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Pro~lems and Solutions

on Mechanics

Major American Universities Ph.D.

Qualif~~n~ Questions and Solutions

Problems and Solutions

on Mechanics

Co~~il~d by:

The Physics Coaching Class

University of Science and

Technology

of China

Refereed by:

~iangYuan-qi, Gu En-pu, Cheng

Jia-fu,

ti Ze-hua,Ylang De-tian

Edited by:

Lirn Yung-kuo

World Scientific

~e~~e~s~ye~offdo~ 'Singapore e~~ng~ong

Pub~j.~he(/ hy

World Scienrific Publishing Co. Pte. 1-td

5 Toh Tuck Link, Singapore 596224

(A'A @c,e; Suite 202, 1060 Main Street, Kivcr Wge, NJ 07661 UK c@j%'r,c: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Catafoguing-in-Publi~tion Data

A Latalogue record for this book is available frum the British Library

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~onic or mec~iuiii~~il~ including ph~~t~jc~~py~ng, recording or any inform~fio~t storuge und reirievul

sy.ytern vim' known or to be invented, with~)i~t wriiren permission from the Publisher. For photocopying of material in this ,volume, please pay a copying fee through the Copyright

Clearance Center, Inc., 222

Rosewood Drive, Danvers, MA 01923, USA. In this case permission to ph~toeopy is not required from the publisher.

ISBN 981 -02-1295-X

981-02-1298-4 (pbk)

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

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,

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- De-tian)

Fm Yang-mei)

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

CONTENTS

Preface

Introduction

Part I Newtonian Mechanics

1. Dynamics of a Point Mass (1001-1108)

2. Dynamics of a System of Point

Masses (1109-1144)

3. Dynamics of Rigid Bodies (1145-1223)

4. Dynamics of Deformable Bodies (1224-1272)

Part I1 Analytical Mechanics

1. Lagrange's Equations (2001-2027)

2. Small Oscillations (2028-2067)

3. Hamilton's Canonical Equations 12068-2084)

Part I11 Special Relativity

Special Relativity (3001- 3054)

V vii 1 3 185
237
367
459
46 1
521
619
657
659
Index to Problems 751

PART I

NEWTONIAN MECHANICS

1. DYNAMICS OF A POINT MASS (1001-1108)

1001
A man of weight w is in an elevator of weight w. The elevator accelerates (a) What is the apparent weight of the man? (b) The man climbs a vertical ladder within the elevator at a speed v relative to the elevator. What is the man's rate of expenditure of energy (power output)? ( Wisconsin)

Solution:

vertically up at a rate a and at a certain instant has a speed V. (a) The apparent weight of the man is 20

F=wf-a=w

9 g being the acceleration of gravity. (b) The man's rate of expenditure of energy is 1002
An orbiting space station is observed to remain always vertically above the same point on the earth. Where on earth is the observer? Describe the orbit of the space station as completely as possible. ( Wisconsin)

Solution:

The observer must be on the equator of the earth. The orbit of the space station is a large circle in the equatorial plane with center at the center of the earth. The radius of the orbit can be figured out using the orbiting period of

24 hours* as follows. Let the radius of the orbit be R

and that of the earth be &. *For a more accurate calculation, the orbiting period should be taken as 23 hours

56 minutes and 4 seconds.

3

4 Problems €4 Solutions on Mechanics

We have

mu2 GMm

R R2 '

where v is the speed of the space station, G is the universal constant of gravitation, m and M are the masses of the space station and the earth respectively, giving As GM

02 = -

R' GMm mg=-, we have

GM=Gg.

Hence 2 - v -- R For circular motion with constant speed v, the orbiting period is 2xR T=-. 21
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