The cambridge handbook of physics formulas

The Cambridge Handbook of Physics Formulas GRAHAM WOAN Department of Physics & Astronomy University of Glasgow PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA http://www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia ´ 13, 28014 Madrid, Spain Ruiz de Alarcon c Cambridge University Press 2000 This book is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 2000 Printed in the United States of America Typeface Times Roman 10/12 pt System LATEX 2ε [tb] A catalog record for this book is available from the British Library Library of Congress Cataloging in Publication Data Woan, Graham, 1963– The Cambridge handbook of physics formulas / Graham Woan p cm ISBN 0-521-57349-1 – ISBN 0-521-57507-9 (pbk.) Physics – Formulas QC61.W67 I Title 1999 530′ 02′ 12 – dc21 ISBN 521 57349 hardback ISBN 521 57507 paperback 99-15228 CIP Contents Preface page vii How to use this book 1 Units, constants, and conversions 1.1 Introduction, • 1.2 SI units, • 1.3 Physical constants, • 1.4 Converting between units, 10 • 1.5 Dimensions, 16 • 1.6 Miscellaneous, 18 Mathematics 19 2.1 Notation, 19 • 2.2 Vectors and matrices, 20 • 2.3 Series, summations, and progressions, 27 • 2.4 Complex variables, 30 • 2.5 Trigonometric and hyperbolic formulas, 32 • 2.6 Mensuration, 35 • 2.7 Differentiation, 40 • 2.8 Integration, 44 • 2.9 Special functions and polynomials, 46 • 2.10 Roots of quadratic and cubic equations, 50 • 2.11 Fourier series and transforms, 52 • 2.12 Laplace transforms, 55 • 2.13 Probability and statistics, 57 • 2.14 Numerical methods, 60 Dynamics and mechanics 63 3.1 Introduction, 63 • 3.2 Frames of reference, 64 • 3.3 Gravitation, 66 • 3.4 Particle motion, 68 • 3.5 Rigid body dynamics, 74 • 3.6 Oscillating systems, 78 • 3.7 Generalised dynamics, 79 • 3.8 Elasticity, 80 • 3.9 Fluid dynamics, 84 Quantum physics 89 4.1 Introduction, 89 • 4.2 Quantum definitions, 90 • 4.3 Wave mechanics, 92 • 4.4 Hydrogenic atoms, 95 • 4.5 Angular momentum, 98 • 4.6 Perturbation theory, 102 • 4.7 High energy and nuclear physics, 103 Thermodynamics 5.1 Introduction, 105 • 5.2 Classical thermodynamics, 106 • 5.3 Gas laws, 110 • 5.4 Kinetic theory, 112 • 5.5 Statistical thermodynamics, 114 • 5.6 Fluctuations and noise, 116 • 5.7 Radiation processes, 118 105 Solid state physics 123 6.1 Introduction, 123 • 6.2 Periodic table, 124 • 6.3 Crystalline structure, 126 • 6.4 Lattice dynamics, 129 • 6.5 Electrons in solids, 132 Electromagnetism 135 7.1 Introduction, 135 • 7.2 Static fields, 136 • 7.3 Electromagnetic fields (general), 139 • 7.4 Fields associated with media, 142 • 7.5 Force, torque, and energy, 145 • 7.6 LCR circuits, 147 • 7.7 Transmission lines and waveguides, 150 • 7.8 Waves in and out of media, 152 • 7.9 Plasma physics, 156 Optics 161 8.1 Introduction, 161 • 8.2 Interference, 162 • 8.3 Fraunhofer diffraction, 164 • 8.4 Fresnel diffraction, 166 • 8.5 Geometrical optics, 168 • 8.6 Polarisation, 170 • 8.7 Coherence (scalar theory), 172 • 8.8 Line radiation, 173 Astrophysics 175 9.1 Introduction, 175 • 9.2 Solar system data, 176 • 9.3 Coordinate transformations (astronomical), 177 • 9.4 Observational astrophysics, 179 • 9.5 Stellar evolution, 181 • 9.6 Cosmology, 184 Index 187 Chapter Dynamics and mechanics 3.1 Introduction Unusually in physics, there is no pithy phrase that sums up the study of dynamics (the way in which forces produce motion), kinematics (the motion of matter), mechanics (the study of the forces and the motion they produce), and statics (the way forces combine