Problems and Solutions on Mechanics: Major American University PhD Qualifying Questions and Solutions

770 Pages·1994·17.01 MB·English

by  Yung-Kuo Lim

Solutions
Pro~lemsa nd
on Mechanics

Major American Universities Ph.D.
Q u a l i f ~ ~Qune~st ions and Solutions
Problems and Solutions
on Mechanics
C o ~ ~ ibly:~ d
The Physics Coaching Class
University of Science and
Technology of China
Refereed by:
~iangYuan-qiG, u En-pu, Cheng
Jia-fu, ti Ze-hua,Ylang De-tian
Edited by:
Lirn Yung-kuo
World Scientific
~ e ~ ~ e ~ s ~ y e'Si~ngoapfofrde eo~~~ n g ~ o n g
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World Scienrific Publishing Co. Pte. 1-td
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First published 1994
Reprinted 2001,2002
Major American Zlniversities Ph.D. Qualifying Questions and Solutions
PROBLEMS AND SOLUTIONS ON MECHANICS
Copyright 0 1994 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or pctrts thereof, muy not be reproduced in uny form or by uny meuns,
elel~oniocr mec~iuiii~i~ncillu~d ing ph~~t~jc~~rpecyo~rdningg, or any inform~fios~totr uge und reirievul
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ISBN 981 -02-1295-X
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Printed in Singapore.
PREFACE
This series of physics problems and solutions which consists of seven
volumes - Mechanics, Electromagnetism, Optics, Atomic, Nuclear and
Particle Physics, Thermodynamics and Statistical Physics, Quantum Me-
chanics, Solid State Physics - contains a selection of 2550 problems from
the graduate-school entrance and qualifying examination papers of seven
US. universities - California University Berkeley Campus, Columbia Uni-
versity, Chicago University, Massachusetts Institute of Technology, New
York State University Buffalo Campus, Princeton University, Wisconsin
University - as well as the CUSPEA and C.C. Ting’s papers for selection
of Chinese students for further studies in U.S.A. and their solutions which
represent the effort of more than 70 Chinese physicists plus some 20 more
who checked the solutions.
The series is remarkable for its comprehensive coverage. In each area
the problems span a wide spectrum of topics while many problems overlap
several areas. The problems themselves are remarkable for their versati-
lity in applying the physical laws and principles, their up-to-date realistic
situations, and their scanty demand on mathematical skills. Many of the
problems involve order-of-magnitude calculations which one often requires
in an experimental situation for estimating a quantity from a simple model.
In short, the exercises blend together the objectives of enhancement of one’s
understanding of the physical principles and ability of practical application.
The solutions as presented generally just provide a guidance to solving
the problems, rather than step by step manipulation, and leave much to
the students to work out for themselves, of whom much is demanded of the
basic knowledge in physics. Thus the series would provide an invaluable
complement to the textbooks.
The present volume for Mechanics which consists of three parts -
Newtonian Mechanics, Analytical Mechanics, and Special Relativity -
contains 410 problems. 27 Chinese physicists were involved in the task
of preparing and checking the solutions.
V
vi Preface
In editing, no attempt has been made to unify the physical terms, units,
and symbols. Rather , they are left to the setters’ and solvers’ own prefer-
ence so as to reflect the realistic situation of the usage today. Great pains
has been taken to trace the logical steps from the first principles to the final
solutions, frequently even to the extent of rewritting the entire solution.
In addition, a subject index has been included to facilitate the location of
topics. These editorial efforts hopefully will enhance the value of the volume
to the students and teachers alike.
Yung-Kuo Lim
Editor
INTRODUCTION
Solving problems in course work is an exercise of the mental faculties,
and examination problems are usually chosen from, or set similar to, such
problems. Working out problems is thus an essential and important aspect
of the study of physics
The series on Problems and Solutions in Physics comprises seven vol-
umes and is the result of months of work of a number of Chinese physicists.
The subjects of the volumes and the respective coordinators are as follows:
1. Mechanics (Qiang Yuan-qi, Gu En-pu, Cheng Jiefu, Li Ze-hua, Yang
De-tian)
2. EZectromagnetism (Zhao Sh-ping, You Jun-han, Zhu Jun-jie)
3. Optics (Bai Gui-ru, Guo Guang-can)
4. Atomic, Nuclear and Particle Physics (Jin Huai-cheng, Yang Baezhong,
Fm Yang-mei)
5. Thermodynamics and Statistical Physics (Zheng Jiu-ren)
6. Quantum Mechanics (Zhang Yong-de, Zhu Dong-pei, Fan Hong-yi)
7. Solid State Physics and Miscellaneous Topics (Zhang Jia-lu, Zhou You-
yuan, Zhang Shi-ling)
These volumes, which cover almost all aspects of university physics,
contain some 2550 problems solved in detail.
The problems have been carefully chosen from a total of 3100 problems
collected from the China-U.S.A. Physics Examination and Application
Programme, the Ph.D. Qualifying Examination on Experimental High
Energy Physics sponsored by Chao-chong Ting, and the graduate qualifying
examinations of seven world-renowned American universities: Columbia
University, the University of California at Berkeley, Massachusetts Institute
of Technology, the University of Wisconsin, the University of Chicago,
Princeton University, and the State University of New York at Buffalo.
Generally speaking, examination problems in physics in American uni-
versities do not require too much mathematics. They can be characterized
vii
viii Introduction
to a large extent as follows. Many problems are concerned with the
various frontier subjects and overlapping domains of topics, having been
selected from the setters’ own research encounters. These problems show a
“modern” flavor. Some problems involve a wide field and require a sharp
mind for their analysis, while others require simple and practical methods
demanding a fine “touch of physics.” We believe that these problems, as
a whole, reflect to some extent the characteristics of American science and
culture, as well as give a glimpse of the philosophy underlying American
education.
That being so, we consider it worthwhile to collect and solve these
problems and introduce them to physics students and teachers everywhere,
even though the work is both tedious and strenuous. About a hundred
teachers and graduate students took part in this time-consuming task.
This volume on Mechanics which contains 410 problems is divided into
three parts: Part I consists of 272 problems on Newtonian Mechanics;
Part 11, 84 problems on Analytical Mechanics; Part 111, 54 problems on
Special Relativity.
A small fraction of the problems is of the nature of mechanics as in
general physics, while the majority properly belongs to theoretical me-
chanics, with some on relativity. A wide range of knowledge is required
for solving some of the problems which demand a good understanding
of electromagnetism, optics, particle physics, mathematical physics, etc.
We consider such problems particularly beneficial to the student as they
show the interrelationship of different areas of physics which one is likely
to encounter in later life. Twenty seven physicists contributed to this
volume, notably Ma Qian-cheng, Deng You-ping, Yang Zhong-xia, Ji Shu,
Yang De-tian, Wang Ping, Li Xiao-ping, Qiang Yuan-qi, Chen Wei-zu, Hou
Bi-hui, and Chm Ze-xian.
7 August 1991