-5 hexagonal tiling honeycomb is the same as the runcitruncated order-6 dodecahedral honeycomb. === Omnitruncated order-5 hexagonal tiling honeycomb === Omnitruncated order-5 hexagonal tiling honeycomb Omnitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,2,3{6,3,5} Coxeter diagram Cells tr{6,3} 40px tr{5,3} 40px {}x{10} 40px {}x{12} 40px Faces square {4} hexagon {6} decagon {10} dodecagon {12} Vertex figure 80px irregular tetrahedron Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The omnitruncated order-5 hexagonal tiling honeycomb, t0,1,2,3{6,3,5}, has truncated trihexagonal tiling, truncated icosidodecahedron, decagonal prism, and dodecagonal prism facets, with an irregular tetrahedral vertex figure. 480px === Alternated order-5 hexagonal tiling honeycomb === Alternated order-5 hexagonal tiling honeycomb Alternated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Semiregular honeycomb Schläfli symbol h{6,3,5} Coxeter diagram ↔ Cells {3[3]} 40px {3,5} 40px Faces triangle {3} Vertex figure 40px truncated icosahedron Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive, edge-transitive, quasiregular The alternated order-5 hexagonal tiling honeycomb, h{6,3,5}, ↔ , has triangular tiling and icosahedron facets, with a truncated icosahedron vertex figure. In general relativity and differential geometry, the Bel–Robinson tensor is a tensor defined in the abstract index notation by: :T_{abcd}=C_{aecf}C_{b} {}^{e} {}_{d} {}^{f} + \frac{1}{4}\epsilon_{ae}{}^{hi} \epsilon_{b}{}^{ej}{}_{k} C_{hicf} C_{j}{}^{k}{}_{d}{}^{f} Alternatively, :T_{abcd} = C_{aecf}C_{b} {}^{e} {}_{d} {}^{f} - \frac{3}{2} g_{a[b} C_{jk]cf} C^{jk}{}_{d}{}^{f} where C_{abcd} is the Weyl tensor. It contains triangular tiling facets in a hexagonal tiling vertex figure. === Cantic order-6 hexagonal tiling honeycomb === Cantic order-6 hexagonal tiling honeycomb Cantic order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h2{6,3,6} Coxeter diagrams ↔ Cells t{3,6} 40px r{6,3} 40px h2{6,3} 40px Faces triangle {3} hexagon {6} Vertex figure 80px triangular prism Coxeter groups \overline{VP}_3, [6,3[3]] Properties Vertex- transitive, edge-transitive The cantic order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the rectified triangular tiling honeycomb, ↔ , with trihexagonal tiling and hexagonal tiling facets in a triangular prism vertex figure. ===Runcic order-6 hexagonal tiling honeycomb=== Runcic order-6 hexagonal tiling honeycomb Runcic order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h3{6,3,6} Coxeter diagrams ↔ Cells rr{3,6} 40px {6,3} 40px {3[3]} 40px {3}x{} 40px Faces triangle {3} square {4} hexagon {6} Vertex figure 80px triangular cupola Coxeter groups \overline{VP}_3, [6,3[3]] Properties Vertex- transitive The runcic hexagonal tiling honeycomb, h3{6,3,6}, , or , has hexagonal tiling, rhombitrihexagonal tiling, triangular tiling, and triangular prism facets, with a triangular cupola vertex figure. ===Runicantic order-6 hexagonal tiling honeycomb=== Runcicantic order-6 hexagonal tiling honeycomb Runcicantic order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h2,3{6,3,6} Coxeter diagrams ↔ Cells tr{6,3} 40px t{6,3} 40px h2{6,3} 40px {}x{3} 40px Faces triangle {3} square {4} hexagon {6} dodecagon {12} Vertex figure 80px rectangular pyramid Coxeter groups \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The runcicantic order-6 hexagonal tiling honeycomb, h2,3{6,3,6}, , or , contains truncated trihexagonal tiling, truncated hexagonal tiling, trihexagonal tiling, and triangular prism facets, with a rectangular pyramid vertex figure. == See also == * Convex uniform honeycombs in hyperbolic space * Regular tessellations of hyperbolic 3-space * Paracompact uniform honeycombs == References == *Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . The order-6 dodecahedral honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures: It is also part of a sequence of regular polytopes and honeycombs with dodecahedral cells: === Rectified order-6 dodecahedral honeycomb === Rectified order-6 dodecahedral honeycomb \--> Type Paracompact uniform honeycomb Schläfli symbols r{5,3,6} t1{5,3,6} Coxeter diagrams ↔ Cells r{5,3} 40px {3,6} 40px Faces triangle {3} pentagon {5} Vertex figure 80px hexagonal prism Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive, edge-transitive The rectified order-6 dodecahedral honeycomb, t1{5,3,6} has icosidodecahedron and triangular tiling cells connected in a hexagonal prism vertex figure. : 480px Perspective projection view within Poincaré disk model It is similar to the 2D hyperbolic pentaapeirogonal tiling, r{5,∞} with pentagon and apeirogonal faces. : 180px === Truncated order-6 dodecahedral honeycomb === Truncated order-6 dodecahedral honeycomb \--> Type Paracompact uniform honeycomb Schläfli symbols t{5,3,6} t0,1{5,3,6} Coxeter diagrams ↔ Cells t{5,3} 40px {3,6} 40px Faces triangle {3} decagon {10} Vertex figure 80px hexagonal pyramid Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive The truncated order-6 dodecahedral honeycomb, t0,1{5,3,6} has truncated dodecahedron and triangular tiling cells connected in a hexagonal pyramid vertex figure. 480px === Bitruncated order-6 dodecahedral honeycomb === The bitruncated order-6 dodecahedral honeycomb is the same as the bitruncated order-5 hexagonal tiling honeycomb. === Cantellated order-6 dodecahedral honeycomb === Cantellated order-6 dodecahedral honeycomb Cantellated order-6 dodecahedral honeycomb Type Paracompact uniform honeycomb Schläfli symbols rr{5,3,6} t0,2{5,3,6} Coxeter diagrams ↔ Cells rr{5,3} 40px rr{6,3} 40px {}x{6} 40px Faces triangle {3} square {4} pentagon {5} hexagon {6} Vertex figure 80px wedge Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive The cantellated order-6 dodecahedral honeycomb, t0,2{5,3,6}, has rhombicosidodecahedron, trihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure. 480px === Cantitruncated order-6 dodecahedral honeycomb === Cantitruncated order-6 dodecahedral honeycomb Cantitruncated order-6 dodecahedral honeycomb Type Paracompact uniform honeycomb Schläfli symbols tr{5,3,6} t0,1,2{5,3,6} Coxeter diagrams ↔ Cells tr{5,3} 40px t{3,6} 40px {}x{6} 40px Faces square {4} hexagon {6} decagon {10} Vertex figure 80px mirrored sphenoid Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive The cantitruncated order-6 dodecahedral honeycomb, t0,1,2{5,3,6} has truncated icosidodecahedron, hexagonal tiling, and hexagonal prism facets, with a mirrored sphenoid vertex figure. 480px === Runcinated order-6 dodecahedral honeycomb === The runcinated order-6 dodecahedral honeycomb is the same as the runcinated order-5 hexagonal tiling honeycomb. === Runcitruncated order-6 dodecahedral honeycomb === Runcitruncated order-6 dodecahedral honeycomb Runcitruncated order-6 dodecahedral honeycomb Type Paracompact uniform honeycomb Schläfli symbols t0,1,3{5,3,6} Coxeter diagrams Cells t{5,3} 40px rr{6,3} 40px {}x{10} 40px {}x{6} 40px Faces square {4} hexagon {6} decagon {10} Vertex figure 80px isosceles-trapezoidal pyramid Coxeter groups \overline{HV}_3, [5,3,6] Properties Vertex-transitive The runcitruncated order-6 dodecahedral honeycomb, t0,1,3{5,3,6} has truncated dodecahedron, rhombitrihexagonal tiling, decagonal prism, and hexagonal prism facets, with an isosceles- trapezoidal pyramid vertex figure. 480px === Runcicantellated order-6 dodecahedral honeycomb === The runcicantellated order-6 dodecahedral honeycomb is the same as the runcitruncated order-5 hexagonal tiling honeycomb. === Omnitruncated order-6 dodecahedral honeycomb === The omnitruncated order-6 dodecahedral honeycomb is the same as the omnitruncated order-5 hexagonal tiling honeycomb. == See also == * Convex uniform honeycombs in hyperbolic space * Regular tessellations of hyperbolic 3-space * Paracompact uniform honeycombs == References == *Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . Thus, infinitely many hexagonal tilings meet at each vertex of this honeycomb.Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III == Related tilings == The order-6 hexagonal tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface. : 240px It contains and that tile 2-hypercycle surfaces, which are similar to the paracompact tilings and (the truncated infinite-order triangular tiling and order-3 apeirogonal tiling, respectively): : 120px 120px == Symmetry == 120px|thumb|left|Subgroup relations: ↔ The order-6 hexagonal tiling honeycomb has a half-symmetry construction: . The order-6 dodecahedral honeycomb is similar to the 2D hyperbolic infinite-order pentagonal tiling, {5,∞}, with pentagonal faces, and with vertices on the ideal surface. : 240px == Related polytopes and honeycombs == The order-6 dodecahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. The polar circles of the triangles of a complete quadrilateral form a coaxal system. The order-6 hexagonal tiling honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures: It is also part of a sequence of regular polychora and honeycombs with hexagonal tiling cells: It is also part of a sequence of regular polychora and honeycombs with regular deltahedral vertex figures: === Rectified order-6 hexagonal tiling honeycomb === Rectified order-6 hexagonal tiling honeycomb Rectified order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols r{6,3,6} or t1{6,3,6} Coxeter diagrams ↔ ↔ ↔ ↔ Cells {3,6} 40px r{6,3} 40px Faces triangle {3} hexagon {6} Vertex figure 80px hexagonal prism Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] \overline{PP}_3, [3[3,3]] Properties Vertex-transitive, edge-transitive The rectified order-6 hexagonal tiling honeycomb, t1{6,3,6}, has triangular tiling and trihexagonal tiling facets, with a hexagonal prism vertex figure. it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q{6,3,6}, ↔ . 480px It is analogous to 2D hyperbolic order-4 apeirogonal tiling, r{∞,∞} with infinite apeirogonal faces, and with all vertices on the ideal surface. : 240px ==== Related honeycombs==== The order-6 hexagonal tiling honeycomb is part of a series of honeycombs with hexagonal prism vertex figures: It is also part of a matrix of 3-dimensional quarter honeycombs: q{2p,4,2q} === Truncated order-6 hexagonal tiling honeycomb === Truncated order-6 hexagonal tiling honeycomb Truncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t{6,3,6} or t0,1{6,3,6} Coxeter diagram ↔ Cells {3,6} 40px t{6,3} 40px Faces triangle {3} dodecagon {12} Vertex figure 80px hexagonal pyramid Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The truncated order-6 hexagonal tiling honeycomb, t0,1{6,3,6}, has triangular tiling and truncated hexagonal tiling facets, with a hexagonal pyramid vertex figure.Twitter Rotation around 3 fold axis 480px === Bitruncated order-6 hexagonal tiling honeycomb === Bitruncated order-6 hexagonal tiling honeycomb Bitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol bt{6,3,6} or t1,2{6,3,6} Coxeter diagram ↔ Cells t{3,6} 40px Faces hexagon {6} Vertex figure 80px tetrahedron Coxeter groups 2\times\overline{Z}_3, 6,3,6 \overline{VP}_3, [6,3[3]] \overline{V}_3, [3,3,6] Properties Regular The bitruncated order-6 hexagonal tiling honeycomb is a lower symmetry construction of the regular hexagonal tiling honeycomb, ↔ . Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III == Symmetry== A lower-symmetry construction of index 120, [6,(3,5)*], exists with regular dodecahedral fundamental domains, and an icosahedral Coxeter-Dynkin diagram with 6 axial infinite-order (ultraparallel) branches. == Images== The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact order-5 apeirogonal tiling, {∞,5}, with five apeirogonal faces meeting around every vertex. : 180px == Related polytopes and honeycombs == The order-5 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. There are 15 uniform honeycombs in the [6,3,5] Coxeter group family, including this regular form, and its regular dual, the order-6 dodecahedral honeycomb. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has six such hexagonal tilings meeting at each edge. It contains hexagonal tiling facets, with a tetrahedron vertex figure. 480px === Cantellated order-6 hexagonal tiling honeycomb === Cantellated order-6 hexagonal tiling honeycomb Cantellated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol rr{6,3,6} or t0,2{6,3,6} Coxeter diagram ↔ Cells r{3,6} 40px rr{6,3} 40px {}x{6} 40px Faces triangle {3} square {4} hexagon {6} Vertex figure 80px wedge Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The cantellated order-6 hexagonal tiling honeycomb, t0,2{6,3,6}, has trihexagonal tiling, rhombitrihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure. 480px === Cantitruncated order-6 hexagonal tiling honeycomb === Cantitruncated order-6 hexagonal tiling honeycomb Cantitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol tr{6,3,6} or t0,1,2{6,3,6} Coxeter diagram ↔ Cells tr{3,6} 40px t{3,6} 40px {}x{6} 40px Faces triangle {3} square {4} hexagon {6} dodecagon {12} Vertex figure 80px mirrored sphenoid Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The cantitruncated order-6 hexagonal tiling honeycomb, t0,1,2{6,3,6}, has hexagonal tiling, truncated trihexagonal tiling, and hexagonal prism cells, with a mirrored sphenoid vertex figure. 480px === Runcinated order-6 hexagonal tiling honeycomb === Runcinated order-6 hexagonal tiling honeycomb Runcinated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,3{6,3,6} Coxeter diagram ↔ Cells {6,3} 40px40px {}×{6} 40px Faces triangle {3} square {4} hexagon {6} Vertex figure 80px triangular antiprism Coxeter groups 2\times\overline{Z}_3, 6,3,6 Properties Vertex-transitive, edge-transitive The runcinated order-6 hexagonal tiling honeycomb, t0,3{6,3,6}, has hexagonal tiling and hexagonal prism cells, with a triangular antiprism vertex figure. 480px It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr{6,6}, with square and hexagonal faces: : 240px === Runcitruncated order-6 hexagonal tiling honeycomb === Runcitruncated order-6 hexagonal tiling honeycomb Runcitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,3{6,3,6} Coxeter diagram Cells t{6,3} 40px rr{6,3} 40px {}x{6}40px {}x{12} 40px Faces triangle {3} square {4} hexagon {6} dodecagon {12} Vertex figure 80px isosceles-trapezoidal pyramid Coxeter groups \overline{Z}_3, [6,3,6] Properties Vertex-transitive The runcitruncated order-6 hexagonal tiling honeycomb, t0,1,3{6,3,6}, has truncated hexagonal tiling, rhombitrihexagonal tiling, hexagonal prism, and dodecagonal prism cells, with an isosceles- trapezoidal pyramid vertex figure. 480px === Omnitruncated order-6 hexagonal tiling honeycomb === Omnitruncated order-6 hexagonal tiling honeycomb Omnitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,2,3{6,3,6} Coxeter diagram Cells tr{6,3} 40px {}x{12} 40px Faces square {4} hexagon {6} dodecagon {12} Vertex figure 80px phyllic disphenoid Coxeter groups 2\times\overline{Z}_3, 6,3,6 Properties Vertex-transitive The omnitruncated order-6 hexagonal tiling honeycomb, t0,1,2,3{6,3,6}, has truncated trihexagonal tiling and dodecagonal prism cells, with a phyllic disphenoid vertex figure. 480px ===Alternated order-6 hexagonal tiling honeycomb=== Alternated order-6 hexagonal tiling honeycomb Alternated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h{6,3,6} Coxeter diagrams ↔ Cells {3,6} 40px {3[3]} 40px Faces triangle {3} Vertex figure 80px hexagonal tiling Coxeter groups \overline{VP}_3, [6,3[3]] Properties Regular, quasiregular The alternated order-6 hexagonal tiling honeycomb is a lower- symmetry construction of the regular triangular tiling honeycomb, ↔ . A less favorable property is that the second derivative of the flow field at the dipole's edge is not continuous. This subgroup corresponds to a Coxeter diagram with six order-3 branches and three infinite-order branches in the shape of a triangular prism: . == Related polytopes and honeycombs == The order-6 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs in 3-space.

