Vector mechanics for engineers

4, to many practical applications involving the equilibrium of rigid bodies tBoth texts are also available in a single volume, Vector Mechanics for Engineers: Statics and Dynamics, sixth

THIRD SI J\l\ETRIC EDITION.

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Vector Mechanics for Engineers: Statics

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Beer, Ferdinand P.,

(date)-Vector mechanics for engineers: statics

About the Authors

"How did you happen to write your books together, with one of you at

Lehigh and the other at UConn, and how do you manage to keep

collabo-rating on their successive revisions?" These are the two questions most

often asked of our two authors.

The answer to the first question is simple Russ Johnston's first

teaching 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 Born in France and educated in France and Switzerland

(he holds an M.S degree from the Sorbonne and an Sc.D degree in the

field of theoretical mechanics from the University of Geneva), Ferd had

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

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

College in The Williams-MIT joint arts and engineering program Born

in Philadelphia, Russ had obtained a B.S degree in civil engineering

from the University of Delaware and an Sc.D degree in the field of

structural engineering from MIT.

Ferd was delighted to discover that the young man who had been

hired chiefly to teach graduate structural engineering courses was not

only willing but eager to help him reorganize the mechanics courses.

Both believed that these courses should be taught from a few basic

prin-ciples 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.

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

Connecti-cut In the meantime, both Ferd and Russ had assumed administrative

responsibilities in their departments, and both were involved in research,

consulting, and supervising graduate students-Ferd in the area of

sto-chastic processes and random vibrations, and Russ in the area of elastic

v

vi About the Authors stability and structural analysis and design Howe\"er 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 re\ising their

texts and began writing the manuscript of the first edition of Mechanics

of Materials.

This brings us to the second question: How did the authors manage

to work together so effectively after Russ Johnston had left Lehigh? Part

of the answer is provided by their phone bills and the money they have spent on postage As the publication date of a new edition approaches, they call each other daily and rush to the post office with express-mail packages There are also visits between the two families At one time there were even joint camping trips, with both families pitching their tents next to each other Now, with the advent of the fax machine, they

do not need to meet so frequently.

Their collaboration has spanned the years of the revolution in

com-puting The first editions of Mechanics for Engineers and of Vector

Me-chanics for Engineers included notes on the proper use of the slide rule.

To guarantee the accuracy of the answers given in the back of the book, the authors themselves used oversize 20-inch slide rules, then mechani- cal desk calculators complemented by tables of trigonometric functions, and later four-function electronic calculators With the advent of the pocket multifunction calculators, all these were relegated to their respec- tive attics, and the notes in the text on the use of the slide rule were replaced by notes on the use of calculators Now problems requiring the use of a computer are included in each chapter of their texts, and Ferd and Russ program on their own computers the solutions of most of the problems they create.

Ferd and Russ's contributions to engineering education have earned them a number of honors and awards They were presented with the Western Electric Fund Award for excellence in the instruction of engi- neering students by their respective regional sections of the American Society for Engineering Education, and they both received the Distin- guished Educator Award from the Mechanics Division of the same soci- ety In 1991 Russ received the Outstanding Civil Engineer Award from the Connecticut Section of the American Society of Civil Engineers, and

in 1995 Ferd was awarded an honorary Doctor of Engineering degree by Lehigh University.

A new collaborator, Elliot Eisenberg, Professor of Engineering at the Pennsylvania State University, has joined the Beer and Johnston team for this new edition Elliot holds a B.S degree in engineering and an M.E degree, both from Cornell University He has focused his scholarly activi- ties on professional service and teaching, and he was recognized for this work in 1992 when the American Society of Mechanical Engineers awarded him the Ben C Sparks Medal for his contributions to mechani- cal engineering and mechanical engineering technology education and for service to that society and to the American Society for Engineering Education.

And finally, there are the contributions of Theodore Wildi to the integrated conversion of this Third SI Metric Edition He is Chair of the CSA Technical Committee on the International System of Units and

author of Metric Units and Conversion Charts, a widely used handbook

for professional engineers.

