Vector Mechanics Engineer Statics Dynamics 9th Edition

BeerJohnston

VECTOR MECHANICSFOR ENGINEERSVECTOR MECHANICS FOR ENGINEERS

MD DALIM #999860 12/18/08 CYAN MAG YELO BLK

ISBN 978-0-07-352940-0MHID 0-07-352940-0Part ofISBN 978-0-07-727555-6MHID 0-07-727555-1

Ninth Edition

www.mhhe.com

BEER | JOHNSTON | MAZUREK | CORNWELL | EISENBERGNinth Edition

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/Volumes/204/MHDQ078/work%0/indd%0

NINTH EDITION

VECTOR MECHANICSFOR ENGINEERSStatics and DynamicsFerdinand P. BeerLate of Lehigh University

E. Russell Johnston, Jr.University of Connecticut

David F. MazurekU.S. Coast Guard Academy

Phillip J. CornwellRose-Hulman Institute of Technology

Elliot R. EisenbergThe Pennsylvania State University

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VECTOR MECHANICS FOR ENGINEERS: STATICS & DYNAMICS, NINTH EDITIONPublished by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of theAmericas, 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 anyform or by any means, or stored in a database or retrieval system, without the prior written consent of TheMcGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage ortransmission, or broadcast for distance learning.Some ancillaries, including electronic and print components, may not be available to customers outside theUnited States.This book is printed on acid-free paper.1 2 3 4 5 6 7 8 9 0 QPV/QPV 0 9ISBN 9780073529400MHID 0073529400Global Publisher: Raghothaman SrinivasanSenior Sponsoring Editor: Bill StenquistDirector of Development: Kristine TibbettsDevelopmental Editor: Lora NeyensSenior Marketing Manager: Curt ReynoldsSenior Project Manager: Sheila M. FrankSenior Production Supervisor: Sherry L. KaneSenior Media Project Manager: Tammy JuranDesigner: Laurie B. JanssenCover/Interior Designer: Ron Bissell(USE) Cover Image: John Peter Photography/AlamyLead Photo Research Coordinator: Carrie K. BurgerPhoto Research: Sabina DowellSupplement Producer: Mary Jane LampeCompositor: Aptara, Inc.Typeface: 10.5/12 New CaledoniaPrinter: Quebecor World Versailles, KYThe credits section for this book begins on page 1291 and is considered an extension of the copyright page.Library of Congress Cataloging-in-Publication DataVector mechanics for engineers. Statics and dynamics / Ferdinand Beer . . . [et al.]. 9th ed.p. cm.Includes index.ISBN 9780073529400 (combined vol. : hc : alk. paper) ISBN 9780073529233(v. 1 Statics : hc : alk. paper) ISBN 9780077249168 (v. 2 Dynamics : hc : alk. paper)1. Mechanics, Applied. 2. Vector analysis. 3. Statics. 4. Dynamics. I. Beer, Ferdinand Pierre, 1915TA350.B3552 2009620.1905dc222008047184www.mhhe.com

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About the AuthorsAs publishers of the books by Ferd Beer and Russ Johnston we areoften asked how they happened to write their books together withone of them at Lehigh and the other at the University of Connecticut.The answer to this question is simple. Russ Johnstons first teaching appointment was in the Department of Civil Engineering andMechanics at Lehigh University. There he met Ferd Beer, who hadjoined that department two years earlier and was in charge of thecourses in mechanics.Ferd was delighted to discover that the young man who hadbeen hired chiefly to teach graduate structural engineering courseswas not only willing but eager to help him reorganize the mechanicscourses. Both believed that these courses should be taught from a fewbasic principles and that the various concepts involved would be bestunderstood and remembered by the students if they were presented tothem in a graphic way. Together they wrote lecture notes in statics anddynamics, to which they later added problems they felt would appealto future engineers, and soon they produced the manuscript of the firstedition of Mechanics for Engineers that was published in June 1956.The second edition of Mechanics for Engineers and the firstedition of Vector Mechanics for Engineers found Russ Johnston atWorcester Polytechnic Institute and the next editions at the Universityof Connecticut. In the meantime, both Ferd and Russ assumed administrative responsibilities in their departments, and both were involvedin research, consulting, and supervising graduate studentsFerd inthe area of stochastic processes and random vibrations and Russ in thearea of elastic stability and structural analysis and design. However,their interest in improving the teaching of the basic mechanics courseshad not subsided, and they both taught sections of these courses asthey kept revising their texts and began writing the manuscript of thefirst edition of their Mechanics of Materials text.Their collaboration spanned more than half a century and manysuccessful revisions of all of their textbooks, and Ferds and Russscontributions to engineering education have earned them a numberof honors and awards. They were presented with the Western ElectricFund Award for excellence in the instruction of engineering studentsby their respective regional sections of the American Society for Engineering Education, and they both received the Distinguished Educator Award from the Mechanics Division of the same society. Starting in2001, the New Mechanics Educator Award of the Mechanics Divisionhas been named in honor of the Beer and Johnston author team.Ferdinand P. Beer. Born in France and educated in France andSwitzerland, Ferd received an M.S. degree from the Sorbonne and anSc.D. degree in theoretical mechanics from the University of Geneva.He came to the United States after serving in the French army during

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About the Authors


the early part of World War II and taught for four years at WilliamsCollege in the Williams-MIT joint arts and engineering program. Following his service at Williams College, Ferd joined the faculty ofLehigh University where he taught for thirty-seven years. He heldseveral positions, including University Distinguished Professorand chairman of the Department of Mechanical Engineering andMechanics, and in 1995 Ferd was awarded an honorary Doctor ofEngineering degree by Lehigh University.E. Russell Johnston, Jr. Born in Philadelphia, Russ holds a B.S. degreein civil engineering from the University of Delaware and an Sc. D. degreein the field of structural engineering from the Massachusetts Institute ofTechnology. He taught at Lehigh University and Worcester PolytechnicInstitute before joining the faculty of the University of Connecticut wherehe held the position of Chairman of the Civil Engineering Departmentand taught for twenty-six years. In 1991 Russ received the OutstandingCivil Engineer Award from the Connecticut Section of the AmericanSociety of Civil Engineers.David F. Mazurek. David holds a B.S. degree in ocean engineeringand an M.S. degree in civil engineering from the Florida Institute ofTechnology and a Ph.D. degree in civil engineering from the University of Connecticut. He was employed by the Electric Boat Division ofGeneral Dynamics Corporation and taught at Lafayette College priorto joining the U.S. Coast Guard Academy, where he has been since1990. He has served on the American Railway Engineering and Maintenance of Way Associations Committee 15Steel Structures for thepast eighteen years. His professional interests include bridge engineering, tall towers, structural forensics, and blast-resistant design.Phillip J. Cornwell. Phil holds a B.S. degree in mechanical engineering from Texas Tech University and M.A. and Ph.D. degrees inmechanical and aerospace engineering from Princeton University. Heis currently a professor of mechanical engineering at Rose-HulmanInstitute of Technology where he has taught since 1989. His presentinterests include structural dynamics, structural health monitoring,and undergraduate engineering education. Since 1995, Phil has spenthis summers working at Los Alamos National Laboratory where heis a mentor in the Los Alamos Dynamics Summer School and doesresearch in the area of structural health monitoring. Phil received anSAE Ralph R. Teetor Educational Award in 1992, the Deans Outstanding Scholar Award at Rose-Hulman in 2000, and the Board ofTrustees Outstanding Scholar Award at Rose-Hulman in 2001.Elliot R. Eisenberg. Elliot holds a B.S. degree in engineering and anM.E. degree, both from Cornell University. He has focused his scholarly activities on professional service and teaching, and he was recognized for this work in 1992 when the American Society of MechanicalEngineers awarded him the Ben C. Sparks Medal for his contributionsto mechanical engineering and mechanical engineering technologyeducation and for service to the American Society for EngineeringEducation. Elliot taught for thirty-two years, including twenty-nineyears at Penn State where he was recognized with awards for bothteaching and advising.

