Beer Johnston VECTOR MECHANICS FOR ENGINEERS V E C T O R M E C H A Apago PDF Enhancer M N D D IC A L S I M # F 9 O 9 9 8 R 6 0 1 2 E /1 N 8 /0 8 G C Y I N A N E M E A R G Y S E L O B ISBN 978-0-07-352940-0 L MHID 0-07-352940-0 K Part of ISBN 978-0-07-727555-6 MHID 0-07-727555-1 Ninth Edition | | | | BEER JOHNSTON MAZUREK CORNWELL EISENBERG www.mhhe.com Ninth Edition bbeeee2299440000__ffmm__ii--xxxxiivv..iinndddd PPaaggee ii 1122//1188//0088 33::3399::2277 PPMM uusseerr--ss117722 //VVoolluummeess//220044//MMHHDDQQ007788//wwoorrkk%%00//iinndddd%%00 NINTH EDITION VECTOR MECHANICS FOR ENGINEERS Statics and Dynamics Ferdinand P. Beer Late of Lehigh University Apag Eo. RuPsDseFll JoEhnnhstoann, cJr.e r University of Connecticut David F. Mazurek U.S. Coast Guard Academy Phillip J. Cornwell Rose-Hulman Institute of Technology Elliot R. Eisenberg The Pennsylvania State University bbeeee2299440000__ffmm__ii--xxxxiivv..iinndddd PPaaggee iiii 1122//1188//0088 33::3399::2288 PPMM uusseerr--ss117722 //VVoolluummeess//220044//MMHHDDQQ007788//wwoorrkk%%00//iinndddd%%00 VECTOR MECHANICS FOR ENGINEERS: STATICS & DYNAMICS, NINTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions © 2007, 2004, and 1997. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 QPV/QPV 0 9 ISBN 978–0–07–352940–0 MHID 0–07–352940–0 Global Publisher: Raghothaman Srinivasan Senior Sponsoring Editor: Bill Stenquist Director of Development: Kristine Tibbetts Developmental Editor: Lora Neyens Apago PDF Enhancer Senior Marketing Manager: Curt Reynolds Senior Project Manager: Sheila M. Frank Senior Production Supervisor: Sherry L. Kane Senior Media Project Manager: Tammy Juran Designer: Laurie B. Janssen Cover/Interior Designer: Ron Bissell (USE) Cover Image: ©John Peter Photography/Alamy Lead Photo Research Coordinator: Carrie K. Burger Photo Research: Sabina Dowell Supplement Producer: Mary Jane Lampe Compositor: Aptara®, Inc. Typeface: 10.5/12 New Caledonia Printer: Quebecor World Versailles, KY The credits section for this book begins on page 1291 and is considered an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Vector mechanics for engineers. Statics and dynamics / Ferdinand Beer . . . [et al.]. — 9th ed. p. cm. Includes index. ISBN 978–0–07–352940–0 (combined vol. : hc : alk. paper) — ISBN 978–0–07–352923–3 (v. 1 — “Statics” : hc : alk. paper) — ISBN 978–0–07–724916–8 (v. 2 — “Dynamics” : hc : alk. paper) 1. Mechanics, Applied. 2. Vector analysis. 3. Statics. 4. Dynamics. I. Beer, Ferdinand Pierre, 1915– TA350.B3552 2009 620.1905—dc22 2008047184 www.mhhe.com bbeeee2299440000__ffmm__ii--xxxxiivv..iinndddd PPaaggee iiiiii 1122//1188//0088 33::3399::2288 PPMM uusseerr--ss117722 //VVoolluummeess//220044//MMHHDDQQ007788//wwoorrkk%%00//iinndddd%%00 About the Authors A s publishers of the books by Ferd Beer and Russ Johnston we are often asked how they happened to write their books together with one of them at Lehigh and the other at the University of Connecticut. T he answer to this question is simple. Russ Johnston’s first teach- ing appointment was in the Department of Civil Engineering and M echanics at Lehigh University. There he met Ferd Beer, who had joined that department two years earlier and was in charge of the courses in mechanics. F erd was delighted to discover that the young man who had been hired chiefly to teach graduate structural engineering courses was not only willing but eager to help him reorganize the mechanics courses. Both believed that these courses should be taught from a few basic principles and that the various concepts involved would be best understood and remembered by the students if they were presented to them in a graphic way. Together they wrote lecture notes in statics and dynamics, to which they later added problems they felt would appeal to future engineers, and soon they produced the manuscript of the first edition of Mechanics for Engineers that was published in June 1956. T he second edition of M echanics for Engineers and the first Apago PDF Enhancer edition of V ector Mechanics for Engineers found Russ Johnston at Worcester Polytechnic Institute and the next editions at the University of Connecticut. In the meantime, both Ferd