1.41

4.76

0.87

-1270

-0.55

A

Frozen-core $\mathrm{SCF} / \mathrm{DZP}$ and CI-SD/DZP calculations on $\mathrm{H}_2 \mathrm{O}$ at its equilibrium geometry gave energies of -76.040542 and -76.243772 hartrees. Application of the Davidson correction brought the energy to -76.254549 hartrees. Find the coefficient of $\Phi_0$ in the normalized CI-SD wave function.

As with other perturbative approaches, the Davidson correction is not reliable when the electronic structure of CISD and the reference Hartree–Fock wave functions are significantly different (i.e. when a_0^2 is not close to 1). A solution of these equations yields the Hartree–Fock wave function and energy of the system. The Davidson correction is an energy correction often applied in calculations using the method of truncated configuration interaction, which is one of several post-Hartree-Fock ab initio quantum chemistry methods in the field of computational chemistry. It uses the formula :\Delta E_Q = (1 - a_0^2)(E_{\rm CISD} - E_{\rm HF}), \ :E_{\rm CISDTQ} \approx E_{\rm CISD} + \Delta E_Q, \ where a0 is the coefficient of the Hartree-Fock wavefunction in the CISD expansion, ECISD and EHF are the energies of the CISD and Hartree-Fock wavefunctions respectively, and ΔEQ is the correction to estimate ECISDTQ, the energy of the CISDTQ wavefunction. Developing post-Hartree–Fock methods based on a ROHF wave function is inherently more difficult than using a UHF wave function, due to the lack of a unique set of molecular orbitals. Initially, both the Hartree method and the Hartree–Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. In atomic structure theory, calculations may be for a spectrum with many excited energy levels and consequently the Hartree–Fock method for atoms assumes the wave function is a single configuration state function with well-defined quantum numbers and that the energy level is not necessarily the ground state. In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The Davidson correction does not give information about the wave function. An alternative to Hartree–Fock calculations used in some cases is density functional theory, which treats both exchange and correlation energies, albeit approximately. Finding the Hartree–Fock one-electron wave functions is now equivalent to solving the eigenfunction equation : \hat F(1)\phi_i(1) = \epsilon_i \phi_i(1), where \phi_i(1) are a set of one-electron wave functions, called the Hartree–Fock molecular orbitals. === Linear combination of atomic orbitals === Typically, in modern Hartree–Fock calculations, the one-electron wave functions are approximated by a linear combination of atomic orbitals. They add electron correlation which is a more accurate way of including the repulsions between electrons than in the Hartree–Fock method where repulsions are only averaged. == Details == In general, the SCF procedure makes several assumptions about the nature of the multi-body Schrödinger equation and its set of solutions: * For molecules, the Born–Oppenheimer approximation is inherently assumed. Charlotte Froese Fischer (born 1929) is a Canadian-American applied mathematician and computer scientist noted for the development and implementation of the Multi-Configurational Hartree–Fock (MCHF) approach to atomic-structure calculations and its application to the description of atomic structure and spectra. In computational chemistry, post–Hartree–Fock (post-HF) methods are the set of methods developed to improve on the Hartree–Fock (HF), or self-consistent field (SCF) method. The starting point for the Hartree–Fock method is a set of approximate one-electron wave functions known as spin-orbitals. The Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules, nanostructures and solids but it has also found widespread use in nuclear physics. However, different choices of reference orbitals have shown to provide similar results, and thus many different post-Hartree–Fock methods have been implemented in a variety of electronic structure packages. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting exchange. (See Hartree–Fock–Bogoliubov method for a discussion of its application in nuclear structure theory). Another option is to use modern valence bond methods. == Software packages == For a list of software packages known to handle Hartree–Fock calculations, particularly for molecules and solids, see the list of quantum chemistry and solid state physics software. == See also == Related fields * Quantum chemistry * Molecular physics * Quantum chemistry computer programs * Fock symmetry Concepts * Roothaan equations * Koopmans' theorem * Post-Hartree–Fock * Direct inversion of iterative subspace People * Vladimir Aleksandrovich Fock * Clemens Roothaan * George G. Hall * John Pople * Reinhart Ahlrichs == References == == Sources == * * * == External links == * The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Restricted open-shell Hartree–Fock (ROHF) is a variant of Hartree–Fock method for open shell molecules. However, neither Davidson correction itself nor the corrected energies are size-consistent or size-extensive.

0.2244

30

'-2.0'

0.9731

35.2

D

Let $w$ be the variable defined as the number of heads that show when two coins are tossed simultaneously. Find $\langle w\rangle$.

thumb|upright=1.5|Coin of Epander. thumb|upright=1.5|Coin of Bhradrayasha. thumb|upright=1.35|Coin of Tennes. thumb|upright=1.2|Coin of Sabaces. It was designed by Nico de Haas, a Dutch national-socialist, and struck in 1941 and 1942. ==Mintage== Year Mintage Notes 1941 27,600,000 1942 ==References== Category:Netherlands in World War II Category:Coins of the Netherlands Category:Modern obsolete currencies Category:Currencies of Europe Category:Zinc and aluminum coins minted in Germany and occupied territories during World War II Japanese coins from this period are read clockwise from right to left: :"Year" ← "Number representing year of reign" ← "Emperors name" (Ex: 年 ← 五 ← 和昭) Year of reign Japanese date Gregorian date Mintage 5th 五 1930 6th 六 1931 7th 七 1932 Unknown ==Collecting== The value of any given coin is determined by survivability rate and condition as collectors in general prefer uncleaned appealing coins. The -cent coin minted in the Netherlands during World War II was made of zinc, and worth , or .025, of the Dutch guilder. Japanese coins from this period are read clockwise from right to left: :"Year" ← "Number representing year of reign" ← "Emperors name" (Ex: 年 ← 六 ← 正大) Year of reign Japanese date Gregorian date Mintage 1st 元 1912 2nd 二 1913 3rd 三 1914 4th 四 1915 5th 五 1916 6th 六 1917 7th 七 1918 8th 八 1919 9th 九 1920 === Shōwa === The following are mintage figures for coins minted between the 5th and the 7th year of Emperor Shōwa's reign. This system was officially put into place on May 10, 1871 setting standards for the 20 yen coin. Japanese coins from this period are read clockwise from right to left :"Year" ← "Number representing year of reign" ← "Emperors name" (Ex: 年 ← 七十三 ← 治明) thumb|right|20 yen coin from 1870 (year 3) Design 1 - (1870 - 1892) thumb|right|20 yen coin from 1897 (year 30) Design 2 - (1897 - 1912) Year of reign Japanese date Gregorian date Mintage 3rd 三 1870 5th 五 1872 6th 六 1873 9th 九 1876 10th 十 1877 13th 三十 1880 25th 五十二 1892 Not circulated 30th 十三 1897 37th 七十三 1904 38th 八十三 1905 39th 九十三 1906 40th 十四 1907 41st 一十四 1908 42nd 二十四 1909 43rd 三十四 1910 44th 四十四 1911 45th 五十四 1912 ===Taishō=== The following are mintage figures for the coins that were minted from the 1st to the 9th year of Taishō's reign. Many of these coins were then melted or destroyed as a result of the wars between 1931 and 1945. An auction held in 2011 featuring one of these coins sold it for $230,000 (USD). Some of these coins were kept away in bank vaults for decades before being released as part of a hoard in the mid 2000s. ==Weight and size== Image Minted Size Weight 150px 1870–1880 35.06mm 33.33g 150px 1897–1932 28.78mm 16.66g ==Circulation figures== ===Meiji=== The following are mintage figures for the coins that were minted between the 3rd and 45th (last) year of Meiji's reign. Gold coins of the 20 yen denomination were last minted in 1932, it is unknown how many Shōwa era coins were later melted. Twenty yen coins dated 1877 (year 10) have an extremely low mintage of just 29 coins struck. These coins which are dated from 1870 to 1876 (year 3 to 9) are all priced in five digit dollar amounts (USD) in average condition. These new standards lowered both the size and weight of the coin, the new diameter was set at 28.78mm (previously 35.06mm), and the weight was lowered from 33.3g down to 16.6g. Coinage of the 20 yen piece had all but stopped by 1877, and those struck in 1880 were only done so as part of presentation sets for visiting dignitaries and heads of state. For this denomination all 20 yen coins are scarce as the amount remaining today are dependent on how many were saved or kept away. Twenty Yen coins spanned three different Imperial eras before mintage was halted in 1932. The was a denomination of Japanese yen. These coins were minted in gold, and during their lifespan were the highest denomination of coin that circulated in the country.

