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

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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.)
1. Physics – Formulas.
QC61.W67

I. Title.

1999

530′ .02′ 12 – dc21

ISBN 0 521 57349 1 hardback
ISBN 0 521 57507 9 paperback

99-15228
CIP


Contents

Preface

page vii

How to use this book

1

1

3

Units, constants, and conversions
1.1 Introduction, 3 • 1.2 SI units, 4 • 1.3 Physical constants, 6
• 1.4 Converting between units, 10 • 1.5 Dimensions, 16
• 1.6 Miscellaneous, 18

2

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

3

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

4

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

5

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


6

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

7

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

8

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

9

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 3 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.

1 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

1
T = T ′ + mu′ · v + mv 2
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 = 0.

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′ + 2 x′ ; t′ = γ t − 2 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 = 0 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 =
;
1 + 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
1 − 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 = 0 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 2

′2

p′x = γ(px − vE/c2 )
p′y = py
p′z = pz
E ′ = γ(E − vpx )
′2 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 = 0 in relative motion along x.
components, using the (1,−1,−1,−1) signature.

b Covariant

Propagation of lighta
v
ν′
= γ 1 + cosα
ν
c

Doppler
effect

′

cosθ + v/c
1 + (v/c)cosθ′
cosθ − v/c
cosθ′ =
1 − (v/c)cosθ

cosθ =
Aberration

b

Relativistic
beamingc

(3.22)

P (θ) =

(3.23)
(3.24)

sinθ
2γ 2 [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 = 0 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 = 1.
0
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
0

1

′1

′0

x0 = γ[x′ + (v/c)x′ ];
1

x = γ[x + (v/c)x ];
2

′2

x =x ;

xi

covariant vector
components
contravariant components

0

x′ = γ[x0 − (v/c)x1 ]
′1

1

0

x = γ[x − (v/c)x ]
′3

x =x

3

(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 = 0 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 2 r ′⊥

y sinλ − ˙z cosλ)
m¨
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

˙ sinλ
m¨
y = Fy − 2mωe x
˙ cosλ

m¨z = 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
2
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

− 2 rˆ (r > a)
r
g(r) =

− GMr rˆ (r < a)
3
 a
GM

−
(r > a)
r
φ(r) =

 GM (r 2 − 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 2 )
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)

1
Γαβγ = g αδ (gδβ,γ + gδγ,β − gβγ,δ )
2
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 = 0

(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 = − 1 −

1 µν
g R
2

2M

r

dt2 + 1 −

2M
r

+ r 2 (dθ 2 + sin2 θ dφ2 )

−1

dr 2

(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 2
2Mr sin2 θ
dt
−
2a
dt dφ
̺2

̺2
(r 2 + a2 )2 − a2 ∆sin2 θ 2
̺2 2
2
dr + ̺2 dθ2
+
sin
θ
dφ
+
̺2
∆
ds2 = −

J

angular momentum
(along z)

a
≡ J/M
∆
≡ r2 − 2Mr + a2
(3.62)
2
̺
≡ r2 + a2 cos2 θ
a General relativity conventionally uses “geometrized units” in which G = 1 and c = 1. 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 = m¨r = p˙

(3.63)

F
m

force
mass of particle

r

particle position vector

Momentum

p = m˙r

(3.64)

p

momentum

Kinetic energy

1
T = mv 2
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 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

2

v = u + 2as
1
s = ut + at2
2
u+v
t
s=
2

(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 = µr×˙r

(3.84)

J

angular momentum

Lagrangian

1
L = µ|˙r |2 − U(|r|)
2

(3.85)

L

Lagrangian

U

potential energy of interaction

Ballisticsa
Velocity

v = v0 cosα xˆ + (v0 sinα − gt) yˆ
(3.86)
v 2 = v02 − 2gy

(3.87)

Trajectory

gx2
y = xtanα − 2
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 =
ra

1/2

GM
rb

1/2

2rb
ra + rb

1/2

−1
(3.97)

∆vhb =

1−

2ra
ra + rb

∆v
Mi
Mf

∆vah

velocity increment, a to h

∆vhb
ra
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 2
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
= 1 + ecosφ ,
r
a(1 − e2 )
r=
1 + 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
=
2
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 2
mα2

1/2

= 1−

(3.107)
b2
a2

m

A

1/2

r0

φ
M

(3.108)
ae

J 2 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 2 )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 < 0 then e < 1 and the orbit is an ellipse (a circle if e = 0). If E = 0, then e = 1
and the orbit is a parabola. If E > 0 then e > 1 and the orbit becomes a hyperbola (see Rutherford scattering on next
page).


