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Analytical Mechanics, Seventh Edition BROOKS/COLE Australia•Canada•Mexico.Singapore •Spain United Kingdom .UnitedStates THOIVISON BROOKS/COLE Publisher: David Harris Production Service: Nesbitt Graphics, Inc. Acquisitions Editor Chris Hall Text Designer: John Edeen Development Editor: Rebecca Heider Copy Editor: Nesbitt Graphics, Inc. Editorial Assistant: Seth Dobrin Illustrator: Nesbitt Graphics, Inc. Technology Project Manager: Sam Subity Cover Designer: Belinda Fernandez Marketing Manager: Kelley McMlister Cover Image: Graham Watson Advertising Project Manager: Stacey Purviance Cover Printer: Lehigh Press Project Manager, Editorial Compositor: International Typesetting Production: Beinda Krohmer and Composition Art Director: Rob Hugel Printer The Maple-Vail Book Print/Media Buyer: Doreen Suruki Manufacturing Group Permissions Editor: Sarah Harkrader COPYRIGHT © 2005 Brooks/Cole, a division of Thomson Brooks/Cole Thomson Learning, Inc. Thomson LearningrM is a 10 Davis Drive trademark used herein under license. Belmont, CA 94002 USA ALL RIGHTS RESERVED. No part of this work covered by the copyright hereon may be Asia reproduced or used in any form or by any means— Thomson Learning graphic, electronic, or mechanical, including but 5 Shenton Way #01-01 not limited to photocopying, recording, taping, Web UIC Building distribution, information networks, or information Singapore 068808 storage and retrieval systems—without the written permission of the publisher. Australia/New Zealand Thomson Learning Printed in the United States of America 102 Dodds Street 1 2 3 4 5 6 7 08 07 06 05 04 Southbank, Victoria 3006 Australia For more information about our products, Canada contact us at: Nelson Thomson Learning Academic 1120 Birchmount Road Resource Center Toronto, Ontario M1K 5G4 1-800-423-0563 Canada Forpermission to use material from this text or Europe/Middle East /Africa product, submit a request online at Thomson Learning http://www.thomsonrights.com High Holborn House Any additional questions about permissions can 50/51 Bedford Row be submitted by email to London WC1R 4LR [email protected]. United Kingdom Latin America Library of Congress Control Number: 2003115137 Thomson Learning Seneca, 53 ISBN 0-534-49492-7 ColoniaPolanco 11560 Mexico D.E International Student Edition: ISBN 0-534-40813-3 Mexico (Not for sale in the United States) Spain/Portugal Paraninfo Calle Magallanes, 25 28015 Madrid, Spain Preface Thistextbook is intended for an undergraduate course in classical mechanics taken by stu- dents majoring in physics, physical science, or engineering. We assume that the student has taken a year of calculus-based general physics and a year of differential/integral cal- culus. We recommend that a course in differential equations and matrix algebra be taken prior to, or concurrently with, this course in classical mechanics. The seventh edition of this text adheres to the same general philosophy of the pre- vious editions: it centers on the development and exposition of Newtonian mechanics with the more advanced Lagrangian and Hamiltonian formalism introduced and used only in the last two chapters. New material has been added to, and old material has been elimi- nated from, some of the chapters. We have expended much effort to stamp out annoying typographical errors, inadvertent mistakes, and unclear presentations. Explanations of some of the more difficult concepts have been expanded, and many figures and examples have been added with the intent of achieving greater clarification. Several sections have been greatly modified, and some new ones have been added. Those sections that are to be used with software tools such as Mathcad and Mathematica as part of the problem solv- ing strategy have been streamlined with many of the details relegated to an appendix. A brief synopsis of each chapter follows: • Chapter 1. A brief introduction to dimensional analysis and vector algebra; con- cepts of velocity and acceleration. • Chapter 2. Newton's laws of motion; motion in one dimension. Expansion of dis- cussion of inertial frames of reference. Introduction to solving problems numeri- cally using Mathcad: vertical fall through a fluid. • Chapter 3. Harmonic motion, resonance, the driven oscillator. Numerical solution of non-linear oscillator problems. • Chapter 4. Motion of a particle in three dimensions. Potential energy and conser- vative forces. Introduction to solving problems numerically using Mat hematica; projectile motion in a resistive medium; Mickey Mantle's "tape measure" homerun. • Chapter 5. The analysis of motion in a noninertial frame of reference and fictitious forces. Numerical solution of projectile motion in a rotating frame of reference. • Chapter 6. Gravitation. Expanded discussion of central forces. Conic sections and