-233

-2

8.3147

0.4207

1

E

Calculate the force on an alpha particle passing a gold atomic nucleus at a distance of $0.00300 Å$.

Initially the alpha particles are at a very large distance from the nucleus. :\frac{1}{2} mv^2 = \frac{1}{4\pi \epsilon_0} \cdot \frac{q_1 q_2}{r_\text{min}} Rearranging: :r_\text{min} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2 q_1 q_2}{mv^2} For an alpha particle: * (mass) = = * (for helium) = 2 × = * (for gold) = 79 × = * (initial velocity) = (for this example) Substituting these in gives the value of about , or 27 fm. The true radius of the nucleus is not recovered in these experiments because the alphas do not have enough energy to penetrate to more than 27 fm of the nuclear center, as noted, when the actual radius of gold is 7.3 fm. Rutherford realized this, and also realized that actual impact of the alphas on gold causing any force-deviation from that of the coulomb potential would change the form of his scattering curve at high scattering angles (the smallest impact parameters) from a hyperbola to something else. This was not seen, indicating that the surface of the gold nucleus had not been "touched" so that Rutherford also knew the gold nucleus (or the sum of the gold and alpha radii) was smaller than 27 fm. == Extension to situations with relativistic particles and target recoil == The extension of low-energy Rutherford-type scattering to relativistic energies and particles that have intrinsic spin is beyond the scope of this article. In Rutherford's gold foil experiment conducted by his students Hans Geiger and Ernest Marsden, a narrow beam of alpha particles was established, passing through very thin (a few hundred atoms thick) gold foil. Applying the inverse-square law between the charges on the alpha particle and nucleus, one can write: Assumptions: 1\. The distance from the center of the alpha particle to the center of the nucleus () at this point is an upper limit for the nuclear radius, if it is evident from the experiment that the scattering process obeys the cross section formula given above. The classical Rutherford scattering process of alpha particles against gold nuclei is an example of "elastic scattering" because neither the alpha particles nor the gold nuclei are internally excited. For any central potential, the differential cross- section in the lab frame is related to that in the center-of-mass frame by \frac{d\sigma}{d\Omega}_L=\frac{(1+2s\cos\Theta+s^2)^{3/2}}{1+s\cos\Theta} \frac{d\sigma}{d\Omega} To give a sense of the importance of recoil, we evaluate the head-on energy ratio F for an incident alpha particle (mass number \approx 4) scattering off a gold nucleus (mass number \approx 197): F \approx 0.0780. This same result can be expressed alternatively as : \frac{d\sigma}{d\Omega} = \left( \frac{ Z_1 Z_2 \alpha (\hbar c)} {4 E_{\mathrm{K}10} \sin^2 \frac{\Theta}{2} } \right)^2, where is the dimensionless fine structure constant, is the initial non-relativistic kinetic energy of particle 1 in MeV, and . ==Details of calculating maximal nuclear size== For head-on collisions between alpha particles and the nucleus (with zero impact parameter), all the kinetic energy of the alpha particle is turned into potential energy and the particle is at rest. The Rutherford formula (see below) further neglects the recoil kinetic energy of the massive target nucleus. When a (positive) alpha particle approached sufficiently close to the nucleus, it was repelled strongly enough to rebound at high angles. In classical physics, alpha particles do not have enough energy to escape the potential well from the strong force inside the nucleus (this well involves escaping the strong force to go up one side of the well, which is followed by the electromagnetic force causing a repulsive push-off down the other side). Rutherford showed, using the method outlined below, that the size of the nucleus was less than about (how much less than this size, Rutherford could not tell from this experiment alone; see more below on this problem of lowest possible size). It was determined that the atom's positive charge was concentrated in a small area in its center, making the positive charge dense enough to deflect any positively charged alpha particles that came close to what was later termed the nucleus. The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. For the case of light alpha particles scattering off heavy nuclei, as in the experiment performed by Rutherford, the reduced mass, essentially the mass of the alpha particle and the nucleus off of which it scatters, is essentially stationary in the lab frame. It was found that some of the alpha particles were deflected at much larger angles than expected (at a suggestion by Rutherford to check it) and some even bounced almost directly back. Because the mass of an alpha particle is about 8000 times that of an electron, it became apparent that a very strong force must be present if it could deflect the massive and fast moving alpha particles. With a typical kinetic energy of 5 MeV; the speed of emitted alpha particles is 15,000 km/s, which is 5% of the speed of light. For the more extreme case of an electron scattering off a proton, s \approx 1/1836 and F \approx 0.00218. == See also == *Rutherford backscattering spectrometry ==References== == Textbooks == * == External links == * E. Rutherford, The Scattering of α and β Particles by Matter and the Structure of the Atom, Philosophical Magazine. Rutherford hypothesized that, assuming the "plum pudding" model of the atom was correct, the positively charged alpha particles would be only slightly deflected, if at all, by the dispersed positive charge predicted. The diameter of the nucleus is in the range of () for hydrogen (the diameter of a single proton) to about for uranium.

+4.1

4.4

0.405

269

3.2

C

When an electron in a certain excited energy level in a one-dimensional box of length $2.00 Å$ makes a transition to the ground state, a photon of wavelength $8.79 \mathrm{~nm}$ is emitted. Find the quantum number of the initial state.

