Intermediate Dynamics

INTERMEDIATE
DYNAMICS
Intermediate Dynamics
Intermediate Dynamics
Second Edition
PATRICK HAMILL
Organization
A One-Semester Course
Exercises and Problems
Acknowledgments
A Brief Review of Introductory Concepts
1.1 Kinematics
У Jv = y* aJr.
1.1.1 Motion in a Straight Line at Constant Acceleration
Exercise 1.1
1 .2 Newton’s Second Law
a = F/m.
F = ma. (1.5)
1.3 Work and Energy
1.4 Momentum
1.5 Rotational Motion
1.5.1 Rotational Kinematics
1.5.2 Rotational Dynamics
1.6 Statics
1.7 Rotational Kinetic Energy
1.8 Angular Momentum
I = |1| = |r x p| = |rxmv| = rmv sin0,
L = £1, = J2(ri x p,).
1.9 Rotational Equivalents
N = /a,
1.10 Summary
N = r x F.
1 = r x p.
1.11 Problems
Computational Projects
Kinematics
2.1 Galileo Galilei (Historical Note)
2.2 The Principle of Inertia
A body will remain in uniform motion as long as no net external force acts on it.
2.3 Basic Concepts in Kinematics
2.3.1 Motion in One Dimension with Constant Acceleration
Exercise 2.1
2.3.2 Projectile Motion
2.3.3 Rotation About a Fixed Axis
= -^[(e-c,)(cf+1)-e0(1)]
2.3.4 The Relation between Linear and Rotational Motion
vT = (») x r. (2.7)
2.4 The Position of a Particle on a Plane
2.5 Unit Vectors
c = a x b
a • b = | a | | b | cos(a,b). (2.9)
к x i = j, i x к = —j.
i J к
2.6 Kinematics in Two Dimensions
2.6.1 Cartesian Coordinates
2.6.2 Plane Polar Coordinates
r = r(0), and
2.7 Kinematics in Three Dimensions
2.7.1 Cartesian Coordinates
2.7.2 Cylindrical Coordinates
Ф = Ф(0).
p • p = 1, ф • ф = 1, к • к = 1,
V - Г - £(pp + zk) = pp + + zk.
2.7.3 Spherical Coordinates
Ф = Ф(0).
Эф . Эф .
v = — = — (rr) = —- r + r —, dt dt dt dt
2.8 Summary
v = xi + yj + zk, a = xi + yj + zk.
r = rr,
r = pp + zk,
r = rr,
2.9 Problems
Army
_ 10
Newton’s Laws: Determining the Motion
3.1 Isaac Newton (Historical Note)
3.2 The Law of Inertia
A body in motion will remain in uniform motion and a body at rest will remain at rest unless acted upon by a net external force.
3.3 Newtons Second Law and the Equation of Motion
The rate of change of the momentum of an isolated body is equal to the net external force applied to it.
^=F,
3.4 Newton s Third Law: Action Equals Reaction
To every action there is always opposed an equal reaction.
<^P1 = ^P2 dt dt
3.5 Is Rotational Velocity Absolute or Relative?
3.6 Determining the Motion
Constant Force
Force as a Function of Time
Force as a Function of Velocity
Force as a Function of Position
3.7 Simple Harmonic Motion
v = v0 (x -x0)
co =
3.8 Closed-Form Solutions
3.9 Numerical Solutions (Optional)
3.10 Summary
Jo
Jo
3.11 Problems
Lagrangians and Hamiltonians
4.1 Joseph Louis Lagrange (Historical Note)
4.2 The Equation of Motion by Inspection
4.3 The Lagrangian
4.4 Lagrange’s Equations
4.5 Degrees of Freedom
4.6 Generalized Momentum
4.6.1 Ignorable Coordinates
4.8 The Calculus of Variations
Л, \Эу dx dy J
Cj 1-C2
4.8.1 Hamilton’s Principle
Jo
4.8.2 Relation to Newton’s Second Law
4.9 The Hamiltonian and Hamilton’s Equations
4.10 Summary
4.11 Problems
Computational Projects
Energy
5.1 The Work—Energy Theorem
7-(v2) = 3“(v*v) = dt
5.2 Work Along a Path: The Line Integral
W= F ds = F(A)«—JA.
5.3 Potential Energy
212г r
5.3.1 The Del Operator
5.3.2 The Gradient
5.3.3 The Relationship between Force and Potential Energy
F is conservative iff V x F = 0.