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ContentsPreface

xv

List of Symbols xxiii

1

Introduction

1.11.21.31.41.51.6

What Is Mechanics? 2Fundamental Concepts and Principles 2Systems of Units 5Conversion from One System of Units to AnotherMethod of Problem Solution 11Numerical Accuracy 13

2

Statics of Particles

2.1

Introduction

1

10

14

16

Forces in a Plane 162.22.32.42.52.62.72.82.92.102.11

Force on a Particle. Resultant of Two Forces 16Vectors 17Addition of Vectors 18Resultant of Several Concurrent Forces 20Resolution of a Force into Components 21Rectangular Components of a Force. Unit Vectors 27Addition of Forces by Summing x and y Components 30Equilibrium of a Particle 35Newtons First Law of Motion 36Problems Involving the Equilibrium of a Particle.Free-Body Diagrams 36

Forces in Space

45

2.12 Rectangular Components of a Force in Space 452.13 Force Defined by Its Magnitude and Two Points on ItsLine of Action 482.14 Addition of Concurrent Forces in Space 492.15 Equilibrium of a Particle in Space 57Review and Summary 64Review Problems 67Computer Problems 70

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33.13.23.33.43.53.63.73.83.93.103.113.123.133.143.153.163.173.183.193.20*3.21

Rigid Bodies: EquivalentSystems of Forces 72Introduction 74External and Internal Forces 74Principle of Transmissibility. Equivalent Forces 75Vector Product of Two Vectors 77Vector Products Expressed in Terms ofRectangular Components 79Moment of a Force about a Point 81Varignons Theorem 83Rectangular Components of theMoment of a Force 83Scalar Product of Two Vectors 94Mixed Triple Product of Three Vectors 96Moment of a Force about a Given Axis 97Moment of a Couple 108Equivalent Couples 109Addition of Couples 111Couples Can Be Represented by Vectors 111Resolution of a Given Force into a Force at Oand a Couple 112Reduction of a System of Forces to One Force andOne Couple 123Equivalent Systems of Forces 125Equipollent Systems of Vectors 125Further Reduction of a System of Forces 126Reduction of a System of Forces to a Wrench 128

Review and Summary 146Review Problems 151Computer Problems 154

4

Equilibrium of Rigid Bodies

4.14.2

Introduction 158Free-Body Diagram 159

156

Equilibrium in Two Dimensions 1604.34.44.54.64.7

Reactions at Supports and Connectionsfor a Two-Dimensional Structure 160Equilibrium of a Rigid Body in Two DimensionsStatically Indeterminate Reactions. PartialConstraints 164Equilibrium of a Two-Force Body 181Equilibrium of a Three-Force Body 182

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Equilibrium in Three Dimensions 1894.84.9

Equilibrium of a Rigid Body in Three DimensionsReactions at Supports and Connections for aThree-Dimensional Structure 189

189


55.1

Distributed Forces: Centroidsand Centers of Gravity 218Introduction

220

Areas and Lines5.25.35.45.55.65.7*5.8*5.9

220

Center of Gravity of a Two-Dimensional Body 220Centroids of Areas and Lines 222First Moments of Areas and Lines 223Composite Plates and Wires 226Determination of Centroids by Integration 236Theorems of Pappus-Guldinus 238Distributed Loads on Beams 248Forces on Submerged Surfaces 249

Volumes

258

5.10 Center of Gravity of a Three-Dimensional Body.Centroid of a Volume 2585.11 Composite Bodies 2615.12 Determination of Centroids of Volumes byIntegration 261Review and Summary 274Review Problems 278Computer Problems 281

66.1

Analysis of StructuresIntroduction

284

286

Trusses 2876.26.36.4*6.5*6.66.7*6.8

Definition of a Truss 287Simple Trusses 289Analysis of Trusses by the Method of Joints 290Joints under Special Loading Conditions 292Space Trusses 294Analysis of Trusses by the Method of Sections 304Trusses Made of Several Simple Trusses 305

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Frames and Machines 316

Contents

6.9 Structures Containing Multiforce Members 3166.10 Analysis of a Frame 3166.11 Frames Which Cease to Be Rigid When Detachedfrom Their Supports 3176.12 Machines 331Review and Summary 345Review Problems 348Computer Problems 350

7*7.1*7.2

Forces in Beams and CablesIntroduction 354Internal Forces in Members

352

354

Beams 362*7.3*7.4*7.5*7.6

Various Types of Loading and Support 362Shear and Bending Moment in a Beam 363Shear and Bending-Moment Diagrams 365Relations among Load, Shear, and Bending Moment

*7.7*7.8*7.9*7.10

Cables with Concentrated Loads 383Cables with Distributed Loads 384Parabolic Cable 385Catenary 395

Cables

383


8

Friction

410

8.18.2

Introduction 412The Laws of Dry Friction. Coefficientsof Friction 4128.3 Angles of Friction 4158.4 Problems Involving Dry Friction 4168.5 Wedges 4298.6 Square-Threaded Screws 430*8.7 Journal Bearings. Axle Friction 439*8.8 Thrust Bearings. Disk Friction 441*8.9 Wheel Friction. RollingResistance 442*8.10 Belt Friction 449Review and Summary 460Review Problems 463Computer Problems 467

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99.1

Distributed Forces:Moments of InertiaIntroduction


Contents

470

472

Moments of Inertia of Areas

473

9.29.3

Second Moment, or Moment of Inertia, of an Area 473Determination of the Moment of Inertia of anArea by Integration 4749.4 Polar Moment of Inertia 4759.5 Radius of Gyration of an Area 4769.6 Parallel-Axis Theorem 4839.7 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 Mohrs Circle for Moments and Products of Inertia 506

Moments of Inertia of a Mass9.119.129.139.149.15*9.16*9.17*9.18

512

Moment of Inertia of a Mass 512Parallel-Axis Theorem 514Moments of Inertia of Thin Plates 515Determination of the Moment of Inertia of aThree-Dimensional Body by Integration 516Moments of Inertia of Composite Bodies 516Moment of Inertia of a Body with Respect to an Arbitrary Axisthrough O. Mass Products of Inertia 532Ellipsoid of Inertia. Principal Axes of Inertia 533Determination of the Principal Axes and Principal Moments ofInertia of a Body of Arbitrary Shape 535


10

Method of Virtual Work

*10.1*10.2*10.3*10.4*10.5*10.6*10.7*10.8*10.9

Introduction 558Work of a Force 558Principle of Virtual Work 561Applications of the Principle of Virtual Work 562Real Machines. Mechanical Efficiency 564Work of a Force during a Finite Displacement 578Potential Energy 580Potential Energy and Equilibrium 581Stability of Equilibrium 582


556

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1111.1

Kinematics of ParticlesIntroduction to Dynamics

600

602

Rectilinear Motion of Particles 60311.211.311.411.511.6*11.7*11.8

Position, Velocity, and Acceleration 603Determination of the Motion of a Particle 607Uniform Rectilinear Motion 616Uniformly Accelerated Rectilinear Motion 617Motion of Several Particles 618Graphical Solution of Rectilinear-Motion ProblemsOther Graphical Methods 631

630

Curvilinear Motion of Particles 64111.911.1011.1111.1211.1311.14

Position Vector, Velocity, and Acceleration 641Derivatives of Vector Functions 643Rectangular Components of Velocity and AccelerationMotion Relative to a Frame in Translation 646Tangential and Normal Components 665Radial and Transverse Components 668


1212.112.212.312.412.512.612.712.812.912.10*12.11*12.12*12.13

Kinetics of Particles:Newtons Second Law

690

Introduction 692Newtons Second Law of Motion 693Linear Momentum of a Particle. Rate of Changeof Linear Momentum 694Systems of Units 695Equations of Motion 697Dynamic Equilibrium 699Angular Momentum of a Particle. Rate of Changeof Angular Momentum 721Equations of Motion in Terms of Radial andTransverse Components 722Motion under a Central Force. Conservation ofAngular Momentum 723Newtons Law of Gravitation 724Trajectory of a Particle under a Central Force 734Application to Space Mechanics 735Keplers Laws of Planetary Motion 738


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1313.113.213.313.413.513.6*13.713.813.913.1013.1113.1213.1313.1413.15


Kinetics of Particles: Energy andMomentum Methods 754Introduction 756Work of a Force 756Kinetic Energy of a Particle. Principle of Workand Energy 760Applications of the Principle of Work and EnergyPower and Efficiency 763Potential Energy 782Conservative Forces 784Conservation of Energy 785Motion under a Conservative Central Force.Application to Space Mechanics 787Principle of Impulse and Momentum 806Impulsive Motion 809Impact 821Direct Central Impact 821Oblique Central Impact 824Problems Involving Energy and Momentum 827

762


14

Systems of Particles

14.114.2

854

Introduction 856Application of Newtons Laws to the Motion of a Systemof Particles. Effective Forces 85614.3 Linear and Angular Momentum of a System of Particles 85914.4 Motion of the Mass Center of a System of Particles 86014.5 Angular Momentum of a System of Particles about ItsMass Center 86214.6 Conservation of Momentum for a System of Particles 86414.7 Kinetic Energy of a System of Particles 87214.8 Work-Energy Principle. Conservation of Energy for a Systemof Particles 87414.9 Principle of Impulse and Momentum for a Systemof Particles 874*14.10 Variable Systems of Particles 885*14.11 Steady Stream of Particles 885*14.12 Systems Gaining or Losing Mass 888Review and Summary 905Review Problems 909Computer Problems 912

Contents

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1515.115.215.315.415.515.615.715.8*15.915.1015.11*15.12*15.13*15.14*15.15

Kinematics of Rigid Bodies

914

Introduction 916Translation 918Rotation about a Fixed Axis 919Equations Defining the Rotation of a Rigid Bodyabout a Fixed Axis 922General Plane Motion 932Absolute and Relative Velocity in Plane Motion 934Instantaneous Center of Rotation in Plane Motion 946Absolute and Relative Acceleration inPlane Motion 957Analysis of Plane Motion in Terms of a Parameter 959Rate of Change of a Vector with Respect to aRotating Frame 971Plane Motion of a Particle Relative to a Rotating Frame.Coriolis Acceleration 973Motion about a Fixed Point 984General Motion 987Three-Dimensional Motion of a Particle Relative to a RotatingFrame. Coriolis Acceleration 998Frame of Reference in General Motion 999


1616.116.216.316.4*16.516.616.716.8

Plane Motion of Rigid Bodies: Forcesand Accelerations 1024Introduction 1026Equations of Motion for a Rigid Body 1027Angular Momentum of a Rigid Body inPlane Motion 1028Plane Motion of a Rigid Body.DAlemberts Principle 1029A Remark on the Axioms of the Mechanicsof Rigid Bodies 1030Solution of Problems Involving the Motion of aRigid Body 1031Systems of Rigid Bodies 1032Constrained Plane Motion 1052