and Russ assumed admin- istrative responsibilities in their departments, and both were involved in research, consulting, and supervising graduate students—Ferd in the area of stochastic processes and random vibrations and Russ in the area of elastic stability and structural analysis and design. However, their interest in improving the teaching of the basic mechanics courses had not subsided, and they both taught sections of these courses as they kept revising their texts and began writing the manuscript of the first edition of their Mechanics of Materials text. Their collaboration spanned more than half a century and many successful revisions of all of their textbooks, and Ferd’s and Russ’s contributions to engineering education have earned them a number of honors and awards. They were presented with the Western Electric Fund Award for excellence in the instruction of engineering students by their respective regional sections of the American Society for Engi- neering Education, and they both received the Distinguished Educa- tor Award from the Mechanics Division of the same society. Starting in 2001, the New Mechanics Educator Award of the Mechanics Division has been named in honor of the Beer and Johnston author team. Ferdinand P. Beer. Born in France and educated in France and Switzerland, Ferd received an M.S. degree from the Sorbonne and an Sc.D. degree in theoretical mechanics from the University of Geneva. He came to the United States after serving in the French army during iii bbeeee2299440000__ffmm__ii--xxxxiivv..iinndddd PPaaggee iivv 1122//1188//0088 33::3399::2299 PPMM uusseerr--ss117722 //VVoolluummeess//220044//MMHHDDQQ007788//wwoorrkk%%00//iinndddd%%00 iv About the Authors the early part of World War II and taught for four years at Williams College in the Williams-MIT joint arts and engineering program. Fol- lowing his service at Williams College, Ferd joined the faculty of Lehigh University where he taught for thirty-seven years. He held several positions, including University Distinguished Professor and chairman of the Department of Mechanical Engineering and Mechanics, and in 1995 Ferd was awarded an honorary Doctor of Engineering degree by Lehigh University. E. Russell Johnston, Jr. Born in Philadelphia, Russ holds a B.S.d egree in civil engineering from the University of Delaware and an Sc. D. degree in the field of structural engineering from the M assachusetts Institute of Technology. He taught at Lehigh University and Worcester Polytechnic Institute before joining the faculty of the University of Connecticut where he held the position of Chairman of the Civil Engineering Department and taught for twenty-six years. In 1991 Russ received the Outstanding Civil Engineer Award from the Connecticut Section of the American Society of Civil Engineers. David F. Mazurek. David holds a B.S. degree in ocean engineering and an M.S. degree in civil engineering from the Florida Institute of Technology and a Ph.D. degree in civil engineering from the Univer- sity of Connecticut. He was employed by the Electric Boat Division of General Dynamics Corporation and taught at Lafayette College prior to joining the U.S. Coast Guard Academy, where he has been since 1990. He has served on the American Railway Engineering and Main- tenance of Way Association’s Committee 15—Steel Structures for the Apago PDF Enhancer past eighteen years. His professional interests include bridge engi- neering, tall towers, structural forensics, and blast-resistant design. Phillip J. Cornwell. Phil holds a B.S. degree in mechanical engi- neering from Texas Tech University and M.A. and Ph.D. degrees in mechanical and aerospace engineering from Princeton University. He is currently a professor of mechanical engineering at Rose-Hulman Institute of Technology where he has taught since 1989. His present interests include structural dynamics, structural health monitoring, and undergraduate engineering education. Since 1995, Phil has spent his summers working at Los Alamos National Laboratory where he is a mentor in the Los Alamos Dynamics Summer School and does research in the area of structural health monitoring. Phil received an SAE Ralph R. Teetor Educational