1.2 Fundamental Concepts and Principles 2

Forces in a Plane 12

2.2 Force on a Particle Resultant of Two Forces 12

2.3 Vectors 13

2.4 Addition of Vectors 14

2.5 Resultant of Several Concurrent Forces 16

2.6 Resolution of a Force into Components 17

2.7 Rectangular Components of a Force Unit Vectors 23

2.8 Addition of Forces by Summing x andyComponents 26

2.9 Equilibrium of a Particle 31

2.10 Newton's First Law of Motion 32

2.11 Problems Involving the Equilibrium of a Particle.

Free-Body Diagrams 32

Forces in Space 41

2.12 Rectangular Components of a Force in Space 41

2.13 Force Defined by Its Magnitude and Two Points

on Its Line of Action 44

2.14 Addition of Concurrent Forces in Space 45

vii

viii Contents 2.15 Equilibrium of a Particle in Space 53

Review and Summary for Chapter 2 60 Review Problems 63

3

RIGID BODIES: EQUIVALENT SYSTEMS OF FORCES

67 3.1 Introduction 68

3.2 External and Internal Forces 68 3.3 Principle of Transmissibility Equivalent Forces 69 3.4 Vector Product of Two Vectors 71

3.5 Vector Products Expressed in Terms of Rectangular Components 73

3.6 Moment of a Force about a Point 75 3.7 Varignon's Theorem 77

3.8 Rectangular Components of the Moment of a Force 77 3.9 Scalar Product of Two Vectors 87

3.10 Mixed Triple Product of Three Vectors 89 3.11 Moment of a Force about a Given Axis 91 3.12 Moment of a Couple 101

3.13 Equivalent Couples 102 3.14 Addition of Couples 104 3.15 Couples Can Be Represented by Vectors 104 3.16 Resolution of a Given Force Into a Force at 0 and a Couple 105

3.17 Reduction of a System of Forces to One Force and One Couple 116

3.18 Equivalent Systems of Forces 118 3.19 Equipollent Systems of Vectors 118 3.20 Further Reduction of a System of Forces 119

*3.21 Reduction of a System of Forces to a Wrench 121 Review and Summary for Chapter 3 140

Review Problems 145

4

EQUILIBRIUM OF RIGID BODIES

149 4.1 Introduction 150

4.2 Free-Body Diagram 151 Equilibrium in Two Dimensions 152 4.3 Reactions at Supports and Connections for a Two-Dimensional Structure 152 4.4 Equilibrium of a Rigid Body in Two Dimensions 154 4.5 Statically Indeterminate Reactions Partial Constraints 156 4.6 Equilibrium of a Two-Force Body 173

4.7 Equilibrium of a Three-Force Body 174 Equilibrium in Three Dimensions 181 4.8 Equilibrium of a Rigid Body in Three Dimensions 185 4.9 Reactions at Supports and Connections

for a Three-Dimensional Structure 181 Review and Summary for Chapter 4 198 Review Problems 200