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17

Plane Motion of Rigid Bodies: Energyand Momentum Methods 1080

17.117.217.317.417.517.617.717.8

Introduction 1082Principle of Work and Energy for a Rigid Body 1082Work of Forces Acting on a Rigid Body 1083Kinetic Energy of a Rigid Body in Plane Motion 1084Systems of Rigid Bodies 1085Conservation of Energy 1086Power 1087Principle of Impulse and Momentum for the Plane Motionof a Rigid Body 110317.9 Systems of Rigid Bodies 110617.10 Conservation of Angular Momentum 110617.11 Impulsive Motion 111917.12 Eccentric Impact 1119


18

Kinetics of Rigid Bodies inThree Dimensions 1144

*18.1*18.2

Introduction 1146Angular Momentum of a Rigid Body inThree Dimensions 1147*18.3 Application of the Principle of Impulse and Momentum to theThree-Dimensional Motion of a Rigid Body 1151*18.4 Kinetic Energy of a Rigid Body inThree Dimensions 1152*18.5 Motion of a Rigid Body in Three Dimensions 1165*18.6 Eulers Equations of Motion. Extension ofDAlemberts Principle to the Motion of aRigid Body in Three Dimensions 1166*18.7 Motion of a Rigid Body about a Fixed Point 1167*18.8 Rotation of a Rigid Body about a Fixed Axis 1168*18.9 Motion of a Gyroscope. Eulerian Angles 1184*18.10 Steady Precession of a Gyroscope 1186*18.11 Motion of an Axisymmetrical Bodyunder No Force 1187Review and Summary 1201Review Problems 1206Computer Problems 1209


Contents

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1919.1


Mechanical VibrationsIntroduction

1214

Vibrations without Damping19.219.3*19.419.519.619.7

1212

1214

Free Vibrations of Particles. Simple Harmonic Motion 1214Simple Pendulum (Approximate Solution) 1218Simple Pendulum (Exact Solution) 1219Free Vibrations of Rigid Bodies 1228Application of the Principle of Conservation of Energy 1240Forced Vibrations 1250

Damped Vibrations 1260*19.8 Damped Free Vibrations 1260*19.9 Damped Forced Vibrations 1263*19.10 Electrical Analogues 1264Review and Summary 1277Review Problems 1282Computer Problems 1286

Appendix Fundamentals of Engineering ExaminationPhoto Credits 1291Index 1293Answers to Problems 1305

1289

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PrefaceOBJECTIVESThe main objective of a first course in mechanics should be todevelop in the engineering student the ability to analyze any problemin a simple and logical manner and to apply to its solution a few, wellunderstood, basic principles. This text is designed for the first coursesin statics and dynamics offered in the sophomore or junior year, andit is hoped that it will help the instructor achieve this goal.

GENERAL APPROACHVector analysis is introduced early in the text and is used throughoutthe presentation of statics and dynamics. This approach leads to moreconcise derivations of the fundamental principles of mechanics. It alsoresults in simpler solutions of three-dimensional problems in staticsand makes it possible to analyze many advanced problems in kinematics and kinetics, which could not be solved by scalar methods. Theemphasis in this text, however, remains on the correct understandingof the principles of mechanics and on their application to the solutionof engineering problems, and vector analysis is presented chiefly as aconvenient tool.

Practical Applications Are Introduced Early. One of the characteristics of the approach used in this book is that mechanics ofparticles is clearly separated from the mechanics of rigid bodies. Thisapproach makes it possible to consider simple practical applicationsat an early stage and to postpone the introduction of the more difficult concepts. For example: In Statics, the statics of particles is treated first (Chap. 2); afterthe rules of addition and subtraction of vectors are introduced,the principle of equilibrium of a particle is immediately appliedto practical situations involving only concurrent forces. The statics of rigid bodies is considered in Chaps. 3 and 4. In Chap. 3,the vector and scalar products of two vectors are introduced andused to define the moment of a force about a point and aboutan axis. The presentation of these new concepts is followed by athorough and rigorous discussion of equivalent systems of forcesleading, in Chap. 4, to many practical applications involving theequilibrium of rigid bodies under general force systems.

This text is available in separate volumes, Vector Mechanics for Engineers: Statics, ninthedition, and Vector Mechanics for Engineers: Dynamics, ninth edition.In a parallel text, Mechanics for Engineers: fifth edition, the use of vector algebra islimited to the addition and subtraction of vectors, and vector differentiation is omitted.

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In Dynamics, the same division is observed. The basic concepts offorce, mass, and acceleration, of work and energy, and of impulseand momentum are introduced and first applied to problems involving only particles. Thus, students can familiarize themselveswith the three basic methods used in dynamics and learn theirrespective advantages before facing the difficulties associatedwith the motion of rigid bodies.

New Concepts Are Introduced in Simple Terms. Since this textis designed for the first course in statics and dynamics, new conceptsare presented in simple terms and every step is explained in detail.On the other hand, by discussing the broader aspects of the problems considered, and by stressing methods of general applicability, adefinite maturity of approach is achieved. For example: In Statics, the concepts of partial constraints and statical indeterminacy are introduced early and are used throughout statics. In Dynamics, the concept of potential energy is discussed in thegeneral case of a conservative force. Also, the study of the planemotion of rigid bodies is designed to lead naturally to the studyof their general motion in space. This is true in kinematics as wellas in kinetics, where the principle of equivalence of external andeffective forces is applied directly to the analysis of plane motion,thus facilitating the transition to the study of three-dimensionalmotion.

Fundamental Principles Are Placed in the Context of SimpleApplications. The fact that mechanics is essentially a deductivescience based on a few fundamental principles is stressed. Derivationshave been presented in their logical sequence and with all the rigorwarranted at this level. However, the learning process being largelyinductive, simple applications are considered first. For example: The statics of particles precedes the statics of rigid bodies, andproblems involving internal forces are postponed until Chap. 6. In Chap. 4, equilibrium problems involving only coplanar forcesare considered first and solved by ordinary algebra, while problems involving three-dimensional forces and requiring the full useof vector algebra are discussed in the second part of the chapter. The kinematics of particles (Chap. 11) precedes the kinematicsof rigid bodies (Chap. 15). The fundamental principles of the kinetics rigid bodies are firstapplied to the solution of two-dimensional problems (Chaps. 16and 17), which can be more easily visualized by the student, whilethree-dimensional problems are postponed until Chap. 18.

The Presentation of the Principles of Kinetics Is Unified. Theninth edition of Vector Mechanics for Engineers retains the unifiedpresentation of the principles of kinetics which characterized the previous eight editions. The concepts of linear and angular momentum areintroduced in Chap. 12, so that Newtons second law of motion can bepresented not only in its conventional form F 5 ma, but also as a lawrelating, respectively, the sum of the forces acting on a particle and the

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sum of their moments to the rates of change of the linear and angularmomentum of the particle. This makes possible an earlier introductionof the principle of conservation of angular momentum and a moremeaningful discussion of the motion of a particle under a central force(Sec. 12.9). More importantly, this approach can be readily extendedto the study of the motion of a system of particles (Chap. 14) and leadsto a more concise and unified treatment of the kinetics of rigid bodiesin two and three dimensions (Chaps. 16 through 18).

Free-Body Diagrams Are Used Both to Solve EquilibriumProblems and to Express the Equivalence of Force Systems.Free-body diagrams are introduced early, and their importance isemphasized throughout the text. They are used not only to solveequilibrium problems but also to express the equivalence of two systems of forces or, more generally, of two systems of vectors. Theadvantage of this approach becomes apparent in the study of thedynamics of rigid bodies, where it is used to solve three-dimensionalas well as two-dimensional problems. By placing the emphasis onfree-body-diagram equations rather than on the standard algebraicequations of motion, a more intuitive and more complete understanding of the fundamental principles of dynamics can be achieved.This approach, which was first introduced in 1962 in the first editionof Vector Mechanics for Engineers, has now gained wide acceptanceamong mechanics teachers in this country. It is, therefore, used inpreference to the method of dynamic equilibrium and to the equations of motion in the solution of all sample problems in this book.A Four-Color Presentation Uses Color to Distinguish Vectors.Color has been used, not only to enhance the quality of the illustrations,but also to help students distinguish among the various types of vectors they will encounter. While there is no intention to color codethis text, the same color is used in any given chapter to represent vectors of the same type. Throughout Statics, for example, red is usedexclusively to represent forces and couples, while position vectors areshown in blue and dimensions in black. This makes it easier for thestudents to identify the forces acting on a given particle or rigid bodyand to follow the discussion of sample problems and other examplesgiven in the text. In Dynamics, for the chapters on kinetics, red is usedagain for forces and couples, as well as for effective forces. Red is alsoused to represent impulses and momenta in free-body-diagram equations, while green is used for velocities, and blue for accelerations. Inthe two chapters on kinematics, which do not involve any forces, blue,green, and red are used, respectively, for displacements, velocities, andaccelerations.A Careful Balance Between SI and U.S. Customary Units IsConsistently Maintained. Because of the current trend in theAmerican government and industry to adopt the international system of units (SI metric units), the SI units most frequently used inmechanics are introduced in Chap. 1 and are used throughout thetext. Approximately half of the sample problems and 60 percent ofthe homework problems are stated in these units, while the remainder


Preface

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are in U.S. customary units. The authors believe that this approachwill best serve the need of students, who, as engineers, will have tobe conversant with both systems of units.It also should be recognized that using both SI and U.S. customary units entails more than the use of conversion factors. Since the SIsystem of units is an absolute system based on the units of time, length,and mass, whereas the U.S. customary system is a gravitational systembased on the units of time, length, and force, different approaches arerequired for the solution of many problems. For example, when SIunits are used, a body is generally specified by its mass expressed inkilograms; in most problems of statics it will be necessary to determinethe weight of the body in newtons, and an additional calculation willbe required for this purpose. On the other hand, when U.S. customaryunits are used, a body is specified by its weight in pounds and, indynamics problems, an additional calculation will be required to determine its mass in slugs (or lb ? s2/ft). The authors, therefore, believethat problem assignments should include both systems of units.The Instructors and Solutions Manual provides six differentlists of assignments so that an equal number of problems stated inSI units and in U.S. customary units can be selected. If so desired,two complete lists of assignments can also be selected with up to75 percent of the problems stated in SI units.

