-fTl3:

Conversion Factors

(Acceleration)

(Area)

(Density)

(Force)

(Length)

(Mass)

(Moment of force)

(Moment of inertia, area)

(Moment of inertia, mass)

(Momentum, linear)

(Momentum, angular)

(Power)

(Pressure, stress)

(Spring constant)

(Velocity)

(Volume)

(Work, Energy)

X 4

X 4

X

SI Units Used in MechanicsUnit

(I.H52 km ih)

W{ .1Pa

:-'lass

Al~o ~1)t'IIt'd

SI Unit Prefixes

I {jOO 000 000 (/00 1,'1"<1 TI 000 UUOO()()

100000U M100ll k

102 h10 100_1 10 d

0,010.001 10 ,\

0.000001 100.000 UOOOf) I 10

P

Selected Rules for Writing Metric Quantities

j';.wmple: \\

{Rxamp/(':

lJ.;xu/1/jJ/e:

:1 :-Jumher woupm r

(Kxampll': :321.04tiU';xample:

Virginia Polytechnic Instituteand State University

University of Rhode Island

JOHN & SONS, INC.

ACQUISITIONS WITORMARKETING MANAGERDEVELOPMENT EIlITORSENIOR PRODUCTION EIJITORSENIOR DESIGNERTEXT DESIG1-IERCOVER DESIGNERILLUSTRATION EDITORASSOCIATE PHOTO EIJITORmONT COVER PHOTOBACK COVER PHOTO

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of the Guggenheim Mu:;eum, Bilbao,Spain. Reproduced with permission.

Mechanics in Action:

ISBN 0--471-26607-8

PHOTO CREI>ITS

Dr. James Lathrop Meriam, internationally known author of engi-neering mechanics textbooks and distinguished professor of engineering,

merous and significant contributions to the engineering profession, Dr.Meriam is regarded as one of the premier engineering educators of thetwentieth century. Dr. Mcriam (known as Lath to his friends) received

experience came at Pratt and Whitney Aircraft and the General Electric

fornia-Berkeley for twenty-one years. During this period he served asProfessor of Engineering Mechanics, Assistant Dean of Graduate Stud-

iting professor at the University of California-Santa Barbara andretired for a second time in 1990.

Recognition of his superb teaching abilities followed him wherever

Educator Award from the Mechanics Division of the American Society

Dr. Meriam began his Engineering Mechanics textbook series in1950. The Statics and Dynamics texts reshaped undergraduate mechan-

versions and have been translated into many foreign lanWlages. Hisbooks have been characterized by clear and rigorous presentation of thetheory, instructive sample problems, and numerous and realistic home-work exercises. From the outset, a high standard of illustration has dis-tinguished the series.

riod of more than three years, a 23·foot wooden sailboat namedKai, which is Hawaiian for Song of the Sea. Over the next several years,he and his fortunate sailing companions spent many happy hours sailingoff the coast of Santa Barbara. Dr. Meriam also designed and built fourhomes, including a vacation home on the island of KauaL

In addition to his many professional accomplishments, Lath Medamwill be long remembered for his open friendliness, gentlemanly de-meanor, mature judgment and leadership, generosity, and absolute com·

v

PREFACE

find

engineering science

Engineering Mechanics

En-gineering Alechanics

Preface vii

viii Preface

Problems.selves. The solutions to typical statics problems are presented in detail. In addition,

to the main presentation.There are 963 homework exercises, of which approximately 50 percent are new

Introductory ProblemsRepresentative Problems. The first section consists of simple, uncomplicated prob-

lems are generally arranged in order of increasing difficulty. More difficult exercisesRepresentative Problems

••. Computer-Oriented Problems, marked with an asterisk, appear in a special sec-Review Problems

ductory areas in which U.S. units are mentioned for purposes of completeness and

principles and procedures inherent in the design and analysis of engineering struc-tures and mechanical systems.

