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VECTOR MECHANICS

forENGINEERS

TENTH EDITION

Reactions at Supports and Connections for a Two-Dimensional Structure

Support or Connection

Reaction

Number of

Unknowns

í A j¿

Rollers R°cker Frictionless

surface

f

Forcé with known

line of action

1

| i

Short cable

Short link

Forcé with known

line of action

1

frictíonlessrod F nctionless pin in slot

/

w 90° /

/

/

/

/

Force with known

line of action

1

1 J

Frictionless pin Rough

or hinge

i surface

i

a

Foro

i

or

i

e of unknc

direction

1

>wn

2

-1-

Fixed support

í

Forc

^e and cou

1*

pie

3

The first step in the solution of any problem concerning the

equilibrium of a rigid body is to construct an appropriate free-body

diagram of the body. As parí of that process, it is necessary to show

on the diagram the reactions through which the ground and other

bodies oppose a possible motion of the body. The figures on this and

the facing page summarize the possible reactions exerted on two-

and three-dimensional bodies.

Reactions at Supports and Connections for a Three-Dimensional Structure

i r

Frictionless surface

Forcé with known

line of action

(one unknown)

Forcé with known

Cable line ac hon

(one unknown)

X

Roller on

rough surface

Wheel on rail

Two forcé components

Rough surface

Rail and socket

1 *

X Fx

Three forcé components

F 4

■Xr

F r

e

ML

Universal

joint

Three forcé components

and one couple

t

CD ’

¿L.

f* **

■6

Fixed support

Three forcé components

and three couples

i(M„

T

(M,

cD

¿ 1 .

Hinge and bearing supporting radial load only

Two forcé components

(and two couples; see page 191)

i

CD

(M„

(M.

Pin and bracket

Hinge and bearing supporting

axial thrust and radial load

Three forcé components

(and two couples; see page 191)

Tenth Edition

VECTOR MECHANICS

FOR ENGINEERS

Statics and Dynamics

Ferdinand P. Beer

Late of Lehigh University

E. Russell Johnston, Jr.

Late of University of Connecticut

David F. Mazurek

U.S. Coast Guard Academy

Phillip J. Cornwell

Rose-Hulmán Institute of Technology

With the collaboration of

Brian P. Self

California Polytechnic State University—San Luis Obispo

Connect

i Learn

I Succeed ™

Me

Graw

Hill

The McGrawHill Companies

XConnect

\ Learn

1 Succeed

TM

VECTOR MECHANICS FOR ENGINEERS: STATICS AND DYNAMICS, TENTH EDITION

Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas,

New York, NY 10020. Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in

the United States of America. Previous editions © 2010, 2007, and 2004. 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.

1234567890 DOW/DOW 1098765432

ISRN 978-0-07-339813-6

MHID 0-07-339813-6

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All credits appearing on page or at the end of the book are considered to be an extensión of the copyright page.

Library of Congress Cataloging-in-Publication Data

Vector mechanics for engineers: statics and dynamics / Ferdinand Beer . . . [et al.]. — lOth ed.

p. cm.

Ineludes Índex.

ISBN 978-0-07-339813-6 — ISBN 0-07-339813-6 (hard copy : alk. paper) 1. Mechanics, Applied.

2. Vector analysis. 3. Statics. 4. Dynamics. I. Beer, Ferdinand P. (Ferdinand Pierre), 1915-2003.

TA350.V34 2013

620.1-05—dc23

2011034388

www.mhhe.com

About the Authors

As 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.

The answer to this question is simple. Russ Johnston s first teach-

ing appointment was in the Department of Civil Engineering and

Mechanics 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.

Ferd was delighted to discover that the young man who had

been hired chiefly to teach gradúate 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 principies 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.

The second edition of Mechanics for Engineers and the first

edition of Vector 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 gradúate 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 Ferds and Russs

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 División of the same society. Starting in

2001, the New Mechanics Educator Award of the Mechanics División

has been named in honor of the Beer and Johnston author team.

IV

About the Authors

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 carne to the United States after serving in the French army during

the early part of World War II and taught for four years at Williams

College in the Williams-M IT joint arts and engineering program.