Optional Sections Offer Advanced or Specialty Topics. Alarge number of optional sections have been included. These sectionsare indicated by asterisks and thus are easily distinguished from thosewhich form the core of the basic mechanics course. They may be omitted without prejudice to the understanding of the rest of the text.The topics covered in the optional sections in statics includethe reduction of a system of forces to a wrench, applications to hydrostatics, shear and bending-moment diagrams for beams, equilibriumof cables, products of inertia and Mohrs circle, mass products ofinertia and principal axes of inertia for three-dimensional bodies, andthe method of virtual work. An optional section on the determinationof the principal axes and the mass moments of inertia of a body ofarbitrary shape is included (Sec. 9.18). The sections on beams areespecially useful when the course in statics is immediately followedby a course in mechanics of materials, while the sections on the inertiaproperties of three-dimensional bodies are primarily intended for thestudents who will later study in dynamics the three-dimensional motionof rigid bodies.The topics covered in the optional sections in dynamicsinclude graphical methods for the solution of rectilinear-motionproblems, the trajectory of a particle under a central force, thedeflection of fluid streams, problems involving jet and rocket propulsion, the kinematics and kinetics of rigid bodies in three dimensions, damped mechanical vibrations, and electrical analogues.These topics will be found of particular interest when dynamics istaught in the junior year.The material presented in the text and most of the problemsrequire no previous mathematical knowledge beyond algebra, trigonometry, and elementary calculus; all the elements of vector algebra

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necessary to the understanding of the text are carefully presented inChaps. 2 and 3. However, special problems are included, which makeuse of a more advanced knowledge of calculus, and certain sections,such as Secs. 19.8 and 19.9 on damped vibrations, should be assignedonly if students possess the proper mathematical background. In portions of the text using elementary calculus, a greater emphasis isplaced on the correct understanding and application of the conceptsof differentiation and integration than on the nimble manipulationof mathematical formulas. In this connection, it should be mentionedthat the determination of the centroids of composite areas precedesthe calculation of centroids by integration, thus making it possible toestablish the concept of moment of area firmly before introducingthe use of integration.

CHAPTER ORGANIZATION AND PEDAGOGICAL FEATURESChapter Introduction. Each chapter begins with an introductorysection setting the purpose and goals of the chapter and describingin simple terms the material to be covered and its application to thesolution of engineering problems. Chapter outlines provide studentswith a preview of chapter topics.Chapter Lessons. The body of the text is divided into units, eachconsisting of one or several theory sections, one or several sampleproblems, and a large number of problems to be assigned. Each unitcorresponds to a well-defined topic and generally can be covered inone lesson. In a number of cases, however, the instructor will find itdesirable to devote more than one lesson to a given topic. TheInstructors and Solutions Manual contains suggestions on the coverage of each lesson.Sample Problems. The sample problems are set up in much thesame form that students will use when solving the assigned problems.They thus serve the double purpose of amplifying the text and demonstrating the type of neat, orderly work that students should cultivate in their own solutions.Solving Problems on Your Own. A section entitled SolvingProblems on Your Own is included for each lesson, between thesample problems and the problems to be assigned. The purpose ofthese sections is to help students organize in their own minds thepreceding theory of the text and the solution methods of the sampleproblems so that they can more successfully solve the homeworkproblems. Also included in these sections are specific suggestionsand strategies which will enable students to more efficiently attackany assigned problems.Homework Problem Sets. Most of the problems are of a practical nature and should appeal to engineering students. They are primarily designed, however, to illustrate the material presented in the


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text and to help students understand the principles of mechanics.The problems are grouped according to the portions of material theyillustrate and are arranged in order of increasing difficulty. Problemsrequiring special attention are indicated by asterisks. Answers to70 percent of the problems are given at the end of the book. Problemsfor which the answers are given are set in straight type in the text,while problems for which no answer is given are set in italic.

Chapter Review and Summary. Each chapter ends with areview and summary of the material covered in that chapter. Marginal notes are used to help students organize their review work, andcross-references have been included to help them find the portionsof material requiring their special attention.Review Problems. A set of review problems is included at the endof each chapter. These problems provide students further opportunityto apply the most important concepts introduced in the chapter.Computer Problems. Each chapter includes a set of problemsdesigned to be solved with computational software. Many of theseproblems provide an introduction to the design process. In Statics,for example, they may involve the analysis of a structure for variousconfigurations and loading of the structure or the determination ofthe equilibrium positions of a mechanism which may require an iterative method of solution. In Dynamics, they may involve the determination of the motion of a particle under initial conditions, the kinematicor kinetic analysis of mechanisms in successive positions, or thenumerical integration of various equations of motion. Developing thealgorithm required to solve a given mechanics problem will benefitthe students in two different ways: (1) it will help them gain a betterunderstanding of the mechanics principles involved; (2) it will providethem with an opportunity to apply their computer skills to the solution of a meaningful engineering problem.

SUPPLEMENTSAn extensive supplements package for both instructors and studentsis available with the text.

Instructors and Solutions Manual. The Instructors and SolutionsManual that accompanies the ninth edition features typeset, one-perpage solutions to all homework problems. This manual also featuresa number of tables designed to assist instructors in creating a schedule of assignments for their courses. The various topics covered in thetext are listed in Table I, and a suggested number of periods to bespent on each topic is indicated. Table II provides a brief descriptionof all groups of problems and a classification of the problems in eachgroup according to the units used. Sample lesson schedules areshown in Tables III, IV, and V.

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McGRAW-HILL CONNECT ENGINEERINGMcGraw-Hill Connect Engineering is a web-based assignment andassessment platform that gives students the means to better connectwith their coursework, their instructors, and the important conceptsthat they will need to know for success now and in the future. WithConnect Engineering, instructors can deliver assignments, quizzes,and tests easily online. Students can practice important skills at theirown pace and on their own schedule.Connect Engineering for Vector Mechanics for Engineers isavailable at www.mhhe.com/beerjohnston and includes algorithmicproblems from the text, Lecture PowerPoints, an image bank, andanimations.

Hands-on Mechanics. Hands-on Mechanics is a website designedfor instructors who are interested in incorporating three-dimensional,hands-on teaching aids into their lectures. Developed through apartnership between the McGraw-Hill Engineering Team and theDepartment of Civil and Mechanical Engineering at the UnitedStates Military Academy at West Point, this website not only provides detailed instructions for how to build 3-D teaching tools usingmaterials found in any lab or local hardware store but also providesa community where educators can share ideas, trade best practices,and submit their own demonstrations for posting on the site. Visitwww.handsonmechanics.com.

ELECTRONIC TEXTBOOK OPTIONSEbooks are an innovative way for students to save money and createa greener environment at the same time. An ebook can save studentsabout half the cost of a traditional textbook and offers unique featureslike a powerful search engine, highlighting, and the ability to sharenotes with classmates using ebooks.McGraw-Hill offers two ebook options: purchasing a downloadable book from VitalSource or a subscription to the book from CourseSmart. To talk about the ebook options, contact your McGraw-Hillsales rep or visit the sites directly at www.vitalsource.com andwww.coursesmart.com.

ACKNOWLEDGMENTSA special thanks go to our colleagues who thoroughly checked thesolutions and answers of all problems in this edition and then prepared the solutions for the accompanying Instructors and SolutionManual: Amy Mazurek of Williams Memorial Institute and DeanUpdike of Lehigh University.We are pleased to recognize Dennis Ormond of Fine LineIllustrations for the artful illustrations which contribute so much tothe effectiveness of the text.The authors thank the many companies that provided photographs for this edition. We also wish to recognize the determinedefforts and patience of our photo researcher Sabina Dowell.