to produce equilibrium) We will take the phrase dynamics and mechanics to encompass all the above, although it clearly does not! To some extent this is because the equations governing the motion of matter include some of our oldest insights into the physical world and are consequentially steeped in tradition One of the more delightful, or for some annoying, facets of this is the occasional use of arcane vocabulary in the description of motion The epitome must be what Goldstein1 calls “the jabberwockian sounding statement” the polhode rolls without slipping on the herpolhode lying in the invariable plane, describing “Poinsot’s construction” – a method of visualising the free motion of a spinning rigid body Despite this, dynamics and mechanics, including fluid mechanics, is arguably the most practically applicable of all the branches of physics Moreover, and in common with electromagnetism, the study of dynamics and mechanics has spawned a good deal of mathematical apparatus that has found uses in other fields Most notably, the ideas behind the generalised dynamics of Lagrange and Hamilton lie behind much of quantum mechanics H Goldstein, Classical Mechanics, 2nd ed., 1980, Addison-Wesley Dynamics and mechanics 64 3.2 Frames of reference Galilean transformations Time and positiona r = r ′ + vt t = t′ (3.1) (3.2) Velocity u = u′ + v (3.3) Momentum p = p ′ + mv (3.4) Angular momentum ′ J = J + mr × v + v× p t (3.5) Kinetic energy T = T ′ + mu′ · v + mv 2 r,r′ v t,t′ u,u′ velocity in frames S and S ′ p,p ′ particle momentum in frames S and S ′ particle mass m a Frames ′ ′ position in frames S and S ′ velocity of S ′ in S time in S and S ′ J ,J ′ angular momentum in frames S and S ′ (3.6) T ,T ′ kinetic energy in frames S and S ′ (3.7) γ v Lorentz factor velocity of S ′ in S c speed of light S m S′ r r′ vt coincide at t = Lorentz (spacetime) transformationsa Lorentz factor γ = 1− v2 c2 −1/2 Time and position x = γ(x′ + vt′ ); x′ = γ(x − vt) (3.8) ′ ′ y=y ; y =y (3.9) z = z′; z′ = z (3.10) v v (3.11) t = γ t′ + x′ ; t′ = γ t − x c c Differential dX = (cdt,−dx,−dy,−dz) four-vectorb (3.12) ′ S S x,x′ t,t′ X x-position in frames S and S ′ (similarly for y and z) time in frames S and S′ v ′ x x spacetime four-vector a For frames S and S ′ coincident at t = in relative motion along x See page 141 for the transformations of electromagnetic quantities b Covariant components, using the (1,−1,−1,−1) signature Velocity transformationsa Velocity u′x + v ux = ; + u′x v/c2 u′y ; uy = γ(1 + u′x v/c2 ) u′z uz = ; γ(1 + u′x v/c2 ) a For ux − v − ux v/c2 uy u′y = γ(1 − ux v/c2 ) uz u′z = γ(1 − ux v/c2 ) u′x = γ Lorentz factor = [1 − (v/c)2 ]−1/2 v c velocity of S ′ in S speed of light ui ,u′i particle velocity components in frames S and S ′ (3.13) (3.14) (3.15) frames S and S ′ coincident at t = in relative motion along x ′ S S u v ′ x x 3.2 Frames of reference 65 Momentum and energy transformationsa Momentum and energy px = γ(p′x + vE ′ /c2 ); py = p′y ; pz = p′z ; E = γ(E ′ + vp′x ); 2 ′2 p′x = γ(px − vE/c2 ) p′y = py p′z = pz E ′ = γ(E − vpx ) ′2 a For Lorentz factor = [1 − (v/c)2 ]−1/2 v c velocity of S ′ in S speed of light px ,p′x E,E ′ x components of momentum in S and S ′ (sim for y and z) energy in S and S ′ (rest) mass total momentum in S momentum four-vector = m20 c4 (3.20) m0 p P = (E/c,−px ,−py ,−pz ) (3.21) P E −p c =E −p c Four-vectorb (3.16) (3.17) (3.18) (3.19) γ ′ S S v ′ x x frames S and S ′ coincident at t = in relative motion along x components, using the (1,−1,−1,−1) signature b Covariant Propagation