Following his Service at Williams College, Ferd joined the faculty of

Lehigh University where he taught for thirty-seven years. He held sev-

eral positions, including University Distinguished Professor and chair-

man 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. de¬

gree in civil engineering from the University of Delaware and an Sc.D.

degree in the field of structural engineering from the Massachusetts

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 Depart¬

ment of Civil Engineering and taught for twenty-six years. In 1991

Russ received the Outstanding Civil Engineer Award from the Con¬

necticut Section of the American Society of Civil Engineers.

About the Authors

V

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 División

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 &

Maintenance-of-Way Associations Committee 15—Steel Structures

since 1991. Professional interests inelude bridge engineering, struc-

tural forensics, and blast-resistant design. He is a registered Profes¬

sional Engineer in Connecticut and Pennsylvania.

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 and Vice President

of Academic Affairs at Rose-Hulmán Institute of Technology where he

has taught since 1989. Phil received an SAE Ralph R. Teetor Educa-

tional Award in 1992, the Dean s Outstanding Teacher Award at Rose-

Hulman in 2000, and the Board of Trustees’ Outstanding Scholar

Award at Rose-Hulmán in 2001.

Brian P. Self. Brian obtained his B.S. and M.S. degrees in Engineer¬

ing Mechanics from Virginia Tech, and his Ph.D. in Bioengineering

from the University of Utah. He worked in the Air Forcé Research

Laboratories before teaching at the U.S. Air Forcé Academy for seven

years. Brian has taught in the Mechanical Engineering Department

at Cal Poly, San Luis Obispo since 2006. He has been very active in

the American Society of Engineering Education, serving on its Board

from 2008-2010. With a team of five, Brian developed the Dynamics

Concept Inventory to help assess student conceptual understanding.

His professional interests inelude educational research, aviation physi-

ology, and biomechanics.