Illustrations.toimportant to note that color is used consistently for the identification of certainquantities:

• red for forces and moments,· green for velocity and acceleration arrows,

· orange dashes for selected trajectories of moving points.

Subdued colors are used for those parts of an illustration which are not centralto the problem at hand. \Vhenever possible, mechanisms or objects which commonlyhave certain color will be portrayed in that color. All of the fundamental elementsof technical illustration which have been an essential part of this Engineering Me-chanics series of textbooks have been retained. The author wishes to restate theconviction that a high standard of illustration is critical to any written work in the

Features New to this Edition.editions, we have incorporated these improvements:

The theory portions were re\\Titten for clarity and readability, with a higherlevel of friendlinc8s and a more active voice.

Sections have been shortened and more subheads added to make informationeasier to find.

ORGANIZATION

Preface ix

x Preface

representative.

Soll/Ing Mechanics Problems with ....computational software in the solution of mechanics problems. Developed by Brian

Wiley Website (www.wlley.com/college/merlam).

Electronic figureselectronically for use in creating lectures.

Electronic transparencies for over 40 solved problems, similar to those in thetext, are available for use in lecture or in self-study by students.

On·line problem solving,problems in mechanics for students to solve, featuring step-by-step

Extension sample problems build on sample problems from the text and showhow computational tools can be used to investigate a variety of "what if"

Brian Harper at Ohio State University.

Island merits special acknowledgment for his excellent and careful review of theentire text. Professor Palm has inspected the structure of every sentence and, wherenecessary, has made modifications so that the presentation is clear, direct, concise,

text more easily readable, and reorganized the Chapter Review sections so that thestudent can efficiently survey \\-'hat has been presented. Professor Palm has workedunder a number of constraints and has done so in a friendly and timely manner.

Special recognition is again due Dr. A. L. Hale, formerly of Bell TelephoneLaboratories, for his continuing contribution in the form of invaluable suggestionsand accurate checking of the manuscript. Dr. Hale has rendered similar service forall previous versions of this entire series of mechanics books, dating back to the

figures. Dr. Hale carries out an independent solution to each new homework ex-ercise and provides the author with suggestions and needed corrections to the so-lutions which appear in the Instructor's Manual. Dr. Hale is well known for being

a great asset which aids every user of this textbook.

equilibrium problems in the area of biomechanics. These new problems serve to

Preface xi

University of KentuckyKettering University

Ohlahoma State UniversityUniversity of Alaine

University of Central FloridaOhio State University

California State Polytechnic University, PomonaColorado State Universi(vOakland University

Auburn UniversityUniversity of WyomingYoungstown State UniversityRochester Institute of TechnologyUniversity of WyomingOklahoma State University

CONTENTS

CHAPTER 1 INTRODUCTION TO STATICS 3

CHAPTER 2 FORCE SYSTEMS 23

CHAPTER 3 EQUILIBRIUM lD3

CHAPTER 4 STRUCTURES 165

CHAPTER 5 DISTRIBUTED FORCES 225

CHAPTER 6 FRICTION 327

CHAPTER 7 VIRTUAL WORK 385

APPENDICES

A AREA MOMENTS OF INERTIA

B MASS MOMENTS OF INERTIA

C SELECTED TOPICS OF MATHEMATICS

D USEFUL TABLES

INDEX

463

CONTENTS

Chaeter 1INTRODUCTION TO STATICS

MechanicsBasic ConceptsScalars and VectorsConventions for Equations and Diagrams 5