of lighta v ν′ = γ + cosα ν c Doppler effect ′ cosθ + v/c + (v/c)cosθ′ cosθ − v/c cosθ′ = − (v/c)cosθ cosθ = Aberration b Relativistic beamingc (3.22) P (θ) = (3.23) (3.24) sinθ 2γ [1 − (v/c)cosθ]2 (3.25) ν frequency received in S ν′ α γ frequency emitted in S ′ arrival angle in S Lorentz factor = [1 − (v/c)2 ]−1/2 velocity of S ′ in S speed of light v c S y c α x S S ′′ y y θ,θ′ emission angle of light in S and S ′ v θ′ ′c x x P (θ) angular distribution of photons in S frames S and S ′ coincident at t = in relative motion along x travelling in the opposite sense has a propagation angle of π + θ radians c Angular distribution of photons from a source, isotropic and stationary in S ′ π P (θ) dθ = a For b Light Four-vectorsa Covariant and contravariant components x0 = x0 x3 = −x3 (3.26) Scalar product xi yi = x0 y0 + x1 y1 + x2 y2 + x3 y3 (3.27) x2 = −x2 x1 = −x1 xi xi ,x′ i four-vector components in frames S and S ′ Lorentz transformations ′1 ′0 x0 = γ[x′ + (v/c)x′ ]; x = γ[x + (v/c)x ]; ′2 x =x ; xi covariant vector components contravariant components x′ = γ[x0 − (v/c)x1 ] ′1 x = γ[x − (v/c)x ] ′3 x =x (3.28) (3.29) (3.30) v Lorentz factor = [1 − (v/c)2 ]−1/2 velocity of S ′ in S c speed of light γ frames S and S ′ , coincident at t = in relative motion along the (1) direction Note that the (1,−1,−1,−1) signature used here is common in special relativity, whereas (−1,1,1,1) is often used in connection with general relativity (page 67) a For Dynamics and mechanics 66 Rotating frames A Vector transformation dA dt dA dt + ω× A (3.31) Acceleration ˙v = ˙v ′ + 2ω× v ′ + ω× (ω× r ′ ) (3.32) Coriolis force F ′cor = −2mω× v ′ (3.33) F ′cen = −mω× (ω× r ′ ) (3.34) Centrifugal force Motion relative to Earth Foucault’s penduluma a The 3.3 = S S′ = +mω r y sin z cos) mă x = Fx + 2mωe (˙ (3.35) stationary frame rotating frame angular velocity of S ′ in S ′ ˙v ,˙v accelerations in S and S ′ v ′ velocity in S ′ r ′ position in S ′ F ′cor coriolis force m particle mass sin mă y = Fy 2me x cos măz = Fz mg + 2mωe x (3.37) (3.38) Ωf = −ωe sinλ (3.39) ω λ z nongravitational force latitude local vertical axis y x northerly axis easterly axis Ωf pendulum’s rate of turn F ′cen r ′⊥ F ′cen centrifugal force r ′⊥ perpendicular to particle from rotation axis Fi (3.36) any vector S S′ ω m r′ ωe y z x λ ωe Earth’s spin rate sign is such as to make the rotation clockwise in the northern hemisphere Gravitation Newtonian gravitation Newton’s law of gravitation Gm1 m2 F1= rˆ 12 r12 (3.40) Newtonian field equationsa g = −∇φ (3.41) Fields from an isolated uniform sphere, mass M, r from the centre a The ∇2 φ = −∇ · g = 4πGρ  GM  − rˆ (r > a) r g(r) =  − GMr rˆ (r < a)  a GM  − (r > a) r φ(r) =   GM (r − 3a2 ) (r < a) 2a3 gravitational force on a mass m is mg (3.42) (3.43) m1,2 F1 r 12 ˆ masses force on m1 (= −F ) vector from m1 to m2 unit vector G g φ ρ constant of gravitation gravitational field strength gravitational potential mass density r M a vector from sphere centre mass of sphere radius of sphere M (3.44) a r 3.3 Gravitation 67 General relativitya Line element Christoffel symbols and covariant differentiation ds2 = gµν dxµ dxν = −dτ2 (3.45) Γαβγ = g αδ (gδβ,γ + gδγ,β − gβγ,δ ) φ;γ = φ,γ ≡ ∂φ/∂xγ Aα;γ = Aα,γ + Γαβγ Aβ (3.46) (3.47) (3.48) Bα;γ = Bα,γ − Γβαγ Bβ (3.49) = Γαµγ Γµβδ − Γαµδ Γµβγ + Γαβδ,γ − Γαβγ,δ Bµ;α;β − Bµ;β;α = R γµαβ Bγ (3.50) ds invariant interval dτ gµν dxµ Γαβγ proper time interval metric tensor differential of xµ Christoffel symbols ,α ;α φ Aα partial diff w.r.t xα covariant diff w.r.t xα scalar