5.3.4 Del in Other Representations
Coordinate Transformations
5.3.5 Cylindrical Coordinates
5.3.6 Spherical Coordinates
5.4 Force, Work, and Potential Energy
V(r) - V(r0) = - f F-ds.
5.5 The Conservation of Energy
5.5.1 A Reflection on the Conservation of Energy
5.6 Energy Diagrams
5.7 The Energy Integral: Solving for the Motion
V m~ Jxo - V(x) “ Jo ~ Sm 2 Io
Exercise 5.22
5.8 The Kinetic Energy of a System of Particles
r, = rc + Г-,
5.9 Work on an Extended Body: Pseudowork
5.10 Summary
5.10.1 Mathematical Concepts
W = / F(A) • ds = / F(A) • — dk.
5.10.2 Physical Concepts
W = F • ds.
Г ,dx = A
5.11 Problems
Computational Projects
Linear Momentum
6.1 The Law of Conservation of Momentum
— = F, dt
6.2 The Motion of a Rocket
6.3 Collisions
Case 1: Head-on Collisions
= 2[i ± i] =
i 1
0 i’
Case 2: Glancing Collisions
6.4 Inelastic Collisions: The Coefficient of Restitution
6.5 Impulse
6.6 Momentum of a System of Particles
f£ = ^ = fF« + f;f;F,,
^F' F“"
= F^*.
6.7 Relative Motion and the Reduced Mass
r = r2 -Г1.
6.8 Collisions in Center of Mass Coordinates (Optional)
TZ IX7 I I I
\ («l2/M)Vrel /
6.9 Summary
J=[ FdZ = Ap = py-pz. Jo
6.10 Problems
Computational Techniques: The Runge—Kutta Method
Fi = f(x,/),
Angular Momentum
7.1 Definition of Angular Momentum
7.2 Conservation of Angular Momentum
7.2.1 Torque
riO = d + r,o'
52 N'O' = 52 r,°'x F/ = 52(r,° -d)x F« <7-2)
52N'O' = I2N'O-dxZF'-
7.2.2 Torque and Angular Momentum
= л(гхр, = л ХР + ГХЛ-
— xp = vxp = m(y x v) = 0.
7.3 Angular Momentum of a System of Particles
7.3.1 Angular Momentum Relative to the Center of Mass
Г; = ГС + Г-,
L = 52 т‘ (Гс + r9 x +Ю
L = Lc + L'.
7.4 Rotation of a Rigid Body about a Fixed Axis
lz = rz x pz = mi(Ti x vz) = ?nzrz x (w x rz). (7.7)
1, = mi - (г, • w) г,] .
7.5 The Moment of Inertia
7.5.1 Two Theorems
Гх31//2
7.6 The Gyroscope
7.7 Angular Momentum is an Axial Vector
7.8 Summary
Lz = x r-).
L = cu.
L = /си,
KT(e) T T T
Ly = Lz => //fc)y =
7.9 Problems
Conservation Laws and Symmetries
8.1 Emmy Noether (Historical Note)
8.2 Symmetry
8.3 Symmetry and the Laws of Physics
For every symmetry there is a corresponding constant of the motion.
8.4 Symmetries and Conserved Physical Quantities
8.4.1 Conservation of Linear Momentum
8.4.2 Conservation of Energy
8.5 Are the Laws of Physics Symmetrical?
8.5.1 Nonconservation of Parity
8.6 Strangeness (Optional)
8.7 Symmetry Breaking
8.8 Problems
Gravity
The Gravitational Field
9.1 Newton’s Law of Universal Gravitation
- G yr,
|r2-ri|2
(9.1)
r2 - ri r = .
|г2 -ril
17 П - Г2
9.1.1 Universality of the Law of Gravitation
9.1.2 Action at a Distance
9.2 The Gravitational Field
9.3 The Gravitational Field of an Extended Body
|x - x'|3 (x-x7)2
M . rj 1 T (4L — x')2 _4L — x'_0
9.4 The Gravitational Potential
V(r)
Jsurface IГ — r'| J0=O Л=0 IГ — r'|
9.5 Field Lines and Equipotential Surfaces
9.6 The Newtonian Gravitational Field Equations
9.6.1 Gauss’s Law
V • g = — 4тгСр.
9.7 The Equations of Poisson and Laplace
Vg = - V -V Ф= -V20.
V20 = 4тгСр.
9.8 Einstein’s Theory of Gravitation (Optional)
9.9 Summary
/ F \ r — rz
\m / |r — r'p
g(r) = -G [ p(r/)———Y—^dTf.