Newton's LawsUnits

Unit Cunversions 11Law of Gravitation

1/7 Accuracy, Limits, and Approximations

Problem Solving in Statics

Formulating Problems and Obt.aining Solutions 16

Numerical Values Symbols 17Solution Methods 17

Chapter Review

15

2FORCE SYSTEMS

SECTION A. TWO-DIMENSIONAL FORCE SYSTEMS 27

SECTION B. THREE-DIMENSIONAL FORCE SYSTEMS 64

QUI LI B R I UM 101

SECTION A. EQUILIBRIUM IN TWO DIMENSIONS 104

3/2 System ISDlatiDn and the Free-BDdy Diagram

3/3 Equilibrium CDnditiDns 115

SECTION B. EQUILIBRIUM IN THREE DIMENSIONS 13B

3/4 Equilibrium CDnditiDns 13B

Chapter Review 156

Chapter

STR U CTU RES 165

4/1 IntrDductiDn 1654/2 Plane Trusses 167

4/3 MethDd Df JDints 168

4/4 MethDd Df SectiDns 179

4/5 Space Trusses 188

4/6 Frames and Machines 195

Chapter Review 215

Chapter

DISTRIBUTED FORCES

5/1 IntrDductiDn

SECTION A. CENTERS OF MASS AND CENTROIDS

225

225

227

Hydrostatic Pressure on Submerged

Hydrostatic Pressure on Flat Surfaces

FRICTION

Factors Affecting Friction

348

6/4 Wedges6/5 Screws

6/6 Journal Bearings6/7 Thrust Bearings; Disk Friction6/8 Flexible Belts6/9 Rolling ResistanceChapter Review

Chapter

7/1 Introduction7/2 Work

7/3 Equilibrium

7/4 Potential Energy and Stability

Chapter Review

Appendices

A/I IntroductionA/2 Definitions

A/3 Composite Areas

359360

369376

405

Contents xvii

xviii Contents

ENGINEERING MECHANICS

STATICSSI VERSION

Structures which support large forces must be designed with the principles of mechanicsforemost in mind. In this view of Sydney Harbor, one can see several examples of suchstructures.

MechanicsBasic ConceptsScalars and Vectors

1/4 Newton's LawsUnitsLaw of GravitationAccuracy, Limits, and ApproximationsProblem Solving in Statics

Chapter Review

MECHANICS

Chapter

forces on objects. No other subject plays a greater role in engineeringanalysis than mechanics. Although the principles of mechanics are few,they have wide application in engineering. The principles of mechanicsare central to research and development in the fields of vibrations, sta-

spacecraft design, automatic control, engine performance, fluid flow,electrical machines and apparatus, and molecular, atomic, and sub-atomic behavior. A thorough understanding of this subject is an essential

of this subject is synonymous with the very beginnings of engineering.The earliest recorded writings in mechanics are those of Archimedes

mulated most of the principles of statics. The first investigation of a

with falling stones. The accurate formulation of the laws of motion, as

also conceived the idea of the infinitesimal in mathematical analysis.Substantial contributions to the development of mechanics were also

others.

principles of mechanics and their application. The principles of mechan-ics as a science are rigorously expressed by mathematics, and thus math-ematics plays an important role in the application of these principles to

which concerns the equilibrium of bodies under the action of forces, anddynamics, which concerns the motion of bodies. Engineering Mechanics

Vol. Statics Vol. Dynamics.

BASIC CONCEPTS

is the geometric region occupied by bodies whose positionsare described by linear and angular measurements relative to a coordi-nate system. For problems, three independent coor-dinates are needed. For problems, only two coordinatesare required.

is the measure of the succession of events and is a basic quan-tity in dynamics. Time is not directly involved in the analysis of statics

Mass is a measure of the inertia of a body, which is its resistance

force between it and other bodies. This force appears in many applica-tions in statics.

is the action of one body on another. A force tends to movea body in the direction of its action. The action of a force is characterizedby its magnitude, by the direction of its action, and by its point of ap-plication. Thus force is a vector quantity, and its pl"Opertiesare discussed

sense, a particle is a body whose dimensions are considered to be nearzero so that we may analyze it as a mass concentrated at a point. We

a body as a particle when its dimensions are irrelevant to the descriptionof its position or the action of forces applied to it.

internal deformations in the structural members of the boom. For thepurpose, then, of determining the external forces which act on the boom,

lation of external forces which act on rigid bodies in equilibrium. Deter-

Scalar quantities

mass. Vector quantities,

free

Conventions for Equations and Diagrams

V.

principle oftmn ..•mi!';sibility.