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The authors gratefully acknowledge the many helpful comments and suggestions offered by users of the previous editions ofVector Mechanics for Engineers.E. Russell Johnston, Jr.David MazurekPhillip CornwellElliot R. Eisenberg

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List of SymbolsaA, B, C, . . .A, B, C, . . .AbcCdeFgGhi, j, kI, Ix, . . .IIxy, . . .Jkkx, ky, kOklLmMMORMOMMOLNOpPQrrRRssStTTU

Constant; radius; distanceReactions at supports and connectionsPointsAreaWidth; distanceConstantCentroidDistanceBase of natural logarithmsForce; friction forceAcceleration of gravityCenter of gravity; constant of gravitationHeight; sag of cableUnit vectors along coordinate axesMoments of inertiaCentroidal moment of inertiaProducts of inertiaPolar moment of inertiaSpring constantRadii of gyrationCentroidal radius of gyrationLengthLength; spanMassCouple; momentMoment about point OMoment resultant about point OMagnitude of couple or moment; mass of earthMoment about axis OLNormal component of reactionOrigin of coordinatesPressureForce; vectorForce; vectorPosition vectorRadius; distance; polar coordinateResultant force; resultant vector; reactionRadius of earthPosition vectorLength of arc; length of cableForce; vectorThicknessForceTensionWork

xxiii

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xxiv

List of Symbols


VVwW, Wx, y, zx, y, za, b, ggddrdULhumrf

Vector product; shearing forceVolume; potential energy; shearLoad per unit lengthWeight; loadRectangular coordinates; distancesRectangular coordinates of centroid or center ofgravityAnglesSpecific weightElongationVirtual displacementVirtual workUnit vector along a lineEfficiencyAngular coordinate; angle; polar coordinateCoefficient of frictionDensityAngle of friction; angle

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a

Introduction

In the latter part of the seventeenthcentury, Sir Isaac Newton stated thefundamental principles of mechanics,which are the foundation of much oftodays engineering.


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H1

1

C H A P T E R

Introduction

1


Chapter 1 Introduction1.11.21.31.41.51.6

What Is Mechanics?Fundamental Concepts andPrinciplesSystems of UnitsConversion from One System ofUnits to AnotherMethod of Problem SolutionNumerical Accuracy

1.1


WHAT IS MECHANICS?

Mechanics can be defined as that science which describes and predictsthe conditions of rest or motion of bodies under the action of forces. Itis divided into three parts: mechanics of rigid bodies, mechanics ofdeformable bodies, and mechanics of fluids.The mechanics of rigid bodies is subdivided into statics anddynamics, the former dealing with bodies at rest, the latter with bodiesin motion. In this part of the study of mechanics, bodies are assumedto be perfectly rigid. Actual structures and machines, however, arenever absolutely rigid and deform under the loads to which they aresubjected. But these deformations are usually small and do not appreciably affect the conditions of equilibrium or motion of the structureunder consideration. They are important, though, as far as the resistance of the structure to failure is concerned and are studied inmechanics of materials, which is a part of the mechanics of deformablebodies. The third division of mechanics, the mechanics of fluids, issubdivided into the study of incompressible fluids and of compressiblefluids. An important subdivision of the study of incompressible fluidsis hydraulics, which deals with problems involving water.Mechanics is a physical science, since it deals with the study ofphysical phenomena. However, some associate mechanics with mathematics, while many consider it as an engineering subject. Both theseviews are justified in part. Mechanics is the foundation of most engineering sciences and is an indispensable prerequisite to their study.However, it does not have the empiricism found in some engineeringsciences, i.e., it does not rely on experience or observation alone; by itsrigor and the emphasis it places on deductive reasoning it resemblesmathematics. But, again, it is not an abstract or even a pure science;mechanics is an applied science. The purpose of mechanics is to explainand predict physical phenomena and thus to lay the foundations forengineering applications.

1.2

FUNDAMENTAL CONCEPTS AND PRINCIPLES

Although the study of mechanics goes back to the time of Aristotle(384322 b.c.) and Archimedes (287212 b.c.), one has to wait untilNewton (16421727) to find a satisfactory formulation of its fundamental principles. These principles were later expressed in a modified form by dAlembert, Lagrange, and Hamilton. Their validityremained unchallenged, however, until Einstein formulated his theoryof relativity (1905). While its limitations have now been recognized,newtonian mechanics still remains the basis of todays engineeringsciences.The basic concepts used in mechanics are space, time, mass, andforce. These concepts cannot be truly defined; they should be acceptedon the basis of our intuition and experience and used as a mental frameof reference for our study of mechanics.The concept of space is associated with the notion of the positionof a point P. The position of P can be defined by three lengths measured from a certain reference point, or origin, in three given directions. These lengths are known as the coordinates of P.

2


To define an event, it is not sufficient to indicate its position inspace. The time of the event should also be given.The concept of mass is used to characterize and compare bodieson the basis of certain fundamental mechanical experiments. Two bodies of the same mass, for example, will be attracted by the earth in thesame manner; they will also offer the same resistance to a change intranslational motion.A force represents the action of one body on another. It can beexerted by actual contact or at a distance, as in the case of gravitationalforces and magnetic forces. A force is characterized by its point ofapplication, its magnitude, and its direction; a force is represented bya vector (Sec. 2.3).In newtonian mechanics, space, time, and mass are absolute concepts, independent of each other. (This is not true in relativisticmechanics, where the time of an event depends upon its position, andwhere the mass of a body varies with its velocity.) On the other hand,the concept of force is not independent of the other three. Indeed, oneof the fundamental principles of newtonian mechanics listed belowindicates that the resultant force acting on a body is related to the massof the body and to the manner in which its velocity varies with time.You will study the conditions of rest or motion of particles andrigid bodies in terms of the four basic concepts we have introduced. Byparticle we mean a very small amount of matter which may be assumedto occupy a single point in space. A rigid body is a combination of alarge number of particles occupying fixed positions with respect toeach other. The study of the mechanics of particles is obviously a prerequisite to that of rigid bodies. Besides, the results obtained for aparticle can be used directly in a large number of problems dealingwith the conditions of rest or motion of actual bodies.The study of elementary mechanics rests on six fundamentalprinciples based on experimental evidence.

The Parallelogram Law for the Addition of Forces. This statesthat two forces acting on a particle may be replaced by a single force,called their resultant, obtained by drawing the diagonal of the parallelogram which has sides equal to the given forces (Sec. 2.2).The Principle of Transmissibility. This states that the conditionsof equilibrium or of motion of a rigid body will remain unchanged if aforce acting at a given point of the rigid body is replaced by a force ofthe same magnitude and same direction, but acting at a different point,provided that the two forces have the same line of action (Sec. 3.3).Newtons Three Fundamental Laws. Formulated by Sir IsaacNewton in the latter part of the seventeenth century, these laws can bestated as follows:FIRST LAW. If the resultant force acting on a particle is zero, theparticle will remain at rest (if originally at rest) or will move with constant speed in a straight line (if originally in motion) (Sec. 2.10).


1.2 Fundamental Concepts and Principles

3


4


SECOND LAW. If the resultant force acting on a particle is not zero,the particle will have an acceleration proportional to the magnitude ofthe resultant and in the direction of this resultant force.As you will see in Sec. 12.2, this law can be stated as

Introduction

F 5 ma

m

r

F

F

MFig. 1.1

(1.1)

where F, m, and a represent, respectively, the resultant force acting onthe particle, the mass of the particle, and the acceleration of the particle, expressed in a consistent system of units.THIRD LAW. The forces of action and reaction between bodies incontact have the same magnitude, same line of action, and oppositesense (Sec. 6.1).

Newtons Law of Gravitation. This states that two particles ofmass M and m are mutually attracted with equal and opposite forces Fand 2F (Fig. 1.1) of magnitude F given by the formulaF5G

Mmr2

(1.2)

where r 5 distance between the two particlesG 5 universal constant called the constant of gravitationNewtons law of gravitation introduces the idea of an action exerted ata distance and extends the range of application of Newtons third law:the action F and the reaction 2F in Fig. 1.1 are equal and opposite,and they have the same line of action.A particular case of great importance is that of the attraction ofthe earth on a particle located on its surface. The force F exerted bythe earth on the particle is then defined as the weight W of the particle. Taking M equal to the mass of the earth, m equal to the mass of theparticle, and r equal to the radius R of the earth, and introducing theconstantGMg5 2(1.3)Rthe magnitude W of the weight of a particle of mass m may be expressed asW 5 mg

Photo 1.1 When in earth orbit, people andobjects are said to be weightless even though thegravitational force acting is approximately 90% ofthat experienced on the surface of the earth. Thisapparent contradiction will be resolved in Chapter12 when we apply Newtons second law to themotion of particles.

(1.4)

The value of R in formula (1.3) depends upon the elevation of thepoint considered; it also depends upon its latitude, since the earth isnot truly spherical. The value of g therefore varies with the position ofthe point considered. As long as the point actually remains on the surface of the earth, it is sufficiently accurate in most engineering computations to assume that g equals 9.81 m/s2 or 32.2 ft/s2.A more accurate definition of the weight W should take into account the rotation of theearth.


