Classical Dynamics of Particles and Systems
5th Edition by Thornton & Marion Chapter 1-14
SOLUTION MANUAL
, CHAPTER 0
Contents
Preface v
Problems Solved in Student Solutions Manual vii
1 Matrices, Vectors, and Vector Calculus 1
2 Newtonian Mechanics—Single Particle 29
3 Oscillations 79
4 Nonlinear Oscillations and Chaos 127
5 Gravitation 149
6 Some Methods in The Calculus of Variations 165
7 Hamilton’s Principle—Lagrangian and Hamiltonian Dynamics 181
8 Central-Force Motion 233
9 Dynamics of a System of Particles 277
10 Motion in a Noninertial Reference Frame 333
11 Dynamics of Rigid Bodies 353
12 Coupled Oscillations 397
13 Continuous Systems; Waves 435
14 Special Theory of Relativity 461
iii
, CHAPTER 1
Matrices, Vectors,
and Vector Calculus
1-1.
x2 = x2′
x1′
45˚
x1
45˚
x3
x3′
Axes x′1 and x′3 lie in the x1 x3 plane.
The transformation equations are:
x1′ = x1 cos 45° − x3 cos 45°
x2′ = x2
x3′ = x3 cos 45° + x1 cos 45°
1 1
x1′ = x1 − x3
2 2
x2′ = x2
1 1
x3′ = x1 − x3
2 2
So the transformation matrix is:
1 1
0 −
2 2
0 1 0
1 1
0
2 2
1
, 2 CHAPTER 1
1-2.
a)
x3
D
E
γ
β x2
O
α B
A C
x1
From this diagram, we have
OE cos α = OA
OE cos β = OB (1)
OE cos γ = OD
Taking the square of each equation in (1) and adding, we find
2 2 2 2
OE cos 2 α + cos 2 β + cos 2 γ = OA + OB + OD (2)
But
2 2 2
OA + OB = OC (3)
and
2 2 2
OC + OD = OE (4)
Therefore,
2 2 2 2
OA + OB + OD = OE (5)
Thus,
cos 2 α + cos 2 β + cos 2 γ = 1 (6)
b)
x3
D
E
D′
E′
θ B B′ x2
O
A A′ C C′
x1
First, we have the following trigonometric relation:
OE + OE′ − 2OE OE′ cos θ = EE′
2 2 2
(7)
5th Edition by Thornton & Marion Chapter 1-14
SOLUTION MANUAL
, CHAPTER 0
Contents
Preface v
Problems Solved in Student Solutions Manual vii
1 Matrices, Vectors, and Vector Calculus 1
2 Newtonian Mechanics—Single Particle 29
3 Oscillations 79
4 Nonlinear Oscillations and Chaos 127
5 Gravitation 149
6 Some Methods in The Calculus of Variations 165
7 Hamilton’s Principle—Lagrangian and Hamiltonian Dynamics 181
8 Central-Force Motion 233
9 Dynamics of a System of Particles 277
10 Motion in a Noninertial Reference Frame 333
11 Dynamics of Rigid Bodies 353
12 Coupled Oscillations 397
13 Continuous Systems; Waves 435
14 Special Theory of Relativity 461
iii
, CHAPTER 1
Matrices, Vectors,
and Vector Calculus
1-1.
x2 = x2′
x1′
45˚
x1
45˚
x3
x3′
Axes x′1 and x′3 lie in the x1 x3 plane.
The transformation equations are:
x1′ = x1 cos 45° − x3 cos 45°
x2′ = x2
x3′ = x3 cos 45° + x1 cos 45°
1 1
x1′ = x1 − x3
2 2
x2′ = x2
1 1
x3′ = x1 − x3
2 2
So the transformation matrix is:
1 1
0 −
2 2
0 1 0
1 1
0
2 2
1
, 2 CHAPTER 1
1-2.
a)
x3
D
E
γ
β x2
O
α B
A C
x1
From this diagram, we have
OE cos α = OA
OE cos β = OB (1)
OE cos γ = OD
Taking the square of each equation in (1) and adding, we find
2 2 2 2
OE cos 2 α + cos 2 β + cos 2 γ = OA + OB + OD (2)
But
2 2 2
OA + OB = OC (3)
and
2 2 2
OC + OD = OE (4)
Therefore,
2 2 2 2
OA + OB + OD = OE (5)
Thus,
cos 2 α + cos 2 β + cos 2 γ = 1 (6)
b)
x3
D
E
D′
E′
θ B B′ x2
O
A A′ C C′
x1
First, we have the following trigonometric relation:
OE + OE′ − 2OE OE′ cos θ = EE′
2 2 2
(7)