Article 1/3 Scalars and Vectors 5

6 Chapter 1 Introduction to Statics

Figure

Working with Vedors

V, V1/2b. vector sum,

Figure 1/3

vector scalar

2•

+V2 + VI'

vedaI' subtraction.

components1/4a

rectangular compo-nents. 1/4b x· y-components,

Article 1/3 Scalars and Vectors 7

2/ V

V'01VI

Ib)

Figure 1/4

x'

V

'"

I

I

x-, y-,

x-,

Figure 1/5

v mVy

l'2 + m2 + n2

Law I. uni-{orm velocity

Law II.

Law III.collinear

vector

isolate

on

Principia (1687)

In mechanics we use four fundamental quantities called dimensions.are length, ma.ss, force, and time. The units used to measure these

quantities cannot all be chosen independently because they must be coo-

in science and technology will be used in this text. The four fundamental

Article 1/5 Unit.

SI UNITS

Ul\'IT SYMBOLQUA:\TITY

LengthTimeForce

51 Units

DIMENSIONALSYl\IBOL

T

aseumts second

kg

U.S. CUSTO:\IARY UNITS

UNIT SYMBOL

slug

second seeunt pound lb

throughout the world, and is a modern version of the metric system. By

gweight

IV(NI (kg) X

U.S. Customary Units

g

W(lb)g 2)

kilopoundton,

absolute

gravitational

2•

exclusively mass-never

non· English-speaking

Primary Standards

Mass.

Time.

for

tfIn

11

m/s"

Unit Conversions

SI In

(~.~O 11>11;j2.2Ihl'11,1:1.2 Nl

I kg~IASS (2.20 Ibm)

10.454 kgl

LENGTH(0.:30;)

Figure 1/6

12 Chapter Introduction to Statics

between selected quantities in the two systems appear inside the backcover for convenient reference. Although these charts are useful for ob-taining a feel for the relative size of SI and U.S. units, in time enbTineers

verting from U.S. units. ]n statics we are primarily concerned with theunits of length and force, with mass needed only when we compute grav-itational force, as explained previously.

systems of units, to aid in visualizing their relative magnitudes.

lAW OF GRAVITATION

In statics as well as dynamics we often need to compute the weightof a body, which is the gravitational force acting on it. This computation

law of gravitation.ton. The law of gravitation is expressed by the equation

where F = the mutual force of attraction between two particles

a universal constant known as the constant of gravitation= the masses of the t\"Y'0particles

the distance between the centers of the particles

The mutual forces F obey the law of action and reaction, since they areequal and opposite and are directed along the line joining the centers of

Gravitational Attradlon of the Earth

Gravitational forces exist between every pair of bodies. On the sur-face of the earth the only gravitational force of appreciable magnitudeis the force due to the attraction of the carth. For example, each of twoiron spheres 100 mm in diameter is attracted to the earth with a grav-

force of mutual attraction between the spheres if they are just touching

of the earth is the only gravitational force we need to consider for mostengineering applications on the earth's surface.

Figure 1/7

Article 1/7 Accuracy. Limits. and Approximations 13

m

\V.

Wg

m

1/7 ACCURACY, LIMITS, AND ApPROXIMATIONS

2.

shown

Differentials

The order of differential quantities frequently causes misunder·standing in the derivation of equations. Higher-order differentials may

mathematical limit is approached. For example, the element of volume:, V h rto be a circular slice a distance x from the vertex and of thickness .lx.The expression for the volume of the element is

V dV:'x dx, (:.x)2 (:.x)3

which gives an exact expression when integrated.