Brief Contents

Introduction 1

Statics of Particles 14

Rigid Bodies: Equivalent Systems of Forces 74

4 Equilibrium of Rigid Bodies 158

Distributed Forces: Centroids and Centers of Gravity 218

6 Analysis of Structures 282

Forces in Beams and Cables 352

8 Friction 410

Distributed Forces: Moments of Inertia 468

10 Method of Virtual Work 556

11 Kinematics of Particles 600

12 Kinetics of Particles: Newton's Second Law 694

13 Kinetics of Particles: Energy and Momentum Methods 762

14 Systems of Particles 866

15 Kinematics of Rigid Bodies 926

16 Plañe Motion of Rigid Bodies: Forces and Accelerations 1040

17 Plañe Motion of Rigid Bodies: Energy and Momentum

Methods 1104

18 Kinetics of Rigid Bodies in Three Dimensions 1172

19 Mechanical Vibrations 1280

Appendix Al

Photo Credits C1

Index II

Answers to Problems AN1

vii

Contents

L

Preface xix

Guided Tour xxiii

What Resources Support This Textbook? xxv

Acknowledgments xxvii

Connect xxviü

List of Symbols xxx

Introduction 1

1.1 What Is Mechamos? 2

1.2 Fundamental Concepts and Principies 2

1.3 Systems of Units 5

1.4 Conversión from One System of Units to Another 10

1.5 Method of Problem Solution 11

1.6 Numerical Accuracy 13

Statics of Particles 14

2.1 Introduction 16

Forces in a Plañe 16

2.2 Forcé 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 Forcé into Components 21

2.7 Rectangular Components of a Forcé. 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 ¡n Space 45

2.12 Rectangular Components of a Forcé in Space 45

2.13 Forcé Defined by Its Magnitude and Two Points on Its

Une of Action 49

2.14 Addition of Concurrent Forces in Space 50

2.15 Equilibrium of a Particle in Space 58

Review and Summary for Chapter 2 66

Review Problems 69

Computer Problems 72

IX

X

Contents

Rigid Bodies: Equivalent

Systems of Forces 74

3.1 Introduction 76

3.2 External and Infernal Forces 76

3.3 Principie of Transmissibility. Equivalent Forces 77

3.4 Vector Product of Two Vectors 79

3.5 Vector Products Expressed ¡n Terms of

Rectangular Components 81

3.6 Moment of a Forcé about a Point 83

3.7 Varignon's Theorem 85

3.8 Rectangular Components of the Moment of a Forcé 85

3.9 Scalar Product of Two Vectors 96

3.10 Mixed Triple Product of Three Vectors 98

3.11 Moment of a Forcé about a Given Axis 99

3.12 Moment of a Couple 110

3.13 Equivalent Couples 111

3.14 Addition of Couples 113

3.15 Couples Can Be Represented by Vectors 113

3.16 Resolution of a Given Forcé into a Forcé at O

and a Couple 114

3.17 Reduction of a System of Forces to One Forcé and

One Couple 125

3.18 Equ ivalent Systems of Forces 126

3.19 Equipollent Systems of Vectors 127

3.20 Further Reduction of a System of Forces 128

* 3.21 Reduction of a System of Forces to a Wrench 130

Review and Summary for Chapter 3 148

Review Problems 153

Computer Problems 156

4 Equilibrium of Rigid Bodies 158

4.1 Introduction 160

4.2 Free-Body Diagram 161

Equilibrium in Two Dimensions 162

4.3 Reactions at Supports and Connections

for a Two-Dimensional Structure 162

4.4 Equilibrium of a Rigid Body in Two Dimensions 164

4.5 Statically Indeterminate Reactions. Partial Constraints 166

4.6 Equilibrium of a Two-Force Body 183

4.7 Equilibrium of a Three-Force Body 184

Equilibrium in Three Dimensions 191

4.8 Equilibrium of a Rigid Body in Three Dimensions 191

4.9 Reactions at Supports and Connections for a

Three-Dimensional Structure 191

Contents

XI

Review and Summary for Chapter 4 210

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

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 for Chapter 5 274

Review Problems 278

Computer Problems 280

6 Analysis of Structures 282

6.1 Introduction 284

Trusses 285

6.2 Definition of a Truss 285

6.3 Simple Trusses 287

6.4 Analysis of Trusses by the Method of Joints 288

* 6.5 Joints Under Special Loading Conditions 290

* 6.6 Space Trusses 292

6.7 Analysis of Trusses by the Method of Sections 302

* 6.8 Trusses Made of Several Simple Trusses 303

Frames and Machines 314

6.9 Structures Containing Multiforce Members 314

6.10 Analysis of a Frame 314

XII

Contents

6.11 Frames Which Cease to Be Rigid When Detached

from Their Supports 315

6.12 Machines 330

Review and Summary for Chapter 6 344

Review Problems 347

Computer Problems 350

7

Forces in Beams and Cables 352

* 7.1

Introduction 354

* 7.2

Internal Forces in Members 354

Beams 361

* 7.3

Varíous Types of Loading and Support

361

* 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

* 7.9

Parabolic Cable 385

* 7.10

Catenary 395

Review and Summary for Chapter 7 403

Review Problems 406

Computer Problems 408

8 Frictíon 410

8.1 Introduction 412

8.2 The Laws of Dry Frictíon. Coefficients

of Frictíon 412

8.3 Angles of Frictíon 413

8.4 Problems Involving Dry Frictíon 416

8.5 Wedges 429

8.6 Square-Threaded Screws 430

* 8.7 Journal Bearings. Axle Frictíon 439

* 8.8 Thrust Bearings. Disk Frictíon 441

* 8.9 Wheel Frictíon. Rolling Resistance 442

* 8.10 Belt Frictíon 449

Review and Summary for Chapter 8 460

Review Problems 463

Computer Problems 466

Contents

1 ^ Distributed Forces:

Moments of Inertia 468

9.1 Introduction 470

Moments of Inertia of Areas 471

9.2 Second Moment, or Moment of Inertia, of an Area 471

9.3 Determinaron of the Moment of Inertia of an

Area by Integration 472

9.4 Polar Moment of Inertia 473

9.5 Radius of Gyration of an Area 474

9.6 Parallel-Axis Theorem 481

9.7 Moments of Inertia of Composite Areas 482

* 9.8 Product of Inertia 495

* 9.9 Principal Axes and Principal Moments of Inertia 496

* 9.10 Mohr's Circle for Moments and Products of Inertia 504

Moments of Inertia of a Mass 510

9.11 Moment of Inertia of a Mass 510

9.12 Parallel-Axis Theorem 512

9.13 Moments of Inertia of Thin Plates 513

9.14 Determinaron of the Moment of Inertia of a

Three-Dimensional Body by Integration 514

9.15 Moments of Inertia of Composite Bodies 514

* 9.16 Moment of Inertia of a Body with Respect to an Arbitrary Axis

Through O. Mass Products of Inertia 530

* 9.17 Ellipsoid of Inertia. Principal Axes of Inertia 531

* 9.18 Determination of the Principal Axes and Principal Moments of

Inertia of a Body of Arbitrary Shape 533

Review and Summary for Chapter 9 545

Review Problems 551

Computer Problems 554

XIII

Method of Virtual Work 556

* 10.1 Introduction 558

* 10.2 Work of a Forcé 558

* 10.3 Principie of Virtual Work 561

* 10.4 Applications of the Principie of Virtual Work 562

* 10.5 Real Machines. Mechanical Efficiency 564

* 10.6 Work of a Forcé 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 for Chapter 10 592