Award in 1992, the Dean’s Out- standing Scholar Award at Rose-Hulman in 2000, and the Board of Trustees Outstanding Scholar Award at Rose-Hulman in 2001. Elliot R. Eisenberg. Elliot holds a B.S. degree in engineering and an M.E. degree, both from Cornell University. He has focused his schol- arly activities on professional service and teaching, and he was recog- nized for this work in 1992 when the American Society of Mechanical Engineers awarded him the Ben C. Sparks Medal for his contributions to mechanical engineering and mechanical engineering technology education and for service to the American Society for Engineering Education. Elliot taught for thirty-two years, including twenty-nine years at Penn State where he was recognized with awards for both teaching and advising. bbeeee2299440000__ffmm__ii--xxxxiivv..iinndddd PPaaggee vv 1122//1188//0088 33::3399::2299 PPMM uusseerr--ss117722 //VVoolluummeess//220044//MMHHDDQQ007788//wwoorrkk%%00//iinndddd%%00 Contents Preface xv List of Symbols xxiii 1 Introduction 1 1.1 What Is Mechanics? 2 1.2 Fundamental Concepts and Principles 2 1.3 Systems of Units 5 1.4 Conversion from One System of Units to Another 10 1.5 Method of Problem Solution 11 1.6 Numerical Accuracy 13 2 Statics of PartAicpleas go1 4 PDF Enhancer 2.1 Introduction 16 Forces in a Plane 16 2.2 Force on a Particle. Resultant of Two Forces 16 2.3 Vectors 17 2.4 Addition of Vectors 18 2.5 Resultant of Several Concurrent Forces 20 2.6 Resolution of a Force into Components 21 2.7 Rectangular Components of a Force. Unit Vectors 27 2.8 Addition of Forces by Summing x and y Components 30 2.9 Equilibrium of a Particle 35 2.10 Newton’s First Law of Motion 36 2.11 Problems Involving the Equilibrium of a Particle. Free-Body Diagrams 36 Forces in Space 45 2.12 Rectangular Components of a Force in Space 45 2.13 Force Defined by Its Magnitude and Two Points on Its Line of Action 48 2.14 Addition of Concurrent Forces in Space 49 2.15 Equilibrium of a Particle in Space 57 Review and Summary 64 Review Problems 67 Computer Problems 70 v bbeeee2299440000__ffmm__ii--xxxxiivv..iinndddd PPaaggee vvii 1122//1188//0088 33::3399::2299 PPMM uusseerr--ss117722 //VVoolluummeess//220044//MMHHDDQQ007788//wwoorrkk%%00//iinndddd%%00 vi Contents 3 Rigid Bodies: Equivalent Systems of Forces 72 3 .1 Introduction 74 3 .2 External and Internal Forces 74 3 .3 Principle of Transmissibility. Equivalent Forces 75 3 .4 Vector Product of Two Vectors 77 3 .5 Vector Products Expressed in Terms of Rectangular Components 79 3 .6 Moment of a Force about a Point 81 3 .7 Varignon’s Theorem 83 3 .8 Rectangular Components of the Moment of a Force 83 3 .9 Scalar Product of Two Vectors 94 3 .10 Mixed Triple Product of Three Vectors 96 3 .11 Moment of a Force about a Given Axis 97 3 .12 Moment of a Couple 108 3 .13 Equivalent Couples 109 3 .14 Addition of Couples 111 3 .15 Couples Can Be Represented by Vectors 111 3 .16 Resolution of a Given Force into a Force at O and a Couple 112 3 .17 Reduction of a System of Forces to One Force and Apago PDF Enhancer One Couple 123 3 .18 Equivalent Systems of Forces 125 3 .19 Equipollent Systems of Vectors 125 3 .20 Further Reduction of a System of Forces 126 * 3.21 Reduction of a System of Forces to a Wrench 128 Review and Summary 146 Review Problems 151 Computer Problems 154 4 Equilibrium of Rigid Bodies 156 4.1 Introduction 158 4.2 Free-Body Diagram 159 Equilibrium in Two Dimensions 160 4.3 Reactions at Supports and Connections for a Two-Dimensional Structure 160 4.4 Equilibrium of a Rigid Body in Two Dimensions 162 4.5 Statically Indeterminate Reactions. Partial Constraints 164 4.6 Equilibrium of a Two-Force Body 181 4.7 Equilibrium of a Three-Force Body 182 bbeeee2299440000__ffmm__ii--xxxxiivv..iinndddd PPaaggee vviiii 1122//1188//0088 33::3399::3300 PPMM uusseerr--ss117722 //VVoolluummeess//220044//MMHHDDQQ007788//wwoorrkk%%00//iinndddd%%00 Equilibrium in Three Dimensions 189 Contents vii 4.8 Equilibrium of a Rigid Body in Three Dimensions 189 4 .9 Reactions at Supports and Connections for a Three-Dimensional Structure 189 Review and Summary 210 Review Problems 213 Computer Problems 216 5 Distributed Forces: Centroids and Centers of