contravariant vector Bα covariant vector R αβγδ Riemann tensor vµ tangent vector (= dxµ /dλ) affine parameter (e.g., τ for material particles) R αβγδ Riemann tensor Geodesic equation (3.51) Rαβγδ = −Rαβδγ ; Rβαγδ = −Rαβγδ Rαβγδ + Rαδβγ + Rαγδβ = (3.52) (3.53) Dv µ =0 Dλ (3.54) where DAµ dAµ ≡ + Γµαβ Aα v β Dλ dλ (3.55) λ Geodesic deviation D2 ξ µ = −R µαβγ v α ξ β v γ Dλ2 (3.56) ξµ geodesic deviation Ricci tensor Rαβ ≡ R σασβ = g σδ Rδασβ = Rβα (3.57) Rαβ Ricci tensor Einstein tensor Gµν = R µν − (3.58) Gµν R Einstein tensor Ricci scalar (= g µν Rµν ) Einstein’s field equations Gµν = 8πT µν (3.59) T µν p stress-energy tensor pressure (in rest frame) Perfect fluid T µν = (p + ρ)uµ uν + pg µν (3.60) ρ uν density (in rest frame) fluid four-velocity Schwarzschild solution (exterior) ds2 = − − µν g R 2M r dt2 + − 2M r + r (dθ + sin2 θ dφ2 ) −1 dr (3.61) spherically symmetric mass (see Section 9.5) (r,θ,φ) spherical polar coords t time M Kerr solution (outside a spinning black hole) ∆ − a2 sin2 θ 2Mr sin2 θ dt − 2a dt dφ ̺2 ̺2 (r + a2 )2 − a2 ∆sin2 θ ̺2 2 dr + ̺2 dθ2 + sin θ dφ + ̺2 ∆ ds2 = − J angular momentum (along z) a ≡ J/M ∆ ≡ r2 − 2Mr + a2 (3.62) ̺ ≡ r2 + a2 cos2 θ a General relativity conventionally uses “geometrized units” in which G = and c = Thus, 1kg = 7.425 × 10−28 m etc Contravariant indices are written as superscripts and covariant indices as subscripts Note also that ds2 means (ds)2 etc Dynamics and mechanics 68 3.4 Particle motion Dynamics definitionsa Newtonian force F = măr = p˙ (3.63) F m force mass of particle r particle position vector Momentum p = m˙r (3.64) p momentum Kinetic energy T = mv 2 (3.65) T v kinetic energy particle velocity Angular momentum J = r× p (3.66) J angular momentum Couple (or torque) G = r× F (3.67) G couple (3.68) R0 mi ri position vector of centre of mass mass of ith particle position vector of ith particle Centre of mass (ensemble of N particles) a In N i=1 mi r i N i=1 mi R0 = the Newtonian limit, v ≪ c, assuming m is constant Relativistic dynamicsa (3.69) γ v c Lorentz factor particle velocity speed of light (3.70) p m0 relativistic momentum particle (rest) mass F force on particle t time (3.72) Er particle rest energy T = m0 c2 (γ − 1) (3.73) T relativistic kinetic energy E = γm0 c2 (3.74) E total energy (= Er + T ) v2 c2 Lorentz factor γ = 1− Momentum p = γm0 v Force F= Rest energy Er = m0 c2 Kinetic energy Total energy −1/2 dp dt 2 = (p c (3.71) + m20 c4 )1/2 (3.75) a It is now common to regard mass as a Lorentz invariant property and to drop the term “rest mass.” The symbol m0 is used here to avoid confusion with the idea of “relativistic mass” (= γm0 ) used by some authors Constant acceleration v = u + at 2 v = u + 2as s = ut + at2 u+v t s= (3.76) (3.77) (3.78) (3.79) u initial velocity v t s a final velocity time distance travelled acceleration 3.4 Particle motion 69 Reduced mass (of two interacting bodies) r m2 m1 centre of mass r2 r1 m1 m2 m1 + m2 m2 r r1 = m1 + m2 −m1 r r2 = m1 + m2 µ= Reduced mass Distances from centre of mass (3.80) µ mi reduced mass interacting masses (3.81) ri position vectors from centre of mass (3.82) r |r| r = r1 − r2 distance between masses Moment of inertia I = µ|r|2 (3.83) I moment of inertia Total angular momentum J = àrìr (3.84) J angular momentum Lagrangian L = µ|˙r |2 − U(|r|) (3.85) L Lagrangian U potential energy of interaction Ballisticsa Velocity v = v0 cosα xˆ + (v0 sinα − gt) yˆ (3.86) v = v02 − 2gy (3.87) Trajectory gx2 y = xtanα − 2v0 cos2 α Maximum height h= v02 sin2 α 2g (3.89) Horizontal range l= v02 