In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. The excited state originates from the interaction between a photon and the quantum system. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. These energy levels are described by the principal quantum number n = 1, 2, 3, ... . After excitation the atom may return to the ground state or a lower excited state, by emitting a photon with a characteristic energy. Photoexcitation is the production of an excited state of a quantum system by photon absorption. In modern physics, the concept of a quantum jump is rarely used; as a rule scientists speak of transitions between quantum states or energy levels. == Atomic electron transition == thumb|Grotrian diagram of a quantum 3-level system with characteristic transition frequencies, \omega12 and \omega13, and excited state lifetimes \Gamma2 and \Gamma3 Atomic electron transitions cause the emission or absorption of photons. The ground state of the hydrogen atom has the atom's single electron in the lowest possible orbital (that is, the spherically symmetric "1s" wave function, which, so far, has been demonstrated to have the lowest possible quantum numbers). By giving the atom additional energy (for example, by absorption of a photon of an appropriate energy), the electron moves into an excited state (one with one or more quantum numbers greater than the minimum possible). The larger the energy separation of the states between which the electron jumps, the shorter the wavelength of the photon emitted. thumb|EMCCD camera and photomultiplier tube signals while driving quantum jumps on the 674 nm transition of 88Sr+ In an ion trap, quantum jumps can be directly observed by addressing a trapped ion with radiation at two different frequencies to drive electron transitions. However, it is not easy to measure them compared to ground-state absorption, and in some cases complete bleaching of the ground state is required to measure excited-state absorption. ==Reaction== A further consequence of excited-state formation may be reaction of the atom or molecule in its excited state, as in photochemistry. == See also == * Rydberg formula * Stationary state * Repulsive state == References == == External links == * NASA background information on ground and excited states Category:Quantum mechanics Although the initial step excites to a broad range of intermediate states, the precision inherent in the optical excitation process means that the laser light only interacts with a specific subset of atoms in a particular state, exciting to the chosen final state. == Hydrogenic potential == thumb|right|Figure 3. In optical excitation, the incident photon is absorbed by the target atom, resulting in a precise final state energy. A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, n. The lifetime of a system in an excited state is usually short: spontaneous or induced emission of a quantum of energy (such as a photon or a phonon) usually occurs shortly after the system is promoted to the excited state, returning the system to a state with lower energy (a less excited state or the ground state). State |3\rangle has a relatively long lifetime \Gamma3 which causes an interruption of the photon emission as the electron gets shelved in state through application of light with frequency \omega13. Quantum- mechanically, a state with abnormally high n refers to an atom in which the valence electron(s) have been excited into a formerly unpopulated electron orbital with higher energy and lower binding energy. A quantum jump is the abrupt transition of a quantum system (atom, molecule, atomic nucleus) from one quantum state to another, from one energy level to another. In quantum many-body systems, ground states of gapped Hamiltonians have exponential decay of correlations.

4

0.22222222

152.67

1838.50666349

3.2

A

For a macroscopic object of mass $1.0 \mathrm{~g}$ moving with speed $1.0 \mathrm{~cm} / \mathrm{s}$ in a one-dimensional box of length $1.0 \mathrm{~cm}$, find the quantum number $n$.

The magnitude of the momentum is given by :p=\frac{h}{2L}\sqrt{n_x^2+n_y^2+n_z^2} \qquad \qquad n_x,n_y,n_z=1,2,3,\ldots where h is Planck's constant and L is the length of a side of the box. The distance from the origin to any point will be :n=\sqrt{n_x^2+n_y^2+n_z^2}=\frac{2Lp}{h} Suppose each set of quantum numbers specify f states where f is the number of internal degrees of freedom of the particle that can be altered by collision. The principal quantum number is related to the radial quantum number, nr, by: n = n_r + \ell + 1 where ℓ is the azimuthal quantum number and nr is equal to the number of nodes in the radial wavefunction. Using a continuum approximation, the number of states with magnitude of momentum between p and p+dp is therefore :dg = \frac{\pi}{2}~f n^2\,dn = \frac{4\pi fV}{h^3}~ p^2\,dp where V=L3 is the volume of the box. In quantum mechanics, the principal quantum number (symbolized n) is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. These integers are the magnetic quantum numbers. In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other except for instantaneous thermalizing collisions. In atomic physics, a magnetic quantum number is a quantum number used to distinguish quantum states of an electron or other particle according to its angular momentum along a given axis in space. The principal quantum number n represents the relative overall energy of each orbital. More complete calculations will be left to separate articles, but some simple examples will be given in this article. ==Thomas–Fermi approximation for the degeneracy of states== For both massive and massless particles in a box, the states of a particle are enumerated by a set of quantum numbers . Integrating the energy distribution function and solving for N gives the particle number :N = \left(\frac{Vf}{\Lambda^3}\right)\textrm{Li}_{3/2}(z) where Lis(z) is the polylogarithm function. The orbital magnetic quantum number takes integer values in the range from -\ell to +\ell, including zero. The spin magnetic quantum number specifies the z-axis component of the spin angular momentum for a particle having spin quantum number . For large values of n, the number of states with magnitude of momentum less than or equal to p from the above equation is approximately : g = \left(\frac{f}{8}\right) \frac{4}{3}\pi n^3 = \frac{4\pi f}{3} \left(\frac{Lp}{h}\right)^3 which is just f times the volume of a sphere of radius n divided by eight since only the octant with positive ni is considered. In particle physics, G-parity is a multiplicative quantum number that results from the generalization of C-parity to multiplets of particles. The four quantum numbers conventionally used to describe the quantum state of an electron in an atom are the principal quantum number n, the azimuthal (orbital) quantum number \ell, and the magnetic quantum numbers and . Using the Thomas−Fermi approximation, the number of particles dNE with energy between E and E+dE is: :dN_E= \frac{dg_E}{\Phi(E)} where dg_E is the number of states with energy between E and E+dE. ==Energy distribution== Using the results derived from the previous sections of this article, some distributions for the gas in a box can now be determined. The magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field (the Zeeman effect) -- hence the name magnetic quantum number. The quantum number m_l refers to the projection of the angular momentum in this arbitrarily-chosen direction, conventionally called the z-direction or quantization axis. The principal quantum number arose in the solution of the radial part of the wave equation as shown below. Magnetic quantum numbers are capitalized to indicate totals for a system of particles, such as or for the total z-axis orbital angular momentum of all the electrons in an atom. ==Derivation== thumb|These orbitals have magnetic quantum numbers m_l=-\ell, \ldots,\ell from left to right in ascending order. It turns out, however, that the above equation for particle number expresses the number of bosons in excited states rather well, and thus: : N=\frac{g_0 z}{1-z}+\left(\frac{Vf}{\Lambda^3}\right)\operatorname{Li}_{3/2}(z) where the added term is the number of particles in the ground state.

8.44

3

2.3613

0.16

3.23

B

For the $\mathrm{H}_2$ ground electronic state, $D_0=4.4781 \mathrm{eV}$. Find $\Delta H_0^{\circ}$ for $\mathrm{H}_2(g) \rightarrow 2 \mathrm{H}(g)$ in $\mathrm{kJ} / \mathrm{mol}$