Jbody |r —r'|
/body |Г- r'|
V x g = 0,
V2 = 0.
9.10 Problems
Computational Technique: Numerical Integration
Central Force Motion: The Kepler Problem
10.1 Johannes Kepler (Historical Note)
10.2 Kepler’s Laws
10.3 Central Forces
F= 6162 r.
The angular momentum is constant.
10.4 The Equation of Motion
F = — r
10.5 Energy and the Effective Potential Energy
10.6 Solving the Radial Equation of Motion
Sophisticated Technique
10.8 The Equation of an Ellipse
10.9 Kepler’s Laws Revisited
The orbit of a planet is an ellipse with the Sun at one of the focal points.
The radius vector of a planet sweeps out equal areas in equal times.
10.10 Orbital Mechanics
10.10.1 Energy and Angular Momentum of a Satellite
h = \/m = r x v.
10.10.2 The Hohmann Transfer Orbit
10.11 A Perturbed Circular Orbit
/( d2/
10.12 Resonances
10.13 Summary
10.14 Problems
11
Harmonic Motion
11.1 Springs and Pendulums
11.2 Solving the Differential Equation
11.2.1 Homogeneous Linear Differential Equations
11.2.2 An Undamped Harmonic Oscillator
Energy of an Undamped Oscillator
11.3 The Damped Harmonic Oscillator
11.3.1 The Underdamped Oscillator
11.3.2 The Overdamped Oscillator
(11.16)
11.4 The Forced Harmonic Oscillator
11.4.1 Statement of the Problem
11.4.2 Inhomogeneous Linear Ordinary Differential Equations
11.4.3 Obtaining the Particular Solution
(D-W(O-«,= | <1L24)
11.4.4 The Forced Undamped Oscillator
11.4.5 The Forced Damped Oscillator
m ("0 - "rf)2 + (Vdb/rnj2
л _ Fo H m ("0 - wd)2 + (<»db/m)2
m ("o “ "rf)2 + b^db/m)2
The Q Factor
Resonance in Electrical Circuits (Optional)
Resonance in Astronomy
11.5 Coupled Oscillators
11.5.1 Normal Modes
Ci = b, c2 = 0.
11.6 Summary
11.7 Problems
12
The Pendulum
12.1 A Simple Pendulum with Arbitrary Amplitude
/(<+cos0)
12.1.1 Solution in Terms of Elliptic Integrals
12.1.2 Solution Expressed as a Series Expansion
12.2 The Physical Pendulum
nx = (52'И/Г/)x g-
I = (12.12)
12.3 The Center of Percussion
J=/*‘"=/‘/p=p/-p"
v = —.
^=N.
12.4 The Spherical Pendulum
12.4.1 The Conical Pendulum
. d& Jeo
12.4.2 The Spherical Pendulum
12.5 Summary
Veff = о >2 • 2n + m^1 C0S6■ 2ml1 sin 0
12.6 Problems
13
Waves
13.1 A Wave in a Stretched String
13.2 Direct Solution of the Wave Equation
13.2.1 Fourier Series (Optional)
13.3 Standing Waves
13.4 Traveling Waves
Э2у Э ГЭ/(м)1 Э ГЭН
ddm«l_ 2Э2/
13.5 Standing Waves as a Special Case of Traveling Waves
13.6 Energy
?)2-'
13.6.1 Energy Flow
13.7 Momentum (Optional)
13.8 Summary
13.9 Problems
Small Oscillations (Optional)
14.1 Introduction
14.2 Statement of the Problem
14.2.1 Static Equilibrium
* = ^ = E^- = E^-
(14.3)
14.2.2 The Mass Matrix
(14.4)
14.2.3 The Potential Matrix
14.2.4 The Lagrange Equations
mi 0 0 m2
14.3 Normal Modes
14.3.1 One-Dimensional Motion: A Simple Pendulum
14.3.2 One-Dimensional Motion: A Particle on a Spring
A Complex Solution
Exercise 14.5
tan<^ = Re(z+)+Re(zl) "
14.3.3 Solving the Coupled Equations
14.4 Matrix Formulation
4(0) = - 52 Р^С^ ^пф5.
14.4.1 The Modal Matrix
(дгуд).. = 521 p^rf MkiPy lJ к I I
14.5 Normal Coordinates
14.6 Coupled Pendulums: An Example
mm. + + — J m - km = o.