sin2α g (3.90) (3.88) v0 initial velocity v α g velocity at t elevation angle gravitational acceleration ˆ t unit vector time h maximum height l range a Ignoring the curvature and rotation of the Earth and frictional losses g is assumed constant yˆ v0 h α l xˆ Dynamics and mechanics 70 Rocketry Escape velocitya 2GM vesc = r Specific impulse Isp = u g Exhaust velocity (into a vacuum) u= 2γRTc (γ − 1)µ Rocket equation (g = 0) Mi ∆v = uln Mf Multistage rocket vesc escape velocity G M r Isp constant of gravitation mass of central body central body radius specific impulse (3.92) u g R γ effective exhaust velocity acceleration due to gravity molar gas constant ratio of heat capacities (3.93) Tc µ combustion temperature effective molecular mass of exhaust gas rocket velocity increment pre-burn rocket mass post-burn rocket mass 1/2 (3.91) 1/2 ≡ ulnM (3.94) M mass ratio N number of stages (3.95) Mi ui mass ratio for ith burn exhaust velocity of ith burn (3.96) t burn time θ rocket zenith angle N ui lnMi ∆v = i=1 In a constant gravitational field Hohmann cotangential transferb ∆v = ulnM − gtcosθ GM ∆vah = 1/2 GM rb 1/2 2rb + rb 1/2 −1 (3.97) ∆vhb = 1− 2ra + rb ∆v Mi Mf ∆vah velocity increment, a to h ∆vhb rb velocity increment, h to b radius of inner orbit radius of outer orbit 1/2 transfer ellipse, h a b (3.98) a From the surface of a spherically symmetric, nonrotating body, mass M between coplanar, circular orbits a and b, via ellipse h with a minimal expenditure of energy b Transfer 3.4 Particle motion 71 Gravitationally bound orbital motiona U(r) potential energy α GMm ≡− r r Potential energy of interaction U(r) = − Total energy J2 α α =− E =− + r 2mr 2a (3.100) Virial theorem (1/r potential) E = U /2 = − T U = −2 T (3.101) (3.102) Orbital equation (Kepler’s 1st law) Rate of sweeping area (Kepler’s 2nd law) r0 = + ecosφ , r a(1 − e2 ) r= + ecosφ Semi-major axis a= Semi-minor axis b= (3.99) or (3.103) (3.104) G M m α constant of gravitation central mass orbiting mass (≪ M) positive constant E J total energy (constant) total angular momentum (constant) T · kinetic energy mean value r0 r e semi-latus-rectum distance of m from M eccentricity J dA = = constant dt 2m (3.105) A area swept out by radius vector (total area = πab) r0 α = 1−e 2|E| (3.106) a semi-major axis b semi-minor axis 2a J r0 = (1 − e2 )1/2 (2m|E|)1/2 Eccentricityb e= 1+ Semi-latusrectum r0 = 2EJ mα2 1/2 = 1− (3.107) b2 a2 m A 1/2 r0 φ M (3.108) ae J b2 = = a(1 − e2 ) mα a r0 rmin = = a(1 − e) 1+e r0 rmax = = a(1 + e) 1−e (3.110) rmin pericentre distance (3.111) rmax apocentre distance Phase cosφ = (3.112) φ orbital phase Period (Kepler’s 3rd law) P = πα P orbital period Pericentre Apocentre a For (3.109) (J/r) − (mα/J) (2mE + m2 α2 /J )1/2 m 2|E|3 1/2 = 2πa3/2 m α r 2b rmax rmin 1/2 (3.113) an inverse-square law of attraction between two isolated bodies in the nonrelativistic limit If m is not ≪ M, all explicit references to m in Equations (3.100) to (3.113) should be replaced by the reduced mass, µ = Mm/(M +m), and r taken as the body separation The distance of mass m from the centre of mass is then rµ/m (see earlier table on Reduced mass) Other orbital dimensions scale similarly b Note that if the total energy, E, is < then e < and the orbit is an ellipse (a circle if e = 0) If E = 0, then e = and the orbit is a parabola If E > then e > and the orbit becomes a hyperbola (see Rutherford scattering on next page) Dynamics and mechanics 72 Rutherford scatteringa y trajectory for α < b x scattering centre χ a rmin trajectory for α > rmin