DISTRIBUTED FORCES: CENTROIDS AND CENTERS

OF GRAVITY

2045.1 Introduction 206

Areas and Lines 206

5.2 Center of Gravity of a Two-Dimensional Body 206

5.3 Centroids of Areas and Lines 208

5.4 First Moments of Areas and Lines 209

5.5 Composite Plates and Wires 212

5.6 Determination of Centroids by Integration 223

5.7 Theorems of Pappus-Guldinus 225

*5.8 Distributed Loads on Beams 236

*5.9 Forces on Submerged Surfaces 237

Volumes 247

5.10 Center of Gravity of a Three-Dimensional Body.

Centroid of a Volume 247

5.11 Composite Bodies 250

5.12 Determination of Centroids of Volumes by Integration 250

Review and Summary for Chapter 5 262

Review Problems 266

6

ANALYSIS OF STRUCTURES

2706.1 Introduction 271

Trusses 272

6.2 Definition of a Truss 272

6.3 Simple Trusses 274

6.4 Analysis of Trusses by the Method of Joints 275

*6.5 Joints under Special Loading Conditions 277

*6.6 Space Trusses 279

6.7 Analysis of Trusses by the Method of Sections 289

*6.8 Trusses Made of Several Simple Trusses 290

Frames and Machines 301

6.9 Structures Containing Multiforce Members 301

6.10 Analysis of a Frame 301

6.11 Frames Which Cease to Be Rigid When Detached

from Their Supports 302

X *7.4 Shear and Bending Moment in a Beam 346

*7.5 Shear and Bending-Moment Diagrams 348

*7.6 Relations among Load, Shear, and Bending Moment 356 Cables 367

*7.7 Cables with Concentrated Loads 367

*7.8 Cables with Distributed Loads 368

*7.9 Parabolic Cable 369

*7.10 Catenary 378 Review and Summary for Chapter 7 386 Review Problems 389

8

FRICTION

3928.1 Introduction 393

8.2 The Laws of Dry Friction Coefficients of Friction 323 8.3 Angles of Friction 396

8.4 Problems Involving Dry Friction 397 8.5 Wedges 413

8.6 Square-Threaded Screws 413

*8.7 Journal Bearings Axle Friction 422

*8.8 Thrust Bearings Disk Friction 424

*8.9 Wheel Friction Rolling Resistance 425

*8.10 Belt Friction 432 Review and Summary for Chapter 8 443 Review Problems 446

9

DISTRIBUTED FORCES: MOMENTS OF INERTIA

451 9.1 Introduction 452

Moments of Inertia of Areas 453 9.2 Second Moment, or Moment of Inertia, of an Area 453 9.3 Determination of the Moment of Inertia of an Area

by Integration 454 9.4 Polar Moment of Inertia 455 9.5 Radius of Gyration of an Area 456 9.6 Parallel-Axis Theorem 463

9.7 Moments of Inertia of Composite Areas 464

*9.8 Product of Inertia 476

*9.9 Principal Axes and Principal Moments of Inertia 477

*9.10 Mohr's Circle for Moments and Products of Inertia 485 Moments of Inertia of Masses 491

9.11 Moment of Inertia of a Mass 491 9.12 Parallel-Axis Theorem 493 9.13 Moments of Inertia of Thin Plates 494 9.14 Determination of the Moment of Inertia of a Three-Dimensional Body by Integration 495

9.15 Moments of Inertia of Composite Bodies 495

*9.16 Moment of Inertia of a Body with Respect to an Arbitrary Axis through O Mass Products of Inertia 510

*9.17 Ellipsoid of Inertia Principal Axes of Inertia 511

*9.18 Determination of the Principal Axes and Principal Moments of

Inertia of a Body of Arbitrary Shape 513

Review and Summary for Chapter 9 524

*10.3 Principle of Virtual Work 539

*10.4 Applications of the Principle of Virtual Work 540

*10.5 Real Machines Mechanical Efficiency 542

*10.6 Work of a Force during a Finite Displacement 556

577

A.1 U.S Customary Units 577

A.2 Conversion from One System of Units to Another 578

Index 583

Answers to Problems 589

Contents xi

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 apply to its solution a few, well-understood basic

principles It is hoped that this text, designed for the first course in statics

offered in the sophomore year, and the volume that follows, Vector

Me-chanics for Engineers: Dynamics, will help the instructor achieve this

goal t

Vector algebra is introduced early in the text and is used in the

presentation and the discussion of the fundamental principles of

me-chanics Vector methods are also used to solve many problems,

particu-larly three-dimensional problems where these techniques result in a

sim-pler and more concise solution The emphasis in this text, however,

remains on the correct understanding of the principles of mechanics and

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

algebra is presented chiefly as a convenient tool.!