Review Problems 595

Computer Problems 598

Contents

xiv

11 Kinematics of Particles 600

11.1 Introduction to Dynamics 602

Rectilinear Motion of Particles 603

11.2 Position, Velocity, and Acceleration 603

11.3 Determination of the Motion of a Particle 607

11.4 Uniform Rectilinear Motion 618

11.5 Uniformly Accelerated Rectilinear Motion 618

11.6 Motion of Several Particles 619

* 11.7 Graphical Solution of Rectilinear-Motion Problems 632

* 11.8 Other Graphical Methods 633

Curvilinear Motion of Particles 643

11.9 Position Vector, Velocity, and Acceleration 643

11.10 Derivatives of Vector Functions 645

11.11 Rectangular Components of Velocity and Acceleration 647

11.12 Motion Relative to a Frame ¡n Translation 648

11.13 Tangential and Normal Components 667

11.14 Radial and Transverse Components 670

Review and Summary for Chapter 11 685

Review Problems 689

Computer Problems 692

12 Kinetics of Particles:

Newton's Second Law 694

12.1 Introduction 696

12.2 Newton's Second Law of Motion 697

12.3 Linear Momentum of a Particle. Rate of Change

of Linear Momentum 698

12.4 Systems of Units 699

12.5 Equations of Motion 701

12.6 Dynamic Equilibrium 703

12.7 Angu lar Momentum of a Particle. Rate of Change

of Angular Momentum 727

12.8 Equations of Motion ¡n Terms of Radial and

Transverse Components 728

12.9 Motion Under a Central Forcé. Conservaron of

Angular Momentum 729

12.10 Newton's Law of Gravitation 730

* 12.11 Trajectory of a Particle Under a Central Forcé 741

* 12.12 Application to Space Mechanics 742

* 12.13 Kepler's Laws of Planetary Motion 745

Review and Summary for Chapter 12 754

Review Problems 758

Computer Problems 761

Contents

XV

13 Kinetics of Porfíeles: Energy and

Momentum Methods 762

13.1 Introduction 764

13.2 Work of a Forcé 764

13.3 Kinetic Energy of a Particle. Principie of Work

and Energy 768

13.4 Applications of the Principie of Work and Energy 770

13.5 Power and Efficiency 771

13.6 Potential Energy 789

* 13.7 Conservative Forces 791

13.8 Conservation of Energy 792

13.9 Motion Under a Conservative Central Forcé.

Application to Space Mechanics 793

13.10 Principie of Impulse and Momentum 814

13.11 Impulsive Motion 817

13.12 Impact 831

13.13 Direct Central Impact 831

13.14 Oblique Central Impact 834

13.15 Problems Involving Energy and Momentum 837

Review and Summary for Chapter 13 854

Review Problems 860

Computer Problems 864

Systems of Particles 866

14.1 Introduction 868

14.2 Application of Newton's Laws to the Motion of a System

of Particles. Effective Forces 868

14.3 Linear and Angular Momentum of a System of Particles 871

14.4 Motion of the Mass Center of a System of Particles 872

14.5 Angular Momentum of a System of Particles About Its

Mass Center 874

14.6 Conservation of Momentum for a System of Particles 876

14.7 Kinetic Energy of a System of Particles 886

14.8 Work-Energy Principie. Conservation of Energy for a System

of Particles 887

14.9 Principie of Impulse and Momentum for a System

of Particles 887

* 14.10 Variable Systems of Particles 897

* 14.11 Steady Stream of Particles 898

* 14.12 Systems Gaining or Losing Mass 900

Review and Summary for Chapter 14 917

Review Problems 921

Computer Problems 924

XVI

Contents

15 Kinematics of Rigid Bodies 926

15*1 Introduction 928

15.2 Translation 930

15.3 Rotation About a Fixed Axis 931

15.4 Equations Defining the Rotation of a Rigid Body

About a Fixed Axis 934

15.5 General Plañe Motion 944