Scattering potential energy Scattering angle χ |α| tan = 2Eb |α| χ α csc − 2E |α| = a(e ± 1) Eccentricity e= Motion trajectoryb 4E 2 y x − =1 α2 b Rutherford scattering formulad a Nonrelativistic (3.115) r α particle separation constant χ E b scattering angle total energy (> 0) impact parameter (3.118) rmin closest approach a hyperbola semi-axis e eccentricity (3.119) 4E b2 +1 α2 x=± U(r) potential energy (3.117) |α| 2E a= (3.114) (3.116) rmin = Semi-axis Scattering centrec (α0) α U(r) = − r < repulsive α > attractive Closest approach a α2 + b2 4E 1/2 = csc χ (3.120) (3.121) x,y position with respect to hyperbola centre 1/2 dσ dN = dΩ n dΩ α 4χ = csc 4E (3.122) dσ dΩ (3.123) (3.124) differential scattering cross section n beam flux density dN number of particles scattered into dΩ Ω solid angle treatment for an inverse-square force law and a fixed scattering centre Similar scattering results from either an attractive or repulsive force See also Conic sections on page 38 b The correct branch can be chosen by inspection c Also the focal points of the hyperbola d n is the number of particles per second passing through unit area perpendicular to the beam 3.4 Particle motion 73 Inelastic collisionsa m1 m2 v1 m1 v2 Before collision Coefficient of restitution Loss of kinetic energyb v2′ After collision v2′ − v1′ = ǫ(v1 − v2 ) ǫ = if perfectly elastic (3.125) (3.126) ǫ = if perfectly inelastic (3.127) T −T′ = − ǫ2 T ǫ vi vi′ coefficient of restitution pre-collision velocities post-collision velocities T ,T ′ total KE in zero momentum frame before and after collision mi particle masses (3.128) m1 − ǫm2 (1 + ǫ)m2 v1 + v2 m1 + m2 m1 + m2 (1 + ǫ)m1 m2 − ǫm1 v2 + v1 v2′ = m1 + m2 m1 + m2 v1′ = Final velocities m2 v1′ (3.129) (3.130) a Along b In the line of centres, v1 ,v2 ≪ c zero momentum frame Oblique elastic collisionsa θ Before collision m1 Directions of motion Relative separation angle tanθ1′ = θ2′ = θ a Collision After collision m1 m2 sin2θ m1 − m2 cos2θ (m21 + m22 − 2m1 m2 cos2θ)1/2 v m1 + m2 2m1 v v2′ = cosθ m1 + m2 v2′ θ1′ v1′ v   > π/2 if m1 < m2 θ1′ + θ2′ = π/2 if m1 = m2   < π/2 if m1 > m2 v1′ = Final velocities m2 θ2′ m2 θ (3.131) (3.132) θi′ mi angle between centre line and incident velocity final trajectories sphere masses (3.133) (3.134) v (3.135) vi′ between two perfectly elastic spheres: m2 initially at rest, velocities ≪ c incident velocity of m1 final velocities Dynamics and mechanics 74 3.5 Rigid body dynamics Moment of inertia tensor Moment of inertia tensora  Iij = (r δij − xi xj ) dm (y + z ) dm  I =  − xy dm − xz dm Parallel axis theorem − xy dm (x2 + z ) dm − yz dm (3.136)  − xz dm  − yz dm  (x2 + y ) dm ⋆ − ma1 a2 I12 = I12 r δij r2 = x2 + y + z Kronecker delta moment of inertia tensor dm mass element I (3.137) (3.138) xi position vector of dm Iij components of I Iij⋆ tensor with respect to centre of mass ,a position vector of centre of mass m mass of body ⋆ + m(a22 + a23 ) I11 = I11 (3.139) Iij = Iij⋆ + m(|a|2 δij − aj ) (3.140) Angular momentum J = Iω (3.141) J angular momentum ω angular velocity Rotational kinetic energy 1 T = ω · J = Iij ωi ωj 2 (3.142) T kinetic energy aI ii are the moments of inertia of the body Iij (i = j) are its products of inertia The integrals are over the body volume Principal axes Principal moment of inertia tensor Angular momentum  I1 I′ =  0 I2  0 I3 I′ (3.143) Ii J principal moment of inertia tensor principal moments of inertia angular momentum J = (I1 ω1 ,I2 ω2 ,I3 ω3 ) (3.144) ωi components of ω along principal axes Rotational kinetic energy T = (I1 ω12 + I2 ω22 + I3 ω32 ) (3.145) T Moment of inertia ellipsoida T = T (ω1 ,ω2 ,ω3 ) ∂T (J is ⊥ ellipsoid surface) Ji = ∂ωi Perpendicular axis theorem I1 + I2 Symmetries a The ≥ I3 = I3 generally flat lamina ⊥ to 3-axis I1 = I2 = I3 asymmetric top I1 = I2 = I3 I1 = I2 = I3 symmetric top spherical top ellipsoid is defined by the surface of constant T kinetic energy (3.146) I3 (3.147) I1 I2 (3.148) lamina (3.149) 3.5 Rigid body dynamics 75 Moments of inertiaa Thin rod, length l I1 = I2 = l ml 12 (3.150) I3 ≃ (3.151) Solid sphere, radius r I1 = I2 = I3 = mr (3.152) Spherical shell, radius r I1 = I2 = I3 = mr (3.153) Solid cylinder, radius r, length l m l2 I1 = I2 = r + I3 = mr 2 Solid cuboid, sides a,b,c I3 I1 r I3 I2 l (3.154) I1 I2 r (3.155) (3.156) (3.157) I3 = m(a2 + b2 )/12 (3.158) I1 = I2 = h m r2 + 20 I3 = mr 10 a (3.159) Solid ellipsoid, semi-axes a,b,c Elliptical lamina, semi-axes a,b I1 = mb2 /4 I2 = ma2 /4 I3 = m(a2 + b2 )/4 (3.164) (3.165) (3.166) a With h I3 I I r (3.161) (3.162) (3.163) Triangular plate b c I1 = m(b + c )/5 I2 = m(c2 + a2 )/5 I3 = m(a2 + b2 )/5 I1 = I2 = mr /4 I3 = mr /2 c I3 I2 (3.160) Disk, radius r I3 I1 I1 = m(b2 + c2 )/12 I2 = m(c2 + a2 )/12 Solid circular cone, base radius r, height hb I2 I1 m I3 = (a2 + b2 + c2 ) 36 I3 a c b I2 I1 I2 b I3 a I1 I2 (3.167) (3.168) r I1 I3 a (3.169) b I3 c respect to principal axes for bodies of mass m and uniform density The radius of gyration is defined as k = (I/m)1/2 b Origin of axes is at the centre of mass (h/4 above the base) c Around an axis through the centre of mass and perpendicular to the plane of the plate Dynamics and mechanics 76 Centres of mass Solid hemisphere, radius r d = 3r/8 from sphere centre (3.170) Hemispherical shell, radius r d = r/2 from sphere centre (3.171) Sector of disk, radius r, angle 2θ sinθ d= r θ from disk centre (3.172) Arc of circle, radius r, angle 2θ d=r from circle centre (3.173) Arbitrary triangular lamina, height d = h/3 perpendicular from base (3.174) Solid cone or pyramid, height h d = h/4 perpendicular from base (3.175) Spherical cap, height h, sphere radius r solid: d = (2r − h)2 from sphere centre 3r − h shell: d = r − h/2 from sphere centre Semi-elliptical lamina, height h ah sinθ θ d= 4h 3π (3.176) (3.177) from base (3.178) is the perpendicular distance between the base and apex of the triangle Pendulums P period Simple pendulum P = 2π l θ2 + + ··· g 16 Conical pendulum P = 2π l cosα g Torsional penduluma Compound pendulumb Equal double pendulumc a Assuming P = 2π lI0 C (3.179) g gravitational acceleration l length θ0 maximum angular displacement (3.180) α cone half-angle (3.181) I0 moment of inertia of bob C torsional rigidity of wire (see page 81) l 1/2 + I2 cos2 γ2 + I3 cos2 γ3 ) (2 ± 2)g α m 1/2 l √ m 1/2 (ma2 + I1 cos2 γ1 P ≃ 2π mga P ≃ 2π l θ0 (3.182) a distance of rotation axis from centre of mass m mass of body Ii principal moments of inertia γi angles between rotation axis and principal axes l I0 a I1 I2 l 1/2 (3.183) the bob is supported parallel to a principal rotation axis b I.e., an arbitrary triaxial rigid body c For very small oscillations (two eigenmodes) I3 m l m 3.5 Rigid body dynamics 77 Tops and gyroscopes J herpolhode ω space cone invariable plane J3 polhode Ωp body cone moment of inertia ellipsoid θ support point a gyroscope prolate symmetric top Euler’s equations ˙ + (I3 − I2 )ω2 ω3 G1 = I1 ω ˙ + (I1 − I3 )ω3 ω1 G2 = I2 ω ˙ + (I2 − I1 )ω1 ω2 G3 = I3 ω Free symmetric topb (I3 < I2 = I1 ) I1 − I3 ω3 I1 J Ωs = I1 Free asymmetric topc Ω2b = a Steady gyroscopic