(р(5))ГЛ1р(,) = 8st
14.7 Many Degrees of Freedom
14.7.1 Statement of the Problem
14.8 Transition to Continuous Systems
14.8.1 The Lagrangian Density: A Continuous String
9?э(Ю
ay
ar _ ay
’W’*'
«W-"'57'
a / 3y\ a / 3y\
14.9 Summary
J = 1 k=l
v -P2y ) Ffcj “ I Q Q I-
()
14.10 Problems
Accelerated Reference Frames
15.1 A Linearly Accelerating Reference Frame
ro = r + ro/.
mfo = F,
15.2 A Rotating Coordinate Frame
= 0.
—- = vr + ft x r.
(15.3)
(15.4)
15.3 Fictitious Forces
+ X r.
Vi = vr + X r.
' d "I
г J "I
+ ft x vr + ft x (ft X r).
"dr"| c
Vi = (&b + ^окф.
— ' + n0 x + ПокФ-
ai = + Qok(-^)P + ^o(^o + &b)r(k x ф)
= ~Pr [(^6 + + ^o)]
15.4 Centrifugal Force and the Plumb Bob
T + mg — 2m£2 x vr — m [й x (Й x r)] = mar,
T + mg = m [2 x (2 x r)],
g, = g - 2 x (2 x r),
15.5 The Coriolis Force
15.5.1 A Falling Body
ar = —2ft x vr.
ac = —2ft x Vj
Exercise 15.5
15.5.2 A Projectile
7““"
15.6 The Foucault Pendulum
A Neat Trick
A More General Way to Solve the Problem
15.7 Application: The Tidal Force (Optional)
Fc = z— R.
Fj = z—r5.
15.8 Summary
ar = F/m — 2ft x vr — ft x (ft x r),
15.9 Problems
16
Rotational Kinematics
16.1 Orientation of a Rigid Body
16.2 Orthogonal Transformations
an
Interpretation
16.2.1 Orthogonal Matrices
(16.12)
(16.13)
1 0 0 \
0 10
0 0 1/
r = Д-1г'.
|лг||Л| = |1|.
-1 0 |l У I=I -У I
16.3 The Euler Angles
(16.28)
x' =
16.4 Euler’s Theorem
Homogeneous Algebraic Equations
Application
Proof of Euler’s Theorem
4R=XR
|R|2 = R* R.
R'* R' = R* R.
R'* R' = X*XR* R.
X(D = X(2) = X(3) = +1
16.5 Infinitesimal Rotations
_4.4“1 = (1 + e) (1 - e) = 1 + e - e = 1.
dx = x' — x = 6x.
16.6 Summary
x' = Лх.
4R = R,
ЛR = XR.
16.7 Problems
1 3 A
17
Rotational Dynamics
17.1 Angular Momentum
lz- = rz x pz = mi (rz x v/),
L = x (<У X Tz)] •
L = 52™' [w(r' ■- r,^r' ■w)] • (17Л)
Tensors
Dyads and Dyadics
AB + CD + .
1 = ii+jj + kk.
С AB = (C A)B.
I = - Г;Г;),
T = y* p(r) p2l — rr] dr.
+ Zzxki + Z^kj + Z^kk) • (cuxi + (oy) + cuzk)
17.2 Kinetic Energy
T = 1
= a> • ^rz x mzvz
17.3 Properties of the Inertia Tensor
17.3.1 Eigenvalues and Principal Axes
(T - /1) • R = 0.
17.3.2 Evaluating the Inertia Tensor
2см = 2?O - M(/?21 - RR).
R = -1+ -1+ -k
2 2J 2
|R| =
¥ О 0 \
0^0. 0 0 /
/±00 = Ma1 I 0 1 0 \ 0 0 5
4y -ly -ly
-ly -ly
4y -ly
-ly 4y
IR = XR,
5y — X
Зу — X -yx/2
R1 =
/ 0
R2 = 1/л/З
\ -VS??
17.4 The Euler Equations of Motion
N = (£
N = Z • —- + x (Z • o>). dt
17.5 Torque-Free Motion
17.6 The Spinning Top (Gyroscope)
a> = + 0z + i^z',
7 2
7 2
/isin#o /isim^o
Exercise 17.15
17.6.1 Precession without Nutation
2
17.6.2 Nutation
17.7 Summary
T = У p(r)(r2l — rr)dr.