The delta function model becomes particularly useful with the double-well Dirac Delta function model which represents a one-dimensional version of the Hydrogen molecule ion, as shown in the following section. == Double delta potential == thumb|300px|right| The symmetric and anti-symmetric wavefunctions for the double-well Dirac delta function model with "internuclear" distance The double-well Dirac delta function models a diatomic hydrogen molecule by the corresponding Schrödinger equation: -\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x), where the potential is now V(x) = -q \left[ \delta \left(x + \frac{R}{2}\right) + \lambda\delta \left(x - \frac{R}{2} \right) \right], where 0 < R < \infty is the "internuclear" distance with Dirac delta-function (negative) peaks located at (shown in brown in the diagram). Similarly to the single band case, we can write for U^{A}_{jj'} : D_{jj'} \equiv U^{A}_{jj'} = E_{j}(0)\delta_{jj'} + \sum_{\alpha\beta} D^{\alpha\beta}_{jj'}k_{\alpha}k_{\beta}, : D^{\alpha\beta}_{jj'} = \frac{\hbar^2}{2 m_0} \left [ \delta_{jj'}\delta_{\alpha\beta} + \sum^{B}_{\gamma} \frac{ p^{\alpha}_{j\gamma}p^{\beta}_{\gamma j'} + p^{\beta}_{j\gamma}p^{\alpha}_{\gamma j'} }{ m_0 (E_0-E_{\gamma}) } \right ]. The third parameter \gamma_3 relates to the anisotropy of the energy band structure around the \Gamma point when \gamma_2 eq \gamma_3 . == Explicit Hamiltonian matrix == The Luttinger-Kohn Hamiltonian \mathbf{D_{jj'}} can be written explicitly as a 8X8 matrix (taking into account 8 bands - 2 conduction, 2 heavy-holes, 2 light-holes and 2 split-off) : \mathbf{H} = \left( \begin{array}{cccccccc} E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ P_z^{\dagger} & P+\Delta & \sqrt{2}Q^{\dagger} & -S^{\dagger}/\sqrt{2} & -\sqrt{2}P_{+}^{\dagger} & 0 & -\sqrt{3/2}S & -\sqrt{2}R \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ \end{array} \right) == Summary == == References == 2\. A flavor of the k·p perturbation theory used for calculating the structure of multiple, degenerate electronic bands in bulk and quantum well semiconductors. The International Conference on Electronic Properties of Two-Dimensional Systems (EP2DS), is a biannual event, where over hundred scientists meet for the presentation of new developments on the special field of two-dimensional electron systems in semiconductors. thumb|right|200px|Formation of a δ bond by the overlap of two d orbitals thumb|right|200px|3D model of a boundary surface of a δ bond in Mo2 In chemistry, delta bonds (δ bonds) are covalent chemical bonds, where four lobes of one involved atomic orbital overlap four lobes of the other involved atomic orbital. Substituting into the Schrödinger equation in Bloch approximation we obtain : H u_{n\mathbf{k}}(\mathbf{r}) = \left( H_0 + \frac{\hbar}{m_{0}}\mathbf{k}\cdot\mathbf{\Pi} + \frac{\hbar^2 k^2}{4m_{0}^{2}c^{2}} abla V \times \mathbf{p} \cdot \bar{\sigma} \right) u_{n\mathbf{k}}(\mathbf{r}) = E_{n}(\mathbf{k}) u_{n\mathbf{k}}(\mathbf{r}) , where : \mathbf{\Pi} = \mathbf{p} + \frac{\hbar}{4m_{0}^{2}c^{2}}\bar{\sigma} \times abla V and the perturbation Hamiltonian can be defined as : H' = \frac{\hbar}{m_0}\mathbf{k}\cdot\mathbf{\Pi}. The energy of the bound state is then E = -\frac{\hbar^2\kappa^2}{2m} = -\frac{m\lambda^2}{2\hbar^2}. === Scattering (E > 0) === right|thumb|350px|Transmission (T) and reflection (R) probability of a delta potential well. The boundary conditions thus give the following restrictions on the coefficients \begin{cases} A_r + A_l - B_r - B_l &= 0,\\\ -A_r + A_l + B_r - B_l &= \frac{2m\lambda}{ik\hbar^2} (A_r + A_l). \end{cases} === Bound state (E < 0) === right|thumb|350px|The graph of the bound state wavefunction solution to the delta function potential is continuous everywhere, but its derivative is not defined at . The delta function model is actually a one-dimensional version of the Hydrogen atom according to the dimensional scaling method developed by the group of Dudley R. HerschbachD.R. Herschbach, J.S. Avery, and O. Goscinski (eds.), Dimensional Scaling in Chemical Physics, Springer, (1992). In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Kevin Hwang (Hangul: 케빈황; born May 1, 1992), also known by his Korean name Hwang Ji-tu (Hangul: 황지투) and better known by his stage name G2 (Hangul: 지투), is a Korean-American rapper and singer. Matching of the wavefunction at the Dirac delta-function peaks yields the determinant \begin{vmatrix} q - d & q e^{-d R} \\\ q \lambda e^{-d R} & q \lambda - d \end{vmatrix} = 0, \quad \text{where } E = -\frac{d^2}{2}. Substituting the definition of into this expression yields -\frac{\hbar^2}{2m} ik (-A_r + A_l + B_r - B_l) + \lambda(A_r + A_l) = 0. They represent an approximation of the two lowest discrete energy states of the three-dimensional H2^+ and are useful in its analysis. Using Löwdin's method, only the following eigenvalue problem needs to be solved : \sum^{A}_{j'} (U^{A}_{jj'}-E\delta_{jj'})a_{j'}(\mathbf{k}) = 0, where : U^{A}_{jj'} = H_{jj'} + \sum^{B}_{\gamma eq j,j'} \frac{H_{j\gamma}H_{\gamma j'}}{E_0-E_{\gamma}} = H_{jj'} + \sum^{B}_{\gamma eq j,j'} \frac{H^{'}_{j\gamma}H^{'}_{\gamma j'}}{E_0-E_{\gamma}} , : H^{'}_{j\gamma} = \left \langle u_{j0} \right | \frac{\hbar}{m_0} \mathbf{k} \cdot \left ( \mathbf{p} + \frac{\hbar}{4 m_0 c^2} \bar{\sigma} \times abla V \right ) \left | u_{\gamma 0} \right \rangle \approx \sum_{\alpha} \frac{\hbar k_{\alpha}}{m_0}p^{\alpha}_{j \gamma}. Note that the lowest energy corresponds to the symmetric solution d_+. Ytterbium hydride is the hydride of ytterbium with the chemical formula YbH2. We now define the following parameters : A_0 = \frac{\hbar^2}{2 m_0} + \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{x}_{x\gamma}p^{x}_{\gamma x} }{ E_0-E_{\gamma} }, : B_0 = \frac{\hbar^2}{2 m_0} + \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{y}_{x\gamma}p^{y}_{\gamma x} }{ E_0-E_{\gamma} }, : C_0 = \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{x}_{x\gamma}p^{y}_{\gamma y} + p^{y}_{x\gamma}p^{x}_{\gamma y} }{ E_0-E_{\gamma} }, and the band structure parameters (or the Luttinger parameters) can be defined to be : \gamma_1 = - \frac{1}{3} \frac{2 m_0}{\hbar^2} (A_0 + 2B_0), : \gamma_2 = - \frac{1}{6} \frac{2 m_0}{\hbar^2} (A_0 - B_0), : \gamma_3 = - \frac{1}{6} \frac{2 m_0}{\hbar^2} C_0, These parameters are very closely related to the effective masses of the holes in various valence bands. \gamma_1 and \gamma_2 describe the coupling of the |X \rangle , |Y \rangle and |Z \rangle states to the other states. Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use atomic units and set \hbar = m = 1. In this compound, the ytterbium atom has an oxidation state of +2 and the hydrogen atoms have an oxidation state of -1. In this model the influence of all other bands is taken into account by using Löwdin's perturbation method. == Background == All bands can be subdivided into two classes: * Class A: six valence bands (heavy hole, light hole, split off band and their spin counterparts) and two conduction bands.

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The contribution of molecular vibrations to the molar internal energy $U_{\mathrm{m}}$ of a gas of nonlinear $N$-atom molecules is (zero-point vibrational energy not included) $U_{\mathrm{m}, \mathrm{vib}}=R \sum_{s=1}^{3 N-6} \theta_s /\left(e^{\theta_s / T}-1\right)$, where $\theta_s \equiv h \nu_s / k$ and $\nu_s$ is the vibrational frequency of normal mode $s$. Calculate the contribution to $U_{\mathrm{m}, \text { vib }}$ at $25^{\circ} \mathrm{C}$ of a normal mode with wavenumber $\widetilde{v} \equiv v_s / c$ of $900 \mathrm{~cm}^{-1}$.