Gravity 218 5 .1 Introduction 220 Areas and Lines 220 5 .2 Center of Gravity of a Two-Dimensional Body 220 5 .3 Centroids of Areas and Lines 222 5 .4 First Moments of Areas and Lines 223 5 .5 Composite Plates and Wires 226 5 .6 Determination of Centroids by Integration 236 5 .7 Theorems of Pappus-Guldinus 238 * 5.8 Distributed Loads on Beams 248 * 5.9 Forces on Submerged Surfaces 249 Apago PDF Enhancer Volumes 258 5 .10 Center of Gravity of a Three-Dimensional Body. Centroid of a Volume 258 5 .11 Composite Bodies 261 5 .12 Determination of Centroids of Volumes by Integration 261 Review and Summary 274 Review Problems 278 Computer Problems 281 6 Analysis of Structures 284 6 .1 Introduction 286 Trusses 287 6 .2 Definition of a Truss 287 6 .3 Simple Trusses 289 6 .4 Analysis of Trusses by the Method of Joints 290 * 6.5 Joints under Special Loading Conditions 292 * 6.6 Space Trusses 294 6 .7 Analysis of Trusses by the Method of Sections 304 * 6.8 Trusses Made of Several Simple Trusses 305 bbeeee2299440000__ffmm__ii--xxxxiivv..iinndddd PPaaggee vviiiiii 1122//1188//0088 33::3399::3300 PPMM uusseerr--ss117722 //VVoolluummeess//220044//MMHHDDQQ007788//wwoorrkk%%00//iinndddd%%00 viii Contents Frames and Machines 316 6 .9 Structures Containing Multiforce Members 316 6 .10 Analysis of a Frame 316 6 .11 Frames Which Cease to Be Rigid When Detached from Their Supports 317 6 .12 Machines 331 Review and Summary 345 Review Problems 348 Computer Problems 350 7 Forces in Beams and Cables 352 * 7.1 Introduction 354 * 7.2 Internal Forces in Members 354 Beams 362 * 7.3 Various Types of Loading and Support 362 * 7.4 Shear and Bending Moment in a Beam 363 * 7.5 Shear and Bending-Moment Diagrams 365 * 7.6 Relations among Load, Shear, and Bending Moment 373 Cables 383 * 7.7 Cables with Concentrated Loads 383 * 7.8 Cables with Distributed Loads 384 Apago PDF Enhancer * 7.9 Parabolic Cable 385 * 7.10 Catenary 395 Review and Summary 403 Review Problems 406 Computer Problems 408 8 Friction 410 8.1 Introduction 412 8.2 The Laws of Dry Friction. Coefficients of Friction 412 8.3 Angles of Friction 415 8.4 Problems Involving Dry Friction 416 8.5 Wedges 429 8.6 Square-Threaded Screws 430 *8.7 Journal Bearings. Axle Friction 439 *8.8 Thrust Bearings. Disk Friction 441 *8.9 Wheel Friction. Rolling Resistance 442 *8.10 Belt Friction 449 Review and Summary 460 Review Problems 463 Computer Problems 467 bbeeee2299440000__ffmm__ii--xxxxiivv..iinndddd PPaaggee iixx 1122//1188//0088 33::3399::3300 PPMM uusseerr--ss117722 //VVoolluummeess//220044//MMHHDDQQ007788//wwoorrkk%%00//iinndddd%%00 9 Contents ix Distributed Forces: Moments of Inertia 470 9.1 Introduction 472 Moments of Inertia of Areas 473 9.2 Second Moment, or Moment of Inertia, of an Area 473 9.3 Determination of the Moment of Inertia of an Area by Integration 474 9.4 Polar Moment of Inertia 475 9.5 Radius of Gyration of an Area 476 9.6 Parallel-Axis Theorem 483 9.7 Moments of Inertia of Composite Areas 484 * 9.8 Product of Inertia 497 * 9.9 Principal Axes and Principal Moments of Inertia 498 * 9.10 Mohr’s Circle for Moments and Products of Inertia 506 Moments of Inertia of a Mass 512 9.11 Moment of Inertia of a Mass 512 9.12 Parallel-Axis Theorem 514 9.13 Moments of Inertia of Thin Plates 515 9.14 Determination of the Moment of Inertia of a Three-Dimensional Body by Integration 516 9.15 Moments of Inertia of Composite Bodies 516 * 9.16 Moment of Inertia of a ABopdya wgith oRe spPectD toF an AErbnitrahrya Anxisc er through O . Mass Products of Inertia 532 * 9.17 Ellipsoid of Inertia. Principal Axes of Inertia 533 * 9.18 Determination of the Principal Axes and Principal Moments of Inertia of a Body of Arbitrary Shape 535 Review and Summary 547 Review Problems 553 Computer Problems 555 10 Method of Virtual Work 556 *10.1 Introduction 558 *10.2 Work of a Force 558 *10.3 Principle of Virtual Work 561 *10.4 Applications of the Principle of Virtual Work 562 *10.5 Real Machines. Mechanical Efficiency 564 *10.6 Work of a Force during a Finite Displacement 578 *10.7 Potential Energy 580 *10.8 Potential Energy and Equilibrium 581 *10.9 Stability of Equilibrium 582 Review and Summary 592 Review Problems 595 Computer Problems 598