All 14 Chapters Covered
SOLUTION MANUAL
, 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
,iv CONTENTS
Preface
This Instructor’s Manual contains the solutions to all the end-of-chapter problems (but not the
appendices) from Classical Dynamics of Particles and Systems, Fifth Edition, by Stephen T. Thornton and
Jerry B. Marion. It is intended for use only by instructors using Classical Dynamics as a textbook, and it is
not available to students in any form. A Student Solutions Manual containing solutions to about 25% of
the end-of-chapter problems is available for sale to students. The problem numbers of those solutions in
the Student Solutions Manual are listed on the next page.
As a result of surveys received from users, I continue to add more worked out examples in the text
and add additional problems. There are now 509 problems, a significant number over the 4th edition.
The instructor will find a large array of problems ranging in difficulty from the simple “plug and
chug” to the type worthy of the Ph.D. qualifying examinations in classical mechanics. A few of the
problems are quite challenging. Many of them require numerical methods. Having this solutions manual
should provide a greater appreciation of what the authors intended to accomplish by the statement of the
problem in those cases where the problem statement is not completely clear. Please inform me when
either the problem statement or solutions can be improved. Specific help is encouraged. The instructor
will also be able to pick and choose different levels of difficulty when assigning homework problems.
And since students may occasionally need hints to work some problems, this manual will allow the
instructor to take a quick peek to see how the students can be helped.
It is absolutely forbidden for the students to have access to this manual. Please do not give
students solutions from this manual. Posting these solutions on the Internet will result in widespread
distribution of the solutions and will ultimately result in the decrease of the usefulness of the text.
The author would like to acknowledge the assistance of Tran ngoc Khanh (5th edition), Warren
Griffith (4th edition), and Brian Giambattista (3rd edition), who checked the solutions of previous
versions, went over user comments, and worked out solutions for new problems.
Without their help, this manual would not be possible. The author would appreciate receiving reports of
suggested improvements and suspected errors. Comments can be sent by email to , the
more detailed the better.
Stephen T. Thornton
Charlottesville, Virginia
v
, CHAPTER 1
Matrices, Vectors,
and Vector Calculus
1-1.
x1
45˚
x1
45˚
x3
x3
Axes x1 and x3 lie in the x1x3 plane. The
transformation equations are:
x1 = x1 cos 45 − x3 cos 45
x2 = x2
x3 = x3 cos 45 + x1 cos 45
1 1
x = x − x
1 1 3
2 2
x2 = x2
1 1
x = x − x
3 1 3
2 2
So the transformation matrix is:
1 1
0 −
2 2
0 1 0
1 1
0
2 2
1
SOLUTION MANUAL
, 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
,iv CONTENTS
Preface
This Instructor’s Manual contains the solutions to all the end-of-chapter problems (but not the
appendices) from Classical Dynamics of Particles and Systems, Fifth Edition, by Stephen T. Thornton and
Jerry B. Marion. It is intended for use only by instructors using Classical Dynamics as a textbook, and it is
not available to students in any form. A Student Solutions Manual containing solutions to about 25% of
the end-of-chapter problems is available for sale to students. The problem numbers of those solutions in
the Student Solutions Manual are listed on the next page.
As a result of surveys received from users, I continue to add more worked out examples in the text
and add additional problems. There are now 509 problems, a significant number over the 4th edition.
The instructor will find a large array of problems ranging in difficulty from the simple “plug and
chug” to the type worthy of the Ph.D. qualifying examinations in classical mechanics. A few of the
problems are quite challenging. Many of them require numerical methods. Having this solutions manual
should provide a greater appreciation of what the authors intended to accomplish by the statement of the
problem in those cases where the problem statement is not completely clear. Please inform me when
either the problem statement or solutions can be improved. Specific help is encouraged. The instructor
will also be able to pick and choose different levels of difficulty when assigning homework problems.
And since students may occasionally need hints to work some problems, this manual will allow the
instructor to take a quick peek to see how the students can be helped.
It is absolutely forbidden for the students to have access to this manual. Please do not give
students solutions from this manual. Posting these solutions on the Internet will result in widespread
distribution of the solutions and will ultimately result in the decrease of the usefulness of the text.
The author would like to acknowledge the assistance of Tran ngoc Khanh (5th edition), Warren
Griffith (4th edition), and Brian Giambattista (3rd edition), who checked the solutions of previous
versions, went over user comments, and worked out solutions for new problems.
Without their help, this manual would not be possible. The author would appreciate receiving reports of
suggested improvements and suspected errors. Comments can be sent by email to , the
more detailed the better.
Stephen T. Thornton
Charlottesville, Virginia
v
, CHAPTER 1
Matrices, Vectors,
and Vector Calculus
1-1.
x1
45˚
x1
45˚
x3
x3
Axes x1 and x3 lie in the x1x3 plane. The
transformation equations are:
x1 = x1 cos 45 − x3 cos 45
x2 = x2
x3 = x3 cos 45 + x1 cos 45
1 1
x = x − x
1 1 3
2 2
x2 = x2
1 1
x = x − x
3 1 3
2 2
So the transformation matrix is:
1 1
0 −
2 2
0 1 0
1 1
0
2 2
1