Dynamics and mechanics

72

Rutherford scatteringa
y

trajectory
for α < 0

b

x

scattering
centre

χ

a

rmin
trajectory
for α > 0

rmin

Scattering potential
energy

Scattering angle

χ
|α|
tan =
2 2Eb
|α|

χ α
csc −
2E
2 |α|
= a(e ± 1)

Eccentricity

e=

Motion trajectoryb

4E 2 2 y 2
x − 2 =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 2 b2
+1
α2

x=±

U(r) potential energy

(3.117)

|α|
2E

a=

(3.114)

(3.116)

rmin =

Semi-axis

Scattering centrec

(α<0)

(α>0)

α
U(r) = −
r
< 0 repulsive
α
> 0 attractive

Closest approach

a

α2
+ b2
4E 2

1/2

= csc

χ
2

(3.120)
(3.121)

x,y position with respect to
hyperbola centre

1/2

dσ 1 dN
=
dΩ n dΩ
α 2 4χ
=
csc
4E
2

(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 = 1 if perfectly elastic

(3.125)
(3.126)

ǫ = 0 if perfectly inelastic

(3.127)

T −T′
= 1 − 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 2 δij − xi xj ) dm

(y 2 + z 2 ) dm

I =  − xy dm
− xz dm

Parallel axis
theorem

− xy dm
(x2 + z 2 ) dm
− yz dm

(3.136)

− xz dm

− yz dm 
(x2 + y 2 ) dm

⋆
− ma1 a2
I12 = I12

r
δij

r2 = x2 + y 2 + z 2

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
ai ,a position vector of
centre of mass
m
mass of body

⋆

+ m(a22 + a23 )
I11 = I11

(3.139)

Iij = Iij⋆ + m(|a|2 δij − ai aj )

(3.140)

Angular
momentum

J = Iω

(3.141)

J

angular momentum

ω

angular velocity

Rotational
kinetic energy

1
1
T = ω · J = Iij ωi ωj

2
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
0

0
I2
0


0
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

1

T = (I1 ω12 + I2 ω22 + I3 ω32 )
2

(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 2
12

(3.150)

I3 ≃ 0

(3.151)

Solid sphere, radius r

2
I1 = I2 = I3 = mr 2
5

(3.152)

Spherical shell, radius r

2
I1 = I2 = I3 = mr 2

3

(3.153)

Solid cylinder, radius r,
length l

m 2 l2
I1 = I2 =
r +
4
3
1
I3 = mr 2
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 =

3
h
m r2 +
20
4

3
I3 = mr 2
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)

2

a With

h

I3 I
2
I r
1

2

(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 /2
c

I3

I2

(3.160)

2

Disk, radius r

I3

I1

I1 = m(b2 + c2 )/12
I2 = m(c2 + a2 )/12

2

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

2 sinθ
d= r
3 θ

from disk centre

(3.172)

Arc of circle, radius r, angle
2θ

d=r

from circle centre

(3.173)

Arbitrary triangular lamina,
height ha

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 =

3 (2r − h)2
from sphere centre
4 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
1 + 0 + ···
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

1
(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

3

herpolhode

ω

space
cone

invariable
plane

J3
polhode

Ωp

body
cone

moment
of inertia
ellipsoid

θ
support point

a

2
gyroscope

prolate symmetric top

Euler’s equations

˙ 1 + (I3 − I2 )ω2 ω3
G1 = I1 ω
˙ 2 + (I1 − I3 )ω3 ω1
G2 = I2 ω
˙ 3 + (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 ) 2
ω3
I1 I2

(3.184)
(3.185)
(3.186)

Gi

external couple (= 0 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 = 0

(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

2

Differential
equation

dx
dx
+ ω02 x = 0
+ 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 2 )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)

2

− 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 2. 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 2 ]1/2
2γωf
tanφ = 2
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

2

ωar
= ω02 − 2γ 2

(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