**Problemas de mecânica aplicada**

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(Parte **1** de 9)

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 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 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., 2 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.

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.

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

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 1, 84 problems on Analytical Mechanics; Part 11, 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

Preface Introduction

Part I Newtonian Mechanics

1. Dynamics of a Point Mass (1001-1108) 2. Dynamics of a System of Point Masses (1109-14) 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)

(Parte **1** de 9)