The principles we have just listed will be introduced in the courseof our study of mechanics as they are needed. The study of the staticsof particles carried out in Chap. 2, will be based on the parallelogramlaw of addition and on Newtons first law alone. The principle of transmissibility will be introduced in Chap. 3 as we begin the study of thestatics of rigid bodies, and Newtons third law in Chap. 6 as we analyzethe forces exerted on each other by the various members forming astructure. In the study of dynamics, Newtons second law and Newtonslaw of gravitation will be introduced. It will then be shown that Newtonsfirst law is a particular case of Newtons second law (Sec. 12.2) and thatthe principle of transmissibility could be derived from the other principles and thus eliminated (Sec. 16.5). In the meantime, however,Newtons first and third laws, the parallelogram law of addition, andthe principle of transmissibility will provide us with the necessary andsufficient foundation for the entire study of the statics of particles,rigid bodies, and systems of rigid bodies.As noted earlier, the six fundamental principles listed above arebased on experimental evidence. Except for Newtons first law and theprinciple of transmissibility, they are independent principles whichcannot be derived mathematically from each other or from any otherelementary physical principle. On these principles rests most of theintricate structure of newtonian mechanics. For more than two centuries a tremendous number of problems dealing with the conditions ofrest and motion of rigid bodies, deformable bodies, and fluids havebeen solved by applying these fundamental principles. Many of thesolutions obtained could be checked experimentally, thus providing afurther verification of the principles from which they were derived. Itis only in the twentieth century that Newtons mechanics was found atfault, in the study of the motion of atoms and in the study of the motionof certain planets, where it must be supplemented by the theory ofrelativity. But on the human or engineering scale, where velocities aresmall compared with the speed of light, Newtons mechanics has yet tobe disproved.

1.3

SYSTEMS OF UNITS

With the four fundamental concepts introduced in the preceding section are associated the so-called kinetic units, i.e., the units of length,time, mass, and force. These units cannot be chosen independently ifEq. (1.1) is to be satisfied. Three of the units may be defined arbitrarily; they are then referred to as basic units. The fourth unit, however, must be chosen in accordance with Eq. (1.1) and is referred to asa derived unit. Kinetic units selected in this way are said to form aconsistent system of units.

International System of Units (SI Units). In this system, whichwill be in universal use after the United States has completed its conversion to SI units, the base units are the units of length, mass, andtime, and they are called, respectively, the meter (m), the kilogram(kg), and the second (s). All three are arbitrarily defined. The second,SI stands for Systme International dUnits (French).


1.3 Systems of Units

5


6

Introduction

a = 1 m /s 2m = 1 kg

F=1N


which was originally chosen to represent 1/86 400 of the mean solarday, is now defined as the duration of 9 192 631 770 cycles of the radiation corresponding to the transition between two levels of the fundamental state of the cesium-133 atom. The meter, originally defined asone ten-millionth of the distance from the equator to either pole, isnow defined as 1 650 763.73 wavelengths of the orange-red light corresponding to a certain transition in an atom of krypton-86. The kilogram, which is approximately equal to the mass of 0.001 m3 of water,is defined as the mass of a platinum-iridium standard kept at the International Bureau of Weights and Measures at Svres, near Paris, France.The unit of force is a derived unit. It is called the newton (N) and isdefined as the force which gives an acceleration of 1 m/s2 to a mass of1 kg (Fig. 1.2). From Eq. (1.1) we write1 N 5 (1 kg)(1 m/s2 ) 5 1 kg ? m/s2

Fig. 1.2

m = 1 kg

a = 9.81 m /s 2W = 9.81 N

(1.5)

The SI units are said to form an absolute system of units. This meansthat the three base units chosen are independent of the location wheremeasurements are made. The meter, the kilogram, and the secondmay be used anywhere on the earth; they may even be used on anotherplanet. They will always have the same significance.The weight of a body, or the force of gravity exerted on that body,should, like any other force, be expressed in newtons. From Eq. (1.4)it follows that the weight of a body of mass 1 kg (Fig. 1.3) isW 5 mg5 (1 kg)(9.81 m/s2 )5 9.81 N

Fig. 1.3

Multiples and submultiples of the fundamental SI units may beobtained through the use of the prefixes defined in Table 1.1. Themultiples and submultiples of the units of length, mass, and force mostfrequently used in engineering are, respectively, the kilometer (km)and the millimeter (mm); the megagram (Mg) and the gram (g); andthe kilonewton (kN). According to Table 1.1, we have

1 km 5 1000 m 1 mm 5 0.001 m1 Mg 5 1000 kg1 g 5 0.001 kg1 kN 5 1000 NThe conversion of these units into meters, kilograms, and newtons,respectively, can be effected by simply moving the decimal pointthree places to the right or to the left. For example, to convert3.82 km into meters, one moves the decimal point three places to theright:3.82 km 5 3820 mSimilarly, 47.2 mm is converted into meters by moving the decimalpoint three places to the left:47.2 mm 5 0.0472 m

Also known as a metric ton.


TABLE 1.1



Sl Prefixes

Multiplication Factor

Prefix

Symbol

1 000 000 000 000 5 10121 000 000 000 5 1091 000 000 5 1061 000 5 103100 5 10210 5 1010.1 5 10210.01 5 10220.001 5 10230.000 001 5 10260.000 000 001 5 10290.000 000 000 001 5 102120.000 000 000 000 001 5 102150.000 000 000 000 000 001 5 10218

teragigamegakilohectodekadecicentimillimicronanopicofemtoatto

TGMkhdadcmmnpfa

The first syllable of every prefix is accented so that the prefix will retain its identity. Thus, thepreferred pronunciation of kilometer places the accent on the first syllable, not the second.The use of these prefixes should be avoided, except for the measurement of areas and volumesand for the nontechnical use of centimeter, as for body and clothing measurements.

Using scientific notation, one may also write3.82 km 5 3.82 3 103 m47.2 mm 5 47.2 3 1023 mThe multiples of the unit of time are the minute (min) and thehour (h). Since 1 min 5 60 s and 1 h 5 60 min 5 3600 s, these multiples cannot be converted as readily as the others.By using the appropriate multiple or submultiple of a given unit,one can avoid writing very large or very small numbers. For example,one usually writes 427.2 km rather than 427 200 m, and 2.16 mmrather than 0.002 16 m.

Units of Area and Volume. The unit of area is the square meter(m2), which represents the area of a square of side 1 m; the unit of volume is the cubic meter (m3), equal to the volume of a cube of side 1 m.In order to avoid exceedingly small or large numerical values in thecomputation of areas and volumes, one uses systems of subunitsobtained by respectively squaring and cubing not only the millimeterbut also two intermediate submultiples of the meter, namely, thedecimeter (dm) and the centimeter (cm). Since, by definition,1 dm 5 0.1 m 5 1021 m1 cm 5 0.01 m 5 1022 m1 mm 5 0.001 m 5 1023 m

It should be noted that when more than four digits are used on either side of the decimalpoint to express a quantity in SI unitsas in 427 200 m or 0.002 16 mspaces, nevercommas, should be used to separate the digits into groups of three. This is to avoidconfusion with the comma used in place of a decimal point, which is the convention inmany countries.

7


8

Introduction


the submultiples of the unit of area are1 dm 2 5 (1 dm) 2 5 (1021 m) 2 5 1022 m 21 cm 2 5 (1 cm) 2 5 (1022 m) 2 5 1024 m 21 mm 2 5 (1 mm) 2 5 (1023 m) 2 5 1026 m 2and the submultiples of the unit of volume are1 dm 3 5 (1 dm) 3 5 (1021 m) 3 5 1023 m 31 cm 3 5 (1 cm) 3 5 (1022 m) 3 5 1026 m 31 mm 3 5 (1 mm) 3 5 (1023 m) 3 5 1029 m 3It should be noted that when the volume of a liquid is being measured,the cubic decimeter (dm3) is usually referred to as a liter (L).Other derived SI units used to measure the moment of a force,the work of a force, etc., are shown in Table 1.2. While these units willbe introduced in later chapters as they are needed, we should note animportant rule at this time: When a derived unit is obtained by dividing a base unit by another base unit, a prefix may be used in thenumerator of the derived unit but not in its denominator. For example,the constant k of a spring which stretches 20 mm under a load of100 N will be expressed ask5

100 N100 N55 5000 N/m20 mm0.020 m

ork 5 5 kN/m

but never as k 5 5 N/mm.TABLE 1.2

Principal SI Units Used in Mechanics

Quantity

Unit

Symbol

Formula

AccelerationAngleAngular accelerationAngular velocityAreaDensityEnergyForceFrequencyImpulseLengthMassMoment of a forcePowerPressureStressTimeVelocityVolumeSolidsLiquidsWork

Meter per second squaredRadianRadian per second squaredRadian per secondSquare meterKilogram per cubic meterJouleNewtonHertzNewton-secondMeterKilogramNewton-meterWattPascalPascalSecondMeter per second

...rad............JNHz...mkg...WPaPas...

m/s2rad/s2rad/sm2kg/m3N?mkg ? m/s2s21kg ? m/sN?mJ/sN/m2N/m2m/s

Cubic meterLiterJoule

...LJ

m31023m3N?m

Supplementary unit (1 revolution 5 2p rad 5 3608).Base unit.