8 smO lxO=8

CDS

Small-Angle Approximations

1/8

and sin are very nearly the same. Also cos () is close to unity.Furthermore, sin () and tan () have almost the same values. Thus, forsmall angles we may write

0 '"

provided that the angles are expressed in radians. These approximationsmay be obtained by retaining only the first terms in the series expan·sions for these three functions. As an example of these approximations,for an angle of 1

a more accurate approximation is desired, the first two terms maybe retained, and they are

(J", + (J",

where the angles must be expressed in radians. (To convert degrees to1T/180'.)

Article 1/8 Problem Solving in Statics 15

de

Making Appropriate Assumptions

model,

Using Graphics

16 Chapter Introduction to Statics

with its physical interpretation, especially when we must visualizethree-dimensional problems.

than with a direct mathematical solution. Graphical solutions areboth a practical way to obtain results, and an aid in our thoughtprocesses. Because graphics represents the physical situation and itsmathematical expression simultaneously, graphics helps us makethe transition between the two.

3. Charts or graphs are valuable aids for representing results in a formwhich is easy to understand.

Formulating Problems and Obtaining Solutions

In statics, as in all engineering problems, we need to use a precise

following sequence of steps.

(b)

State your assumptions and approximations.

(b)

(d) Ensure that your calculations are consistent with the accuracy

(e)

calculations.

Ensure that your answers are reasonable in terms of magni-tudes, directions, common sense, etc.Draw conclusions.

which seem complicated at first often become clear when you approach

The Free-Body Diagram

The subject of statics is based on surprisingly few fundamental con-cepts and involves mainly the application of these basic relations to avariety of situations. In this application the method of analysis is all-

17

Numerical Values versus Symbols

Solution Methods

18 Chapter 1 Introduction to Statics

method of solution is an important aspect of the experience to be gainedfrom the problem work. There are a number in Vol. 1 Staticswhich are designated as Computer-Oriented Problems. These problems

advantage.

CHAPTER REVIEW

This chapter has introduced the concepts, definitions, and units usedin statics, and has given an overview of the procedure used to formulateand solve problems in statics. Now that you have finished this chapter,

1. Express vectors in terms of unit vectors and perpendicular compo-nents, and perform vector addition and subtraction.

2. State Newton's laws of motion.

accuracy.

approximations.6. Describe the methodology used to formulate and solve statics

Determine the weight in newtons of a car whose mass is 1400 kg'. Convertthe mass of the car to slugs and then determine its weight in pounds.

Chapter Review 19

m =

m ~ 1400 kg[ 1 Ibm k ] 3090 Ibm0.45359 g.

The weight in pounds associated with the mass of 3090 lbm is 3090 lb, as cal-culated above. We recall that Ilbm is the amount of mass which under standardconditions has a weight of 1 Ib of force ..We rarely refer to the U.S. mass unitIbm in this textbook series, but rather use the slug for mass. The sole use ofslug, rather than the unnecessary use of two units for mass, will prove to bepowerful and simple-especially in dynamics,

:-.J'otuthat me cdkul.ll.t'd re"ult (9,;.~) slng:-~, ml18t SlIl'l' tlUlt when a calculutNl number is~llb:>equ('nt calnilarinn:>. rctnineo the it~ (9.).9:'Wg3 .. Ilt'l'ded.

Thi~ it in a ] initial ano recalling We must not punch9.i,9 into Ollr l'.:dcu!.uor nnd to mllltipl:~ :l~ .. practlc£' ,\111 result in lo~:-. nUffierlcnl H{;('t!lT\ey. Someindi, idual,; Iikt' to pla<:e a :::;mallllldieatltll1 of tht' :-.toragl' thl' right maq,rill the \~ork direed.vbt.'s\do number

As another route to the last result, we can convert from kg to Ibm. Again usingthe table inside the front cover, we have

From the table of conversion factors inside the front cover of the textbook, wesee that 1 slug is equal to 14.594 kg. Thus, the mass of the car in slugs is

Solution. From relationship 1/3. we have

mg 1400(9.81) 13730 N

1400 k 95.914.594 kg

Finally, its weight in pounds is

W mg (95.9)(32.2) 3090 Ib

Ans.