15.6 Absolute and Relative Velocity ¡n Plañe Motion 946

15.7 Instantaneous Center of Rotation ¡n Plañe Motion 958

15.8 Absolute and Relative Acceleration ¡n Plañe Motion 970

* 15.9 Analysis of Plañe Motion ¡n Terms of a Parameter 972

15.10 Rate of Change of a Vector with Respect to a

Rotating Frame 985

15.11 Plañe Motion of a Particle Relative to a Rotating Frame.

Coriolis Acceleration 987

* 15.12 Motion About a Fixed Point 998

* 15.13 General Motion 1001

* 15.14 Three-Dimensional Motion of a Particle Relative to a Rotating

Frame. Coriolis Acceleration 1013

* 15.15 Frame of Reference ¡n General Motion 1014

Review and Summary for Chapter 15 1026

Review Problems 1033

Computer Problems 1037

16 Plañe Motion of Rigid Bodies: Forces

and Accelerations 1040

16.1 Introduction 1042

16.2 Equations of Motion for a Rigid Body 1043

16.3 Angular Momentum of a Rigid Body ¡n Plañe Motion 1044

16.4 Plañe Motion of a Rigid Body. D'Alembert's Principie 1045

* 16.5 A Remark on the Axioms of the Mechanics

of Rigid Bodies 1046

16.6 Solution of Problems Involving the Motion of a

Rigid Body 1047

16.7 Systems of Rigid Bodies 1048

16.8 Constrained Plañe Motion 1072

Review and Summary for Chapter 16 1097

Review Problems 1099

Computer Problems 1103

17 Plañe Motion of Rigid Bodies: Energy

and Momentum Methods 1104

17.1 Introduction 1106

17.2 Principie of Work and Energy for a Rigid Body 1106

17.3 Work of Forces Acting on a Rigid Body 1107

17.4 Kinetic Energy of a Rigid Body ¡n Plañe Motion 1108

17.5 Systems of Rigid Bodies 1109

17.6 Conservaron of Energy 1110

17.7 Power lili

17.8 Principie of Impulse and Momentum for the Plañe Motion

of a Rigid Body 1129

17.9 Systems of Rigid Bodies 1132

17.10 Conservaron of Angular Momentum 1132

17.11 Impulsive Motion 1147

17.12 Eccentric Impact 1147

Review and Summary for Chapter 17 1163

Review Problems 1167

Computer Problems 1170

Contents

18 Kinetics of Rigid Bodies in

Three Dimensions 1172

* 18.1 Introduction 1174

* 18.2 Angular Momentum of a Rigid Body in

Three Dimensions 1175

* 18.3 Application of the Principie of Impulse and

Momentum to the Three-Dimensional Motion

of a Rigid Body 1179

* 18.4 Kinetic Energy of a Rigid Body

in Three Dimensions 1180

* 18.5 Motion of a Rigid Body in Three Dimensions 1193

* 18.6 Euler's Equations of Motion. Extensión of

DAIembert's Principie to the Motion of a

Rigid Body in Three Dimensions 1194

* 18.7 Motion of a Rigid Body About a Fixed Point 1195

* 18.8 Rotat¡on of a Rigid Body About a Fixed Axis 1196

* 18.9 Motion of a Gyroscope. Eulerian Angles 1212

* 18.10 Steady Precession of a Gyroscope 1214

* 18.11 Motion of an Axisymmetrical Body

Under No Forcé 1215

Review and Summary for Chapter 18 1229

Review Problems 1234

Computer Problems 1238

xvii

Contents

xviii

19 Mechanical Vibrations 1240

19.1 Introduction 1242

Vibrations Without Damping 1242

19.2 Free Vibrations of Partióles. Simple

Harmonio Motion 1242

19.3 Simple Pendulum (Approximate Solution) 1246

* 19.4 Simple Pendulum (Exact Solution) 1247

19.5 Free Vibrations of Rigid Bodies 1256

19.6 Application of the Principie of Conservaron

of Energy 1268

19.7 Forced Vibrations 1278

Damped Vibrations 1290

* 19.8 Damped Free Vibrations 1290

* 19.9 Damped Forced Vibrations 1293

* 19.10 Electrical Analogues 1294

Review and Summary for Chapter 19 1305

Review Problems 1310

Computer Problems 1314

Appendix Al

Photo Credits C1

Index II

Answers to Problems AN1

Preface

L.