orbits. Expanded discussion of orbital energy. Criteria for stable orbits. Rutherford scattering. • Chapter 7. Many particle systems. The three-body problem: numerical solution. Lagrangian points. Conservation laws and collisions. Expanded presentation of rocket motion. • Chapter 8. Rotation of a body about a fixed axis. Expanded discussion of laminar motion. Moments of inertia. III iv Preface •Chapter 9. Rotation of a body in three dimensions. Numerical solutions of prob- lems involving the rotation of bodies with differing principal moments of inertia. Motion of tops and gyroscopes. Stability of a rotating bicycle wheel (why Lance doesn't fall over). • Chapter 10. Lagrangian and Hainiltonian mechanics. Hamilton's and D'Alembert's principles. Conservation laws. • Chapter 11. Coupled oscillators. Normal coordinates and normal modes of oscil- lation. The eigenvalue problem. The loaded string and wave motion. More worked examples have been added to this edition. Most worked examples can be found at the end of each section. The problems found in the first set at the end of each chapter can be solved analytically. A second set contains problems that require numeri- cal techniques, typically by using Mathcad, Mathematica, or any other software tool favored by the student or required by the instructor. The appendices contain information or reference material that should help the time- challenged student solve problems without resorting to time-consuming data searches else- where. Answers to a few selected odd-numbered problems are given at the end of the text. An updated problem solutions manual is available to instructors who adopt the text. Brooks Cole/Thomson Learning may provide complementary aids and supplements to those qualified under our adoption policy. Please contact your sales representative for more information. Acknowledgments Theauthors wish to acknowledge Mathsoft, Inc. for supplying us with current versions of Mathcad. We also acknowledge Wolfram Assoc. for selling us a copy of their current version of Mathematica 4 at a reduced cost. We would also like to thank those who aided in recommendations concerning the generation of this current edition. Linda McDonald of Northpark University and M. Anthony Reynolds of Embry-Riddle University provided in-depth reviews of certain chapters, and Zenaida Uy of Millersville University sent in com- ments as she taught with the book. Thanks also to the following individuals who answered an on-line survey about their Mechanics course: Charles Benesh, Wesleyan College; Mark S. Boley, Western Illinois University; Donald Bord, University of Michigan, Dearborn; Chris Burns, Swarthmore College; Steve Cederbloom, Mount Union College; Kelvin Chu, University of Vermont; Jim Crumley, St. John's University/College of St. Benedict; Vic DeCarlo, DePauw University; William Franz, Randolph-Macon College; Junichiro Fukai, Auburn; John G. Hardie, Christopher Newport University; Jim McCoy, Tarleton State University; Carl E. Mungan, U.S. Naval Academy; Rolfe G. Petschek, Case Western Reserve University; Brian P. Schwartz, Carthage College; C. Gregory Seab, University of New Orleans; Peter Skiff, Bard College; James Wheeler, Lock Haven University of Pennsylvania; E.J. Zita, The Evergreen State College. I would also like to acknowledge my editor, Rebecca Heider, my children, Pat and Katie, and my wife, Nancy Cohn, for their strong support and encouragement during the preparation of this edition. George L. Cassiday Contents Overview I Fundamental Concepts: Vectors 1 2Newtonian Mechanics: Rectilinear Motion of a Particle 47 3Oscillations 82 4 General Motion of a Particle in Three Dimensions 144 5Noninertial Reference Systems 184 6Gravitation and Central Forces 218 7Dynamics of Systems of Particles 275 8Mechanics of Rigid Bodies: Planar Motion 323 9Motion of Rigid Bodies in Three Dimensions 361 10 Lagrangian Mechanics 417 11Dynamics of Oscillating Systems 465 Appendices A-i Answers to Selected Odd-Numbered Problems ANS-i Selected References R-i Index I-i V Contents Fundamental Concepts: Vectors 1 1.1 Introduction 1 1.2 Measure of Space and Time: Units and Dimensions 2 1.3 Vectors 9 1.4 The Scalar Product 15 1.5 The Vector Product 19 1.6 An Example of the Cross Product: Moment of a Force 22 1.7 Triple Products 23 1.8 Change of Coordinate System: The Transformation Matrix 25 1.9 Derivative of a Vector 30 1.10 Position Vector of a Particle: Velocity and Acceleration in Rectangular Coordinates 31 1.11 Velocity and Acceleration in Plane Polar Coordinates 36 1.12 Velocity and Acceleration in Cylindrical and Spherical Coordinates 39 2Newtonian Mechanics: Rectilinear Motion of a Particle 47 2.1 Newton's Law of Motion: Historical Introduction 47 2.2Rectilinear Motion: Uniform Acceleration Under a Constant Force
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