In terms of the vibrational wavenumbers we can write the partition function as Q_\text{vib}(T) = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{ h c \tilde{ u}_j}{k_\text{B} T}} } It is convenient to define a characteristic vibrational temperature \Theta_{i,\text{vib}} = \frac{h u_i}{k_\text{B}} where u is experimentally determined for each vibrational mode by taking a spectrum or by calculation. Q_\text{vib}(T) =\prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} = \prod_j e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}} \sum_n \left( e^{-\frac{\hbar \omega_j}{k_\text{B} T}} \right)^n = \prod_j \frac{e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}}}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } where E_\text{ZP} = \frac{1}{2} \sum_j \hbar \omega_j is total vibrational zero point energy of the system. Using this approximation we can derive a closed form expression for the vibrational partition function. For an isolated molecule of n atoms, the number of vibrational modes (i.e. values of j) is 3n − 5 for linear molecules and 3n − 6 for non-linear ones.G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945 In crystals, the vibrational normal modes are commonly known as phonons. ==Approximations== ===Quantum harmonic oscillator=== The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or normal modes of the system are considered to be a set of uncoupled quantum harmonic oscillators. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom. ==Definition== For a system (such as a molecule or solid) with uncoupled vibrational modes the vibrational partition function is defined by Q_\text{vib}(T) = \prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} where T is the absolute temperature of the system, k_B is the Boltzmann constant, and E_{j,n} is the energy of j-th mode when it has vibrational quantum number n = 0, 1, 2, \ldots . The second formula is adequate for small values of the vibrational quantum number. Number of degrees of vibrational freedom for nonlinear molecules: 3N-6 Number of degrees of vibrational freedom for linear molecules: 3N-5Housecroft, Catherine E., and A. G. Sharpe. It has also been applied to the study of unstable molecules such as dicarbon, C2, in discharges, flames and astronomical objects.Hollas, p. 211. == Principles == Electronic transitions are typically observed in the visible and ultraviolet regions, in the wavelength range approximately 200–700 nm (50,000–14,000 cm−1), whereas fundamental vibrations are observed below about 4000 cm−1.Energy is related to wavenumber by E=hc \bar u, where h=Planck's constant and c is the velocity of light When the electronic and vibrational energy changes are so different, vibronic coupling (mixing of electronic and vibrational wave functions) can be neglected and the energy of a vibronic level can be taken as the sum of the electronic and vibrational (and rotational) energies; that is, the Born–Oppenheimer approximation applies.Banwell and McCash, p. 162. Vibronic spectra of diatomic molecules in the gas phase have been analyzed in detail.Hollas, pp. 210–228 Vibrational coarse structure can sometimes be observed in the spectra of molecules in liquid or solid phases and of molecules in solution. The molecule is excited to another electronic state and to many possible vibrational states v'=0, 1, 2, 3, ... . The vibrational temperature is used commonly when finding the vibrational partition function. The overall molecular energy depends not only on the electronic state but also on vibrational and rotational quantum numbers, denoted v and J respectively for diatomic molecules. The transition energies, expressed in wavenumbers, of the lines for a particular vibronic transition are given, in the rigid rotor approximation, that is, ignoring centrifugal distortion, byBanwell and McCash, p. 171 :G(J^\prime, J^{\prime \prime}) = \bar u _{v^\prime-v^{\prime\prime}}+B^\prime J^\prime (J^\prime +1)-B^{\prime\prime} J^{\prime\prime}(J^{\prime\prime} +1) Here B are rotational constants and J are rotational quantum numbers. In the next approximation the term values are given by : G(v) = \bar u _{electronic} + \omega_e (v+{1 \over 2}) - \omega_e\chi_e (v+{1 \over 2})^2\, where χe is an anharmonicity constant. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. The Renner–Teller effect is observed in the spectra of molecules having electronic states that allow vibration through a linear configuration. Later studies on the same anion were also able to account for vibronic transitions involving low-frequency lattice vibrations. == Notes == == References == == Bibliography == * Chapter: Molecular Spectroscopy 2. : R_{\rm specific} = \frac{R}{M} Just as the molar gas constant can be related to the Boltzmann constant, so can the specific gas constant by dividing the Boltzmann constant by the molecular mass of the gas. To determine the vibrational spectroscopy of linear molecules, the rotation and vibration of linear molecules are taken into account to predict which vibrational (normal) modes are active in the infrared spectrum and the Raman spectrum. == Degrees of freedom == The location of a molecule in a 3-dimensional space can be described by the total number of coordinates. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . The potential at infinite internuclear distance is the dissociation energy for pure vibrational spectra.

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Calculate the magnitude of the spin magnetic moment of an electron.

In turn, calculation of the magnitude of the total spin magnetic moment requires that () be replaced by: Thus, for a single electron, with spin quantum number the component of the magnetic moment along the field direction is, from (), while the (magnitude of the) total spin magnetic moment is, from (), or approximately 1.73 μ. The component of the orbital magnetic dipole moment for an electron with a magnetic quantum number ℓ is given by :(\boldsymbol{\mu}_\text{L})_z = -\mu_\text{B} m_\ell. ==History== The electron magnetic moment is intrinsically connected to electron spin and was first hypothesized during the early models of the atom in the early twentieth century. Again it is important to notice that is a negative constant multiplied by the spin, so the magnetic moment of the electron is antiparallel to the spin. : Number of unpaired electrons Spin-only moment () 1 1.73 2 2.83 3 3.87 4 4.90 5 5.92 === Elementary particles === In atomic and nuclear physics, the Greek symbol represents the magnitude of the magnetic moment, often measured in Bohr magnetons or nuclear magnetons, associated with the intrinsic spin of the particle and/or with the orbital motion of the particle in a system. Note that is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum. In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin and electric charge. The analysis is readily extended to the spin-only magnetic moment of an atom. The spin magnetic dipole moment is approximately one B because g_{\rm s} \approx 2 and the electron is a spin- particle (): The component of the electron magnetic moment is (\boldsymbol{\mu}_\text{s})_z = -g_\text{s}\,\mu_\text{B}\,m_\text{s}\,, where s is the spin quantum number. The spin frequency of the electron is determined by the -factor. : u_s = \frac{g}{2} u_c : \frac{g}{2} = \frac{\bar{ u}_c + \bar{ u}_a}{\bar{ u}_c} ==See also== * Spin (physics) * Electron precipitation * Bohr magneton * Nuclear magnetic moment * Nucleon magnetic moment * Anomalous magnetic dipole moment * Electron electric dipole moment * Fine structure * Hyperfine structure ==References== ==Bibliography== * * Category:Atomic physics Category:Electric dipole moment Category:Magnetic moment Category:Particle physics Category:Physical constants Values of the intrinsic magnetic moments of some particles are given in the table below: : Intrinsic magnetic moments and spins of some elementary particles Particle name (symbol) Magnetic dipole moment (10 J⋅T) Spin quantum number (dimensionless) electron (e−) proton (H+) neutron (n) muon (μ) deuteron (H) 1 triton (H) helion (He) alpha particle (He) 0 0 For the relation between the notions of magnetic moment and magnetization see magnetization. == See also == * Moment (physics) * Electric dipole moment * Toroidal dipole moment * Magnetic susceptibility * Orbital magnetization * Magnetic dipole–dipole interaction == References and notes == ==External links== * Category:Magnetostatics Category:Magnetism Category:Electric and magnetic fields in matter Category:Physical quantities Category:Moment (physics) Category:Magnetic moment Here is the electron spin angular momentum. The magnetic moment of such a particle is parallel to its spin. If the electron is visualized as a classical rigid body in which the mass and charge have identical distribution and motion that is rotating about an axis with angular momentum , its magnetic dipole moment is given by: \boldsymbol{\mu} = \frac{-e}{2m_\text{e}}\,\mathbf{L}\,, where e is the electron rest mass. In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin of elementary particles. The CODATA value for the electron magnetic moment is : ===Orbital magnetic dipole moment=== The revolution of an electron around an axis through another object, such as the nucleus, gives rise to the orbital magnetic dipole moment. The magnetic moment of the electron is : \mathbf{m}_\text{S} = -\frac{g_\text{S} \mu_\text{B} \mathbf{S}}{\hbar}, where is the Bohr magneton, is electron spin, and the g-factor is 2 according to Dirac's theory, but due to quantum electrodynamic effects it is slightly larger in reality: . However, in order to obtain the magnitude of the total spin angular momentum, be replaced by its eigenvalue, where s is the spin quantum number. The magnetic moment of the electron has been measured using a one-electron quantum cyclotron and quantum nondemolition spectroscopy. The value of the electron magnetic moment (symbol μe) is In units of the Bohr magneton (μB), it is , a value that was measured with a relative accuracy of . ==Magnetic moment of an electron== The electron is a charged particle with charge −, where is the unit of elementary charge. The sum of the proton and neutron magnetic moments gives 0.879 µN, which is within 3% of the measured value 0.857 µN. The ratio between the true spin magnetic moment and that predicted by this model is a dimensionless factor , known as the electron -factor: \boldsymbol{\mu} = g_\text{e}\,\frac{(-e)}{~2m_\text{e}~}\,\mathbf{L}\,. The intrinsic electron magnetic dipole moment is approximately equal to the Bohr magneton μ because and the electron's spin is also : \frac{\hbar}{2} = - \mu_\text{B}|}} Equation () is therefore normally written as: Just like the total spin angular momentum cannot be measured, neither can the total spin magnetic moment be measured.