U.S. Customary Units. Most practicing American engineers stillcommonly use a system in which the base units are the units of length,force, and time. These units are, respectively, the foot (ft), the pound(lb), and the second (s). The second is the same as the correspondingSI unit. The foot is defined as 0.3048 m. The pound is defined as theweight of a platinum standard, called the standard pound, which iskept at the National Institute of Standards and Technology outsideWashington, the mass of which is 0.453 592 43 kg. Since the weight ofa body depends upon the earths gravitational attraction, which varieswith location, it is specified that the standard pound should be placedat sea level and at a latitude of 458 to properly define a force of 1 lb.Clearly the U.S. customary units do not form an absolute system ofunits. Because of their dependence upon the gravitational attraction ofthe earth, they form a gravitational system of units.While the standard pound also serves as the unit of mass in commercial transactions in the United States, it cannot be so used in engineering computations, since such a unit would not be consistent withthe base units defined in the preceding paragraph. Indeed, when actedupon by a force of 1 lb, that is, when subjected to the force of gravity,the standard pound receives the acceleration of gravity, g 5 32.2 ft/s2(Fig. 1.4), not the unit acceleration required by Eq. (1.1). The unit ofmass consistent with the foot, the pound, and the second is the masswhich receives an acceleration of 1 ft/s2 when a force of 1 lb is appliedto it (Fig. 1.5). This unit, sometimes called a slug, can be derived fromthe equation F 5 ma after substituting 1 lb and 1 ft/s2 for F and a,respectively. We writeF 5 ma

1 lb 5 (1 slug)(1 ft/s )

m = 1 lba = 32.2 ft /s 2

1 lb5 1 lb ? s2/ft1 ft/s2

a = 1 ft /s 2m = 1 slug(= 1 lb s 2/ft)

Fig. 1.5

(1.6)

Comparing Figs. 1.4 and 1.5, we conclude that the slug is a mass 32.2times larger than the mass of the standard pound.The fact that in the U.S. customary system of units bodies arecharacterized by their weight in pounds rather than by their mass inslugs will be a convenience in the study of statics, where one constantlydeals with weights and other forces and only seldom with masses.However, in the study of dynamics, where forces, masses, and accelerations are involved, the mass m of a body will be expressed in slugswhen its weight W is given in pounds. Recalling Eq. (1.4), we writem5

Wg

(1.7)

where g is the acceleration of gravity ( g 5 32.2 ft/s2).Other U.S. customary units frequently encountered in engineering problems are the mile (mi), equal to 5280 ft; the inch (in.), equal to112 ft; and the kilopound (kip), equal to a force of 1000 lb. The ton isoften used to represent a mass of 2000 lb but, like the pound, must beconverted into slugs in engineering computations.The conversion into feet, pounds, and seconds of quantitiesexpressed in other U.S. customary units is generally more involved and

F = 1 lb

Fig. 1.4

2

and obtain1 slug 5


F = 1 lb

9


10

Introduction


requires greater attention than the corresponding operation in SIunits. If, for example, the magnitude of a velocity is given as v 530 mi/h, we convert it to ft/s as follows. First we writev 5 30

mih

Since we want to get rid of the unit miles and introduce instead theunit feet, we should multiply the right-hand member of the equationby an expression containing miles in the denominator and feet in thenumerator. But, since we do not want to change the value of the righthand member, the expression used should have a value equal to unity.The quotient (5280 ft)/(1 mi) is such an expression. Operating in asimilar way to transform the unit hour into seconds, we writev 5 a30

mi 5280 ft1hbababh1 mi3600 s

Carrying out the numerical computations and canceling out units whichappear in both the numerator and the denominator, we obtainv 5 44

1.4

ft5 44 ft/ss

CONVERSION FROM ONE SYSTEM OF UNITSTO ANOTHER

There are many instances when an engineer wishes to convert into SIunits a numerical result obtained in U.S. customary units or vice versa.Because the unit of time is the same in both systems, only two kineticbase units need be converted. Thus, since all other kinetic units can bederived from these base units, only two conversion factors need beremembered.

Units of Length.

By definition the U.S. customary unit of length is1 ft 5 0.3048 m

(1.8)

It follows that1 mi 5 5280 ft 5 5280(0.3048 m) 5 1609 mor1 mi 5 1.609 kmAlso1 in. 5

112

ft 5

112 (0.3048

(1.9)

m) 5 0.0254 m

or1 in. 5 25.4 mm

(1.10)

Units of Force. Recalling that the U.S. customary unit of force(pound) is defined as the weight of the standard pound (of mass0.4536 kg) at sea level and at a latitude of 458 (where g 5 9.807 m/s2)and using Eq. (1.4), we write



1.5 Method of Problem Solution

W 5 mg1 lb 5 (0.4536 kg)(9.807 m/s2 ) 5 4.448 kg ? m/s2or, recalling Eq. (1.5),1 lb 5 4.448 N

(1.11)

Units of Mass. The U.S. customary unit of mass (slug) is a derivedunit. Thus, using Eqs. (1.6), (1.8), and (1.11), we write1 slug 5 1 lb ? s2/ft 5

1 lb4.448 N55 14.59 N ? s2/m21 ft/s0.3048 m/s2

and, recalling Eq. (1.5),1 slug 5 1 lb ? s2/ft 5 14.59 kg

(1.12)

Although it cannot be used as a consistent unit of mass, we recall thatthe mass of the standard pound is, by definition,1 pound mass 5 0.4536 kg

(1.13)

This constant may be used to determine the mass in SI units (kilograms) of a body which has been characterized by its weight in U.S.customary units (pounds).To convert a derived U.S. customary unit into SI units, one simply multiplies or divides by the appropriate conversion factors. Forexample, to convert the moment of a force which was found to beM 5 47 lb ? in. into SI units, we use formulas (1.10) and (1.11) andwriteM 5 47 lb ? in. 5 47(4.448 N)(25.4 mm)5 5310 N ? mm 5 5.31 N ? mThe conversion factors given in this section may also be used toconvert a numerical result obtained in SI units into U.S. customaryunits. For example, if the moment of a force was found to be M 540 N ? m, we write, following the procedure used in the last paragraphof Sec. 1.3,M 5 40 N ? m 5 (40 N ? m) a

1 lb1 ftbab4.448 N 0.3048 m

Carrying out the numerical computations and canceling out unitswhich appear in both the numerator and the denominator, we obtainM 5 29.5 lb ? ftThe U.S. customary units most frequently used in mechanics arelisted in Table 1.3 with their SI equivalents.

1.5

METHOD OF PROBLEM SOLUTION

You should approach a problem in mechanics as you would approachan actual engineering situation. By drawing on your own experienceand intuition, you will find it easier to understand and formulate theproblem. Once the problem has been clearly stated, however, there is

11


12

Introduction

TABLE 1.3


U.S. Customary Units and Their SI Equivalents

Quantity

U.S. Customary Unit

Acceleration

ft/s2in./s2ft2in2ft ? lbkiplbozlb ? sftin.mioz masslb massslugtonlb ? ftlb ? in.

AreaEnergyForce

ImpulseLength

Mass

Moment of a forceMoment of inertiaOf an areaOf a massMomentumPowerPressure or stressVelocity

VolumeLiquidsWork

in4lb ? ft ? s2lb ? sft ? lb/shplb/ft2lb/in2 (psi)ft/sin./smi/h (mph)mi/h (mph)ft3in3galqtft ? lb

SI Equivalent0.3048 m/s20.0254 m/s20.0929 m2645.2 mm21.356 J4.448 kN4.448 N0.2780 N4.448 N ? s0.3048 m25.40 mm1.609 km28.35 g0.4536 kg14.59 kg907.2 kg1.356 N ? m0.1130 N ? m0.4162 3 106 mm41.356 kg ? m24.448 kg ? m/s1.356 W745.7 W47.88 Pa6.895 kPa0.3048 m/s0.0254 m/s0.4470 m/s1.609 km/h0.02832 m316.39 cm33.785 L0.9464 L1.356 J

no place in its solution for your particular fancy. The solution must bebased on the six fundamental principles stated in Sec. 1.2 or on theorems derived from them. Every step taken must be justified on thatbasis. Strict rules must be followed, which lead to the solution in analmost automatic fashion, leaving no room for your intuition or feeling. After an answer has been obtained, it should be checked. Hereagain, you may call upon your common sense and personal experience.If not completely satisfied with the result obtained, you should carefullycheck your formulation of the problem, the validity of the methodsused for its solution, and the accuracy of your computations.The statement of a problem should be clear and precise. It shouldcontain the given data and indicate what information is required. Aneat drawing showing all quantities involved should be included. Separate diagrams should be drawn for all bodies involved, indicatingclearly the forces acting on each body. These diagrams are known asfree-body diagrams and are described in detail in Secs. 2.11 and 4.2.


The fundamental principles of mechanics listed in Sec. 1.2 willbe used to write equations expressing the conditions of rest or motionof the bodies considered. Each equation should be clearly related toone of the free-body diagrams. You will then proceed to solve theproblem, observing strictly the usual rules of algebra and recordingneatly the various steps taken.After the answer has been obtained, it should be carefully checked.Mistakes in reasoning can often be detected by checking the units. Forexample, to determine the moment of a force of 50 N about a point0.60 m from its line of action, we would have written (Sec. 3.12)M 5 Fd 5 (50 N)(0.60 m) 5 30 N ? mThe unit N ? m obtained by multiplying newtons by meters is the correct unit for the moment of a force; if another unit had been obtained,we would have known that some mistake had been made.Errors in computation will usually be found by substituting thenumerical values obtained into an equation which has not yet beenused and verifying that the equation is satisfied. The importance ofcorrect computations in engineering cannot be overemphasized.