Ans.

Our rc~ult of1:) 7::4 N. Cf>ing the or('ut1t-fi,l,>1.U'('display llACd in h'xt~hook. we l't)und thl' written rl',mlt totour t;iglliticant fi!.l'tJ.n'~, ur ]3730 N.Had I.hp numh!'r bCl,'l1n with an,v

other th.m

good prdctict' with unit {;()IlYI'rsion

,-a uel4,fj!) { kg

the and tlw de-(·quivuh'nt. J\lnkt·

th.lt cfl\lcpHdllon of the unit" leave"tilt unil!'; dc"ired; here the unitt; ofkg cHncf'L leaving df'sired unit..;;;

Use Newton's law of universal gravitation to calculate the weight of a 70~kgperson standing on the surface of the earth. Then repeat the calculation by using

and compare your two results. Use Table D/2 as needed.

Solution. The two results are

70 kg

(6.673 10-")(5.976· 10'4)(70)[6371 . 10

688 N m,

IV mg 70(9.81) 687 N

The discrepancy is due to the fact that Newton's universal gravitational law doesnot take into account the rotation of the earth, On the other hand, the valueg 9.81 m/g:! used in the second equation does account for the earth's rotation.Note that had we used the more accurate valueg 9.80665 m/52 (which likewiseaccounts for the earth's rotation) in the second equation, the discrepancy wouldhave been larger (686 N would have been the result).

cli5tdncl' het\'it'I.mmass conteI''' (If two in·volved I::; ~he rndius of the earth.

2

I + 2

(b)

(c)

2

yI

= 4

V =

Solution

2 ~ 2 + 2 - CDS

S

(b)

=

V105 + 300)

Sri + j 5.43i + 1.328j

+Ans.

13.76°

+

@ Then

VI - V2 4(i CDS 45° + 45°) - 3(i CDS 30° - j 30°)

0.230i + 4.33j

b + (-

win

prinCiple;;,

by +

the direction of V.AnN. Ox ~ 112.6°, -0.385i + 0.923j

fix.

solutions.

V = 14 uunits = 18 units

60°

mine the mat-,'l1itude of the vector difference V' V2- VI and the angle which V' makes with the posi-

b'Tllphical

21.4 176.5°

force is byCalculate the angles made F with the andz-axes.

What is the weight in both newtons and pounds of a75-kg beam'?

736 N, 165.4 Ib

From the gravitational law calculate the weight W(gravitational force with respect to the earth) of an 80-kg man in a spacl->craft traveling in a circular orbit 250km above the earth's surface. l<;xpress W in both new-tons and pounds.

Problems 21

Determine the weight in newtons of a woman whoseweight in pounds is Also, find her ma.ss in slugsand in kiloJ..,'Tams. Determine your own weight innewtons.

An". W 578 N4.04 slul-,TS, 58.9 kg

Suppose that two nondimensional quantities are brivenas A 8.69 and B Using the rules for sib'l1if-icant figures as stated in this chapter, determine thefour quantities IA IA - and

1/9 Compute the mab'l1itude Fofthe force which the earthexerts on the moon. Perform the calculation first innewtons and then convert your result to pounds. Referto Table D/2 for necessary physical quantities.

Ans. F ~ 1.984110 N, F ~ 4.46110"1) Ib

What is the percent error in replacing the sine of 20°by the value of the angle in radians? Repeat for thetangent of 20°, and explain the qualitative ditTerencein the two error percentages.

The properties of force systems must be thoroughly understood by the engineers who designsystems such as this construction crane. Try to visualize the forces present in the various partsof the crane.