17.8 Problems
18
Statics
18.1 Basic Concepts
Nc = r x Fc = r x (F« + Ffe) = Nfl + N&,
18.2 Couples, Resultants, and Equilibrants
Nq = 12 (rQ' x F' ) ’
nQ' = 12 (rQ'< x F') •
Nq' = 12 ( 18.3 Reduction to the Simplest Set of Forces
18.4 The Hanging Cable
18.4.1 A Suspension Bridge
18.4.2 A Hanging Rope
18.5 Stress and Strain
Strain is proportional to stress.
The Stress Tensor
18.6 The Centroid (Optional)
R = s/rJ'
18.7 The Center of Gravity (Optional)
F = Gm-rrcg.
18.8 D’Alembert’s Principle and Virtual Work
18.8.1 A Derivation of Lagrange’s Equations
18.9 Summary
EF'=0’
2>=o.
18.10 Problems
12FiSxi = 12
Computational Techniques: Solving a System of Linear Equations
Лх = В.
x =Л\В.
Fluid Dynamics and Sound Waves (Optional)
19.1 Introduction
19.2 Equilibrium of Fluids (Hydrostatics)
F = mg = (pAV)g,
f = Pg = -Pgk.
An Ideal Gas
19.3 Fluid Kinematics
19.3.1 Viscosity, Laminar Flow, and Turbulence
19.3.2 Eulerian and Lagrangian Formulations
19.3.3 The Convective Derivative
19.3.4 The Equation of Continuity
d „ ГЭ(ргл) 9(pvy) Э(риг)
= V • (pv).
£ + V • (pv) = 0.
I
19.3.5 Application: The Flow of Fluid through a Surface
19.3.6 Solenoidal and Irrotational Fluid Flow
19.4 Equation of Motion: Euler’s Equation
19.4.1 Steady Flow
V • (pv) = 0.
/n • (pv)J5 = 0, 5
Exercise 19.15
19.5 Conservation of Mass, Momentum, and Energy
19.5.1 Conservation of Mass and a Generalized Equation of Continuity
+ V • (pv) = 0.
Эр
19.5.2 Conservation of Momentum
e = f-vp.
19.5.3 Conservation of Energy and Bernoulli’s Equation
1 ?
19.5.4 A Velocity Potential Function
19.6 SoundWaves
dv 1 f
Po
T-(V-v),
V2 ,= Po£V
W-i^ = o,
v = g(m • r — ct),
, 2тг л CD л к = —n = —n.
19.7 Solving the Wave Equation by Separation of Variables
—v += 0-
9 Ct)2
v2[/ + -^U =0, c2
2
i
19.7.1 Sound Waves in Pipes (The Organ)
19.8 Summary
£ + V ■ (pv) = 0.
— + (v- V)v+ -Vp = dt p p
19.9 Problems
The Special Theory of Relativity
20.1 Albert Einstein (Historical Note)
20.2 Experimental Background
20.3 The Postulates of Special Relativity
20.4 The Lorentz Transformations
20.4.1 The Light Clock: A Gedanken Experiment
A?2 =
20.4.2 The Lorentz Transformations
-y/l — V2/c2
20.5 The Addition of Velocities
20.6 Simultaneity and Causality
20.7 The Twin Paradox
20.8 Minkowski Space-Time Diagrams
20.9 4-Vectors
20.9.1 Space-like and Time-like Intervals
20.9.2 The 4-Vector Velocity
20.10 Relativistic Dynamics
20.11 Summary
s2 — (xi - %o)2 + (У1 - yo)2 + (Z1 - zo)2 - C2(tl - to)2.
20.12 Problems
Classical Chaos (Optional)
21.1 Configuration Space and Phase Space
1 9 1 9
2 2
0=A-
21.2 Periodic Motion
21.3 Attractors
21.4 Chaotic Trajectories and Liapunov Exponents
21.5 Poincare Maps
21.6 The Henon—Heiles Hamiltonian
21.7 Summary
21.8 Problem
Appendix A
Formulas and Constants
Universal Constants
Astronomical Constants
Definition of Hyperbolic Functions
Integrals
Expressions involving terms such as a2 ± x2
Expressions involving trigonometric quantities
Expressions involving hyperbolic functions
Expression involving logarithms
Vector Relations
Expansions
Appendix В
Answers to Selected Problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4
J 1 mr2 1 wr2n2 I К
•• _ g sin a у _ m ( g sin a \ 1-;ЛмСО!>2а’ m+M V-swcos2“7
/ g(a-R)
“ V (а-Я)На2
2m LЭх2 Эу2
-^-v2
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21