1.6

NUMERICAL ACCURACY

The accuracy of the solution of a problem depends upon two items:(1) the accuracy of the given data and (2) the accuracy of the computations performed.The solution cannot be more accurate than the less accurate ofthese two items. For example, if the loading of a bridge is known to be75,000 lb with a possible error of 100 lb either way, the relative errorwhich measures the degree of accuracy of the data is100 lb5 0.0013 5 0.13 percent75,000 lbIn computing the reaction at one of the bridge supports, it would thenbe meaningless to record it as 14,322 lb. The accuracy of the solutioncannot be greater than 0.13 percent, no matter how accurate the computations are, and the possible error in the answer may be as large as(0.13/100)(14,322 lb) < 20 lb. The answer should be properly recordedas 14,320 6 20 lb.In engineering problems, the data are seldom known with anaccuracy greater than 0.2 percent. It is therefore seldom justified towrite the answers to such problems with an accuracy greater than 0.2percent. A practical rule is to use 4 figures to record numbers beginning with a 1 and 3 figures in all other cases. Unless otherwise indicated, the data given in a problem should be assumed known with acomparable degree of accuracy. A force of 40 lb, for example, shouldbe read 40.0 lb, and a force of 15 lb should be read 15.00 lb.Pocket electronic calculators are widely used by practicing engineers and engineering students. The speed and accuracy of these calculators facilitate the numerical computations in the solution of manyproblems. However, students should not record more significant figures than can be justified merely because they are easily obtained. Asnoted above, an accuracy greater than 0.2 percent is seldom necessaryor meaningful in the solution of practical engineering problems.


1.6 Numerical Accuracy

13


Many engineering problems can besolved by considering the equilibrium ofa particle. In the case of thisexcavator, which is being loaded ontoa ship, a relation between the tensionsin the various cables involved can beobtained by considering the equilibriumof the hook to which the cables areattached.

14




2

C H A P T E R


15


Chapter 2 Statics of Particles2.12.22.32.42.52.62.72.82.92.102.11

2.122.13

2.142.15

IntroductionForce on a Particle. Resultant ofTwo ForcesVectorsAddition of VectorsResultant of Several ConcurrentForcesResolution of a Force intoComponentsRectangular Components of aForce. Unit VectorsAddition of Forces by SummingX and Y ComponentsEquilibrium of a ParticleNewtons First Law of MotionProblems Involving theEquilibrium of a Particle. FreeBody DiagramsRectangular Components of aForce in SpaceForce Defined by Its Magnitudeand Two Points on Its Line ofActionAddition of Concurrent Forcesin SpaceEquilibrium of a Particle in Space

2.1


INTRODUCTION

In this chapter you will study the effect of forces acting on particles.First you will learn how to replace two or more forces acting on agiven particle by a single force having the same effect as the originalforces. This single equivalent force is the resultant of the originalforces acting on the particle. Later the relations which exist amongthe various forces acting on a particle in a state of equilibrium willbe derived and used to determine some of the forces acting on theparticle.The use of the word particle does not imply that our studywill be limited to that of small corpuscles. What it means is that thesize and shape of the bodies under consideration will not significantlyaffect the solution of the problems treated in this chapter and thatall the forces acting on a given body will be assumed to be appliedat the same point. Since such an assumption is verified in manypractical applications, you will be able to solve a number of engineering problems in this chapter.The first part of the chapter is devoted to the study of forcescontained in a single plane, and the second part to the analysis offorces in three-dimensional space.

FORCES IN A PLANE2.2

FORCE ON A PARTICLE. RESULTANTOF TWO FORCES

A force represents the action of one body on another and is generallycharacterized by its point of application, its magnitude, and its direction. Forces acting on a given particle, however, have the same pointof application. Each force considered in this chapter will thus becompletely defined by its magnitude and direction.The magnitude of a force is characterized by a certain number of units. As indicated in Chap. 1, the SI units used by engineers to measure the magnitude of a force are the newton (N) andits multiple the kilonewton (kN), equal to 1000 N, while the U.S.customary units used for the same purpose are the pound (lb) andits multiple the kilopound (kip), equal to 1000 lb. The directionof a force is defined by the line of action and the sense of theforce. The line of action is the infinite straight line along whichthe force acts; it is characterized by the angle it forms with somefixed axis (Fig. 2.1). The force itself is represented by a segment of

10

lb

A

Fig. 2.1

16

(a)

10

30

lb

30

A

(b)


that line; through the use of an appropriate scale, the length of thissegment may be chosen to represent the magnitude of the force.Finally, the sense of the force should be indicated by an arrowhead.It is important in defining a force to indicate its sense. Two forceshaving the same magnitude and the same line of action but differentsense, such as the forces shown in Fig. 2.1a and b, will have directlyopposite effects on a particle.Experimental evidence shows that two forces P and Q actingon a particle A (Fig. 2.2a) can be replaced by a single force R whichhas the same effect on the particle (Fig. 2.2c). This force is calledthe resultant of the forces P and Q and can be obtained, as shownin Fig. 2.2b, by constructing a parallelogram, using P and Q as twoadjacent sides of the parallelogram. The diagonal that passes throughA represents the resultant. This method for finding the resultant isknown as the parallelogram law for the addition of two forces. Thislaw is based on experimental evidence; it cannot be proved or derivedmathematically.

2.3


2.3 Vectors

PA

Q(a)

PA

R

Q(b)

VECTORS

It appears from the above that forces do not obey the rules of addition defined in ordinary arithmetic or algebra. For example, twoforces acting at a right angle to each other, one of 4 lb and the otherof 3 lb, add up to a force of 5 lb, not to a force of 7 lb. Forces arenot the only quantities which follow the parallelogram law of addition. As you will see later, displacements, velocities, accelerations, andmomenta are other examples of physical quantities possessing magnitude and direction that are added according to the parallelogramlaw. All these quantities can be represented mathematically by vectors, while those physical quantities which have magnitude but notdirection, such as volume, mass, or energy, are represented by plainnumbers or scalars.Vectors are defined as mathematical expressions possessingmagnitude and direction, which add according to the parallelogram law. Vectors are represented by arrows in the illustrationsand will be distinguished from scalar quantities in this text throughthe use of boldface type (P). In longhand writing, a vector may bedenotedS drawing a short arrow above the letter used to reprebysent it (P ) or by underlining the letter ( P ). The last method maybe preferred since underlining can also be used on a typewriteror computer. The magnitude of a vector defines the length of thearrow used to represent the vector. In this text, italic type will beused to denote the magnitude of a vector. Thus, the magnitude ofthe vector P will be denoted by P.A vector used to represent a force acting on a given particlehas a well-defined point of application, namely, the particle itself.Such a vector is said to be a fixed, or bound, vector and cannot bemoved without modifying the conditions of the problem. Otherphysical quantities, however, such as couples (see Chap. 3), arerepresented by vectors which may be freely moved in space; these

RA(c)Fig. 2.2

17


18


PP

Fig. 2.4P


vectors are called free vectors. Still other physical quantities, suchas forces acting on a rigid body (see Chap. 3), are represented byvectors which can be moved, or slid, along their lines of action;they are known as sliding vectors.Two vectors which have the same magnitude and the samedirection are said to be equal, whether or not they also have the samepoint of application (Fig. 2.4); equal vectors may be denoted by thesame letter.The negative vector of a given vector P is defined as a vectorhaving the same magnitude as P and a direction opposite to that ofP (Fig. 2.5); the negative of the vector P is denoted by 2P. Thevectors P and 2P are commonly referred to as equal and oppositevectors. Clearly, we haveP 1 (2P) 5 0

PFig. 2.5

PP+QAFig. 2.6

Q

2.4

ADDITION OF VECTORS

We saw in the preceding section that, by definition, vectors addaccording to the parallelogram law. Thus, the sum of two vectors Pand Q is obtained by attaching the two vectors to the same point Aand constructing a parallelogram, using P and Q as two sides of theparallelogram (Fig. 2.6). The diagonal that passes through A represents the sum of the vectors P and Q, and this sum is denoted byP 1 Q. The fact that the sign 1 is used to denote both vector andscalar addition should not cause any confusion if vector and scalarquantities are always carefully distinguished. Thus, we should notethat the magnitude of the vector P 1 Q is not, in general, equal tothe sum P 1 Q of the magnitudes of the vectors P and Q.Since the parallelogram constructed on the vectors P and Q doesnot depend upon the order in which P and Q are selected, we conclude that the addition of two vectors is commutative, and we write(2.1)

P1Q5Q1P

Some expressions have magnitude and direction, but do not add according to theparallelogram law. While these expressions may be represented by arrows, they cannotbe considered as vectors.A group of such expressions is the finite rotations of a rigid body. Place a closedbook on a table in front of you, so that it lies in the usual fashion, with its front coverup and its binding to the left. Now rotate it through 180 about an axis parallel to thebinding (Fig. 2.3a); this rotation may be represented by an arrow of length equal to180 units and oriented as shown. Picking up the book as it lies in its new position, rotate

=180Fig. 2.3

(a)Finite rotations of a rigid body

180

=(b)



2.4 Addition of Vectors

From the parallelogram law, we can derive an alternativemethod for determining the sum of two vectors. This method, knownas the triangle rule, is derived as follows. Consider Fig. 2.6, wherethe sum of the vectors P and Q has been determined by the parallelogram law. Since the side of the parallelogram opposite Q is equalto Q in magnitude and direction, we could draw only half of theparallelogram (Fig. 2.7a). The sum of the two vectors can thus befound by arranging P and Q in tip-to-tail fashion and then connecting the tail of P with the tip of Q. In Fig. 2.7b, the other half of theparallelogram is considered, and the same result is obtained. Thisconfirms the fact that vector addition is commutative.The subtraction of a vector

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