OBJECTIVES

The main objective of a first course in mechanics should be to

develop in the engineering student the ability to analyze any problem

in a simple and logical manner and to apply to its solution a few,

well-understood, basic principies. It is hoped that this text, as well

as the preceding volume, Vector Mechanics for Engineers: Statics,

will help the instructor achieve this goal.f

GENERAL APPROACH

Vector algebra was introduced at the beginning of the first volume and

is used in the presentation of the basic principies of statics, as well as

in the solution of many problems, particularly three-dimensional prob-

lems. Similarly, the concept of vector differentiation will be introduced

early in this volume, and vector analysis will be used throughout the

presentation of dynamics. This approach leads to more concise deriva-

tions of the fundamental principies of mechanics. It also makes it pos-

sible to analyze many problems in kinematics and kinetics which could

not be solved by scalar methods. The emphasis in this text, however,

remains on the correct understanding of the principies of mechanics

and on their application to the solution of engineering problems, and

vector analysis is presented chiefly as a convenient tool. j

Practico Applications Are Introduced Early. One of the char-

acteristics of the approach used in this book is that mechanics of

partióles is clearly separated from the mechanics of rigid hodies. This

approach makes it possible to consider simple practical applications

at an early stage and to postpone the introduction of the more dif-

ficult concepts. For example:

• In Statics , the statics of partióles is treated first, and the principie

of equilibrium of a partióle was immediately applied to practical

situations involving only concurrent forces. The statics of rigid

bodies is considered later, at which time the vector and scalar

products of two vectors were introduced and used to define the

moment of a forcé about a point and about an axis.

• In Dynamics , the same división is observed. The basic con¬

cepts of forcé, mass, and acceleration, of work and energy, and

of impulse and momentum are introduced and first applied to

problems involving only partióles. Thus, students can familiarize

FORCES IN A PLAÑE

2.2 FORCE ON A PARTICLE. RESULTANT

OF TWO FORCES

A forcé represents the action of one body on another and is generally

characterized by its point of application, its magnitude, and its direc-

tion. Forces acting on a given particle, however, have the same point

of application. Each forcé considered in this chapter will thus be

completely defined by its magnitude and direction.

The magnitude of a forcé is characterized by a certain num-

ber of units. As indicated in Chap. 1, the SI units used by engi¬

neers to measure the magnitude of a forcé are the newton (N) and

its múltiple the kilonewton (kN), equal to 1000 N, while the U.S.

customary units used for the same purpose are the pound (Ib) and

its múltiple the kilopound (kip), equal to 1000 Ib. The direction

of a forcé is defined by the Une of action and the sense of the

forcé. The line of action is the infinite straight line along which

the forcé acts; it is characterized by the angle it forms with some

fixed axis (Fig. 2.1). The forcé itself is represented by a segment of

Fig. 2.1 (a) (b)

fBoth texts also are available in a single volume, Vector Mechanics for Engineers: Statics

and Dynamics, tenth edition.

fin a parallel text, Mechanics for Engineers: Dynamics, fifth edition, the use of vector

algebra is limited to the addition and subtraction of vectors, and vector differentiation is

omitted.

XIX

XX

Preface

17.1 INTRODUCCION

In this chapter the method of work and energy and the method of

impulse and momentum will be used to analyze the plañe motion of

rigid bodies and of systems of rigid bodies.

The method of work and energy will be considered first. In

Secs. 17.2 through 17.5, the work of a forcé and of a couple will be

defined, and an expression for the kinetic energy of a rigid body in

plañe motion will be obtained. The principie of work and energy will

then be used to solve problems involving displacements and veloci-

ties. In Sec. 17.6, the principie of conservation of energy will be

applied to the solution of a variety of engineering problems.

In the second part of the chapter, the principie of impulse and

momentum will be applied to the solution of problems involving veloc-

ities and time (Secs. 17.8 and 17.9) and the concept of conservation

of angular momentum will be introduced and discussed (Sec. 17.10).

In the last part of the chapter (Secs. 17.11 and 17.12), problems

involving the eccentric impact of rigid bodies will be considered. As

was done in Chap. 13, where we analyzed the impact of particles,

the coefficient of restitution between the colliding bodies will be

used together with the principie of impulse and momentum in the

solution of impact problems. It will also be shown that the method

used is applicable not only when the colliding bodies move freely

after the impact but also when the bodies are partially constrained

in their motion.