precession Ωb = (I1 − I3 )(I2 − I3 ) ω3 I1 I2 (3.184) (3.185) (3.186) Gi external couple (= for free rotation) Ii ωi principal moments of inertia angular velocity of rotation (3.187) Ωb body frequency (3.188) Ωs J space frequency total angular momentum Ωp θ J3 precession angular velocity angle from vertical angular momentum around symmetry axis mass (3.189) Ω2p I1′ cosθ − Ωp J3 + mga = (3.190) Ωp ≃ (3.191) Mga/J3 (slow) ′ J3 /(I1 cosθ) (fast) mg m g a gravitational acceleration distance of centre of mass from support point moment of inertia about support point Gyroscopic stability J32 ≥ 4I1′ mgacosθ (3.192) Gyroscopic limit (“sleeping top”) J32 ≫ I1′ mga (3.193) Nutation rate Ωn = J3 /I1′ (3.194) Ωn nutation angular velocity Gyroscope released from rest Ωp = (3.195) t time a Components mga (1 − cosΩn t) J3 I1′ are with respect to the principal axes, rotating with the body body frequency is the angular velocity (with respect to principal axes) of ω around the 3-axis The space frequency is the angular velocity of the 3-axis around J , i.e., the angular velocity at which the body cone moves around the space cone c J close to 3-axis If Ω2 < 0, the body tumbles b b The Dynamics and mechanics 78 3.6 Oscillating systems Free oscillations Differential equation dx dx + ω02 x = + 2γ dt2 dt Underdamped solution (γ < ω0 ) Critically damped solution (γ = ω0 ) Overdamped solution (γ > ω0 ) γ damping factor (per unit mass) ω0 undamped angular frequency (3.197) A amplitude constant where ω = (ω02 − γ )1/2 (3.198) φ ω phase constant angular eigenfrequency x = e−γt (A1 + A2 t) (3.199) Ai amplitude constants x = e−γt (A1 eqt + A2 e−qt ) (3.200) − ω02 )1/2 2πγ an = an+1 ω π ω0 Q= ≃ if 2γ ∆ (3.201) (3.202) ∆ = ln Quality factor oscillating variable time x = Ae−γt cos(ωt + φ) where q = (γ Logarithmic decrementa (3.196) x t Q≫1 (3.203) ∆ logarithmic decrement an nth displacement maximum Q quality factor a The decrement is usually the ratio of successive displacement maxima but is sometimes taken as the ratio of successive displacement extrema, reducing ∆ by a factor of Logarithms are sometimes taken to base 10, introducing a further factor of log10 e Forced oscillations Differential equation Steadystate solutiona d2 x dx + ω02 x = F0 eiωf t + 2γ dt2 dt (3.204) x = Aei(ωf t−φ) , (3.205) where A = F0 [(ω02 − ωf2 )2 + (2γωf )2 ]−1/2 F0 /(2ω0 ) [(ω0 − ωf )2 + γ ]1/2 2γωf tanφ = ω0 − ωf2 ≃ (γ ≪ ωf ) (3.206) (3.207) (3.208) x t γ oscillating variable time damping factor (per unit mass) ω0 F0 undamped angular frequency force amplitude (per unit mass) forcing angular frequency amplitude phase lag of response behind driving force ωf A φ Amplitude resonanceb ωar = ω02 − 2γ (3.209) ωar amplitude resonant forcing angular frequency Velocity resonancec ωvr = ω0 (3.210) ωvr velocity resonant forcing angular frequency Quality factor Q= (3.211) Q quality factor Impedance Z = 2γ + i (3.212) Z impedance (per unit mass) a Excluding ω0 2γ ωf2 − ω02 ωf the free oscillation terms frequency for maximum displacement c Forcing frequency for maximum velocity Note φ = π/2 at this frequency b Forcing ... in physics, there is no pithy phrase that sums up the study of dynamics (the way in which forces produce motion), kinematics (the motion of matter), mechanics (the study of the forces and the. ..PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2... arguably the most practically applicable of all the branches of physics Moreover, and in common with electromagnetism, the study of dynamics and mechanics has spawned a good deal of mathematical