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A particle is subject to the potential energy $V=a x^4+b y^4+c z^4$. If its ground-state energy is $10 \mathrm{eV}$, calculate $\langle V\rangle$ for the ground state.

The average energy in this state would be \langle\psi|H|\psi\rangle = \int dx\, \left(-\frac{\hbar^2}{2m} \psi^* \frac{d^2\psi}{dx^2} + V(x)|\psi(x)|^2\right), where is the potential. The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. Formulae are given in SI units and Gaussian-cgs units. == Definition == The electromagnetic four- potential can be defined as: : SI units Gaussian units A^\alpha = \left( \frac{1}{c}\phi, \mathbf{A} \right)\,\\! For hydrogen (H), an electron in the ground state has energy , relative to the ionization threshold. Also electronvolts may be used, 1 eV = 1.602×10−19 Joules. ==Electrostatic potential energy of one point charge== ===One point charge q in the presence of another point charge Q=== right|A point charge q in the electric field of another charge Q.|thumb|434px The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). An excited state is any state with energy greater than the ground state. An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. In quantum field theory, the ground state is usually called the vacuum state or the vacuum. The solutions are outlined in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box. === Vacuum case states === Let us now consider V(r) = 0. Therefore, the potential energy is unchanged up to order \varepsilon^2, if we deform the state \psi with a node into a state without a node, and the change can be ignored. Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. Now, consider the potential energy. The correction to the potential V(r) is called the centrifugal barrier term. An object may be said to have electric potential energy by virtue of either its own electric charge or its relative position to other electrically charged objects. We can therefore remove all nodes and reduce the energy by O(\varepsilon), which implies that cannot be the ground state. However, the contribution to the potential energy from this region for the state with a node is V^\varepsilon_\text{avg} = \int_{-\varepsilon}^\varepsilon dx\, V(x)|\psi|^2 = |c|^2\int_{-\varepsilon}^\varepsilon dx\, x^2V(x) \simeq \frac{2}{3}\varepsilon^3|c|^2 V(0) + \cdots, lower, but still of the same lower order O(\varepsilon^3) as for the deformed state , and subdominant to the lowering of the average kinetic energy. While both the scalar and vector potential depend upon the frame, the electromagnetic four-potential is Lorentz covariant. When talking about electrostatic potential energy, time-invariant electric fields are always assumed so, in this case, the electric field is conservative and Coulomb's law can be used. In other words, 13.6 eV is the energy input required for the electron to no longer be bound to the atom. (The average energy may then be further lowered by eliminating undulations, to the variational absolute minimum.) === Implication === As the ground state has no nodes it is spatially non- degenerate, i.e. there are no two stationary quantum states with the energy eigenvalue of the ground state (let's name it E_g) and the same spin state and therefore would only differ in their position-space wave functions. The term "electric potential energy" is used to describe the potential energy in systems with time-variant electric fields, while the term "electrostatic potential energy" is used to describe the potential energy in systems with time-invariant electric fields. ==Definition== The electric potential energy of a system of point charges is defined as the work required to assemble this system of charges by bringing them close together, as in the system from an infinite distance. The electrostatic potential energy can also be defined from the electric potential as follows: ==Units== The SI unit of electric potential energy is joule (named after the English physicist James Prescott Joule).

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For an electron in a certain rectangular well with a depth of $20.0 \mathrm{eV}$, the lowest energy level lies $3.00 \mathrm{eV}$ above the bottom of the well. Find the width of this well. Hint: Use $\tan \theta=\sin \theta / \cos \theta$

The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. A potential well is the region surrounding a local minimum of potential energy. Depth in a well is not necessarily measured vertically or along a straight line. In the oil and gas industry, depth in a well is the measurement, for any point in that well, of the distance between a reference point or elevation, and that point. Specification of a differential depth or a thickness: in Figure 2 above, the thickness of the reservoir penetrated by the well might be 57 mMD or 42 mTVD, even though the reservoir true stratigraphic thickness in that area (or isopach) might be only 10 m, and its true vertical thickness (isochore), 14 m. ==See also== *Measured depth *True vertical depth ==References== ==External links== * Determining Lowest Astronomical Tide (LAT) * Seas and Submerged Lands Act 1973 (Australia) * Log Data Acquisition and Quality Control, Ph. A potential hill is the opposite of a potential well, and is the region surrounding a local maximum. ==Quantum confinement== thumb|500 px|Quantum confinement is responsible for the increase of energy difference between energy states and band gap, a phenomenon tightly related to the optical and electronic properties of the materials. But there, this additional width is interpreted as energy dispersion, which is, to the first order, |\Delta r_\pi|_\varepsilon = 2R_P\,\Delta E/E_P. Because wells are not always drilled vertically, there may be two "depths" for every given point in a wellbore: the measured depth (MD) measured along the path of the borehole, and the true vertical depth (TVD), the absolute vertical distance between the datum and the point in the wellbore. Well depth may refer to: *Depth in a well, a measurement of location in oil and gas drilling and production *The charge capacity of each pixel in a charge-coupled device 350px|right|thumb In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. The total energy of the configuration is defined as zero because the charge of the electron is subject to no electric field due to other sources of charge. Configurations reproduced in * * * This webpage contains many more electron configurations with the lowest known energy: https://www.hars.us. Category:Electrostatics Category:Electron Category:Circle packing Category:Unsolved problems in mathematics A well may reach to many kilometers.Sakhalin-1 sets new extended reach drilling record, Rosneft says, 2015 ==Figures== thumb|left|Fig. 1: The specification of depthsthumb|right|Fig. 2: Differential depths: reservoir thickness, isochor, isopach Specification of an absolute depth: in Figure 1 above, point P1 might be at 3207 mMDRT and 2370 mTVDMSL, while point P2 might be at 2530 mMDRT and 2502 mTVDLAT. The problem consists of solving the one- dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. For example, a spherical shell of N=1 represents the uniform distribution of a single electron's charge, -e across the entire shell. ===Randomly distributed point charges=== The global energy of a system of electrons distributed in a purely random manner across the surface of the sphere is given by :U_{\text{rand}}(N)=\frac{N(N-1)}{2} and is, in general, greater than the energy of every Thomson problem solution. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. The angular spread, while also worsening the energy resolution, shows some focusing as the equal negative and positive deviations map to the same final spot. center|thumb|upright=3|Distance from the central trajectory at the exit of a hemispherical electron energy analyzer depending on the electron's kinetic energy, initial position within the 1 mm slit, and the angle at which it enters the radial field after passing through the slit. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance: * constrained global optimization (Altschuler et al. 1994), * steepest descent (Claxton and Benson 1966, Erber and Hockney 1991), * random walk (Weinrach et al. 1990), * genetic algorithm (Morris et al. 1996) While the objective is to minimize the global electrostatic potential energy of each N-electron case, several algorithmic starting cases are of interest. ===Continuous spherical shell charge=== thumb|The extreme upper energy limit of the Thomson Problem is given by N^2/2 for a continuous shell charge followed by N(N − 1)/2, the energy associated with a random distribution of N electrons. Initial angular spread, dependent on the chosen slit and aperture width, worsens the energy resolution.|alt= When two voltages, V_{1} and V_{2}, are applied to the inner and outer hemispheres, respectively, the electric potential in the region between the two electrodes follows from the Laplace equation: : V(r)= - \left[\frac{V_{2}-V_{1}}{R_{2}-R_{1}}\right]\cdot\frac{R_{1}R_{2}}{r} + const. In these cases the measured depth will continue to increase while true vertical depth will decrease toward the toe of the wellbore. ==Depth in practice== * Unit: the usual unit of depth is the metre (m). alt=|thumb|upright=1.2|Hemispherical electron energy analyzer. thumb|400px|right|A generic potential energy well.

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