17.2 PRINCIPLE OF WORK AND ENERGY

FOR A RIGID BODY

The principie of work and energy will now be used to analyze the

plañe motion of rigid bodies. As was pointed out in Chap. 13, the

method of work and energy is particularly well adapted to the solu¬

tion of problems involving velocities and displacements. Its main

advantage resides in the fact that the work of forces and the kinetic

energy of particles are scalar quantities.

In order to apply the principie of work and energy to the analy-

sis of the motion of a rigid body, it will again be assumed that the

rigid body is made of a large number n of particles of mass Am¡.

Recalling Eq. (14.30) of Sec. 14.8, we write

Ti + Vu» = ( 17 . 1 )

where T h 1\ = initial and final valúes of total kinetic energy of particles

forming the rigid body

U\_ > 2 = work of all forces acting on various particles of the body

The total kinetic energy

T = l'Z Am ¡ v f (17.2)

2 ¿=i

is obtained by adding positive scalar quantities and is itself a positive

scalar quantity. You will see later how T can be determined for vari¬

ous types of motion of a rigid body.

themselves with the three basic methods used in dynamics and

learn their respective advantages before facing the difficulties

associated with the motion of rigid bodies.

New Concepts Are Introduced in Simple Terms. Since this

text is designed for the first course in dynamics, new concepts are

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, a defi-

nite maturity of approach has been achieved. For example, the con¬

cept of potential energy is discussed in the general case of a

conservative forcé. Also, the study of the plañe motion of rigid bodies

is designed to lead naturally to the study of their general motion in

space. This is true in kinematics as well as in kinetics, where the

principie of equivalence of external and effective forces is applied

directly to the analysis of plañe motion, thus facilitating the transition

to the study of three-dimensional motion.

Fundamental Principies Are Placed in the Context of Simple

Applications. The fact that mechanics is essentially a deductive

Science based on a few fundamental principies is stressed. Derivations

have been presented in their logical sequence and with all the rigor

warranted at this level. However, the learning process being largely

inductive, simple applications are considered first. For example:

• The kinematics of particles (Chap. 11) precedes the kinematics

of rigid bodies (Chap. 15).

• The fundamental principies of the kinetics of rigid bodies are first

applied to the solution of two-dimensional problems (Chaps. 16

and 17), which can be more easily visualized by the student, while

three-dimensional problems are postponed until Chap. 18.

The Presentation of the Principies of Kinetics Is Unified. The

tenth edition of Vector Mechanics for Engineers retains the unified

presentation of the principies of kinetics which characterized the previ-

ous nine editions. The concepts of linear and angular momentum are

introduced in Chap. 12 so that Newtons second law of motion can be

presented not only in its conventional form F = ma, but also as a law

relating, respectively, the sum of the forces acting on a partióle and the

sum of their moments to the rates of change of the linear and angular

momentum of the partióle. This makes possible an earlier introduction

of the principie of conservation of angular momentum and a more

meaningful discussion of the motion of a partióle under a central forcé

(Sec. 12.9). More importantly, this approach can be readily extended

to the study of the motion of a system of particles (Chap. 14) and leads

to a more concise and unified treatment of the kinetics of rigid bodies

in two and three dimensions (Chaps. 16 through 18).

Free-Body Diagrams Are Used Both to Solve Equilibrium Prob¬

lems and to Express the Equivalence of Forcé Systems.

Free-body diagrams were introduced early in statics, and their impor-

tance was emphasized throughout. They were used not only to solve

equilibrium problems but also to express the equivalence of two

Preface

XXI

Systems of forces or, more generally, of two systems of vectors. The

advantage of this approach becomes apparent in the study of the

dynamics of rigid bodies, where it is used to solve three-dimensional

as well as two-dimensional problems. By placing the emphasis on

“free-body-diagram equations” rather than on the standard algé¬

brale equations of motion, a more intuitive and more complete

understanding of the fundamental principies of dynamics can be

achieved. This approach, which was first introduced in 1962 in the

first edition of Vector Mechamos for Engineers, has now gained

wide acceptance among mechanics teachers in this country. It is,

therefore, used in preference to the method of dynamic equilib-

rium and to the equations of motion in the solution of all sample

problems in this book.

A Careful Balance between SI and U.S. Customary Units Is

Consistently Maintained. Because of the current trend in the

American government and industry to adopt the international system

of units (SI metric units), the SI units most frequently used in

mechanics are introduced in Chap. 1 and are used throughout the

text. Approximately half of the sample problems and 60 percent of

the homework problems are stated in these units, while the remain-

der are in U.S. customary units. The authors believe that this

approach will best serve the need of the students, who, as engineers,

will have to be conversant with both systems of units.

It also should be recognized that using both SI and U.S. cus¬

tomary units entails more than the use of conversión factors. Since

the SI system of units is an absolute system based on the units of

time, length, and mass, whereas the U.S. customary system is a gravi-

tational system based on the units of time, length, and forcé, differ-

ent approaches are required for the solution of many problems. For

example, when SI units are used, a body is generally specified by its

mass expressed in kilograms; in most problems of statics it will be

necessary to determine the weight of the body in newtons, and an

additional calculation will be required for this purpose. On the other

hand, when U.S. customary units are used, a body is specified by its

weight in pounds and, in dynamics problems, an additional calcula¬

tion will be required to determine its mass in slugs (or Ib • s 2 /ft). The

authors, therefore, believe that problem assignments should inelude

both systems of units.

The Instructor s and Solutions Manual provides six different

lists of assignments so that an equal number of problems stated in

SI units and in U.S. customary units can be selected. If so desired,

two complete lists of assignments can also be selected with up to

75 percent of the problems stated in SI units.

1.3 SYSTEMS OF UNITS

With the four fundamental concepts introduced in the preceding sec-

tion are associated the so-called kinetic units, i.e., the units of length,

time, mass, and forcé. These units cannot be chosen independently if

Eq. (1.1) is to be satisfied. Three of the units may be defined arbi-

trarily; they are then referred to as hasic units. The fourth unit, how-

ever, must be chosen in accordance with Eq. (1.1) and is referred to as

a derived unit. Kinetic units selected in this way are said to form a

consistent system of units.

International System of Units (SI Unitst). In this system, which

will be in universal use after the United States has completed its con¬

versión to SI units, the base units are the units of length, mass, and

time, and they are called, respectively, the meter (m), the kilogram

(kg), and the second (s). All three are arbitrarily defined. The second,

fSI stands for Systéme International d’Unités (French).

national Bureau ofWeights and Measures at Sévres, near Paris, France.

The unit of forcé is a derived unit. It is called the newton (N) and is

defined as the forcé which gives an acceleration of 1 m/s 2 to a mass of

1 kg (Fig. 1.2). From Eq. (1.1) we write

1 N = (1 kg)(l m/s 2 ) = 1 kg • m/s 2 (1.5)

The SI units are said to form an absolute system of units. This means

that the three base units chosen are independent of the location where

measurements are made. The meter, the kilogram, and the second

may be used anywhere on the earth; they may even be used on another

planet. They will always have the same significance,

The weight of a body, or the/orce of gravity exerted on that body,

should, like any other forcé, be expressed in newtons. From Eq. (1.4)

it follows that the weight of a body of mass 1 kg (Fig. 1.3) is

W = mg

= (1 kg)(9.81 m/s 2 )

= 9.81 N

Múltiples and submultiples of the fundamental SI units may be

obtained through the use of the prefixes defined in Table 1.1. The

múltiples and submultiples of the units of length, mass, and forcé most

frequently used in engineering are, respectively, the kilometer (km)

and the millimeter (mm); the megagram] (Mg) and the g ram (g); and

the kilonewton (kN). According to Table 1.1, we have

1 km = 1000 m 1 mm = 0.001 m

1 Mg = 1000 kg 1 g = 0.001 kg

1 kN = 1000 N

The conversión of these units into meters, kilograms, and newtons,

respectively, can be effected by simply moving the decimal point

three places to the right or to the left. For example, to convert

3.82 km into meters, one moves the decimal point three places to the

right:

3.82 km = 3820 ir

Similarly, 47.2 n

point three place

o meters by moving the decimal

Optional Sections Offer Advanced or Specialty Topics. A

large number of optional sections have been included. These sections

are indicated by asterisks and thus are easily distinguished from those

which form the core of the basic dynamics course. They can be omit-

ted without prejudice to the understanding of the rest of the text.

The topics covered in the optional sections inelude graphical

methods for the solution of rectilinear-motion problems, the trajectory

XXII

Preface

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