One of the characteristics of the approach used in these volumes is

that the mechanics of particles has been clearly separated from the

me-chanics of rigid bodies This approach makes it possible to consider

sim-ple practical applications at an early stage and to postpone the

introduc-tion of more difficult concepts In this volume, for example, the statics of

particles is treated first (Chap 2); after the rules of addition and

subtrac-tion of vectors have been introduced, the principle of equilibrium of a

particle is immediately applied to practical situations involving only

con-current 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

and used to define the moment of a force about a point and about an axis.

The presentation of these new concepts is followed by a thorough and

rigorous discussion of equivalent systems of forces leading, in Chap 4, to

many practical applications involving the equilibrium of rigid bodies

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

Statics and Dynamics, sixth edition.

JIn a parallel text, Mechanics for Engineers: Statics, fourth edition, the use of vector

algebra is limited to the addition and subtraction of vectors.

xiii

xiv Preface

under general force systems. Inthe volume on dnlamics, the same sion is observed The basic concepts of force, m~s, and acceleration, of work and energy, and of impulse and momentum are introduced and first applied to problems involving only particles Thus students can familiar- ize themselves with the three basic methods used in dmamics and learn their respective advantages before facing the difficulti~s associated with the motion of rigid bodies.

divi-Since this text is designed for a first course in statics, 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 consid- ered, a definite maturity of approach is achieved For example, the con- cepts of partial constraints and of static indeterminacy are introduced early in the text and then are used throughout.

The fact that mechanics is essentially adeductive science based on a few fundamental principles is stressed Derivations are presented in their logical sequence and with all the rigor warranted at this level However, the learning process being largely inductive, simple applications are con- sidered first Thus, the statics of particles precedes the statics of rigid bodies, and problems involving internal forces are postponed until Chap 6 Also, in Chap 4, equilibrium problems involving only coplanar forces are considered first and are solved by ordinary algebra, while prob- lems involving three-dimensional forces, which require the full use of vector algebra, are discussed in the second part of the chapter.

Free-body diagrams are introduced early, and their importance is emphasized throughout the text Color has been used to distinguish forces from other elements of the free-body diagrams This makes it easier for the students to identify the forces acting on a given particle or rigid body and to follow the discussion of sample problems and other examples given in the text Free-body diagrams are used not only to solve equilibrium problems but also to express the equivalence of two systems

of forces or, more generally, of two systems of vectors This approach is particularly useful as a preparation for the study of the dynamics of rigid bodies As will be shown in the volume on dynamics, by placing the emphasis on "free-body-diagram equations" rather than on the standard algebraic equations of motion, a more intuitive and more complete un- derstanding of the fundamental principles of dynamics can be achieved Because of the current trend among engineers to adopt the interna-

tional system of units (SI units), the SI units most frequently used in

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

A large number of optional sections are included These sections are indicated by asterisks and thus are easily distinguished from those which form the core of the basic statics course They may be omitted without prejudice to the understanding of the rest of the text Among the topics covered in these additional sections are the reduction of a system of forces to a wrench, applications to hydrostatics, shear and bending- moment diagrams for beams, equilibrium of cables, products of inertia and Mohr's circle, mass products of inertia and principal axes of inertia for three-dimensional bodies, and the method of virtual work An op- tional section on the determination of the principal axes and moments of inertia of a body of arbitrary shape has also been included in this new edition (Sec 9.18) The sections on beams are especially useful when the

course in statics is immediately followed by a course in mechanics of

materials, while the sections on the inertia properties of

three-dimen-sional bodies are primarily intended for the students who will later study

in dynamics the three-dimensional motion of rigid bodies.

The material presented in the text and most of the problems require

no previous mathematical knowledge beyond algebra, trigonometry, and

elementary calculus, and all the elements of vector algebra necessary to

the understanding of the text are carefully presented in Chaps 2 and 3.

In general, a greater emphasis is placed on the correct understanding of

the basic mathematical concepts involved than on the nimble

manipula-tion of mathematical formulas In this connection, it should be

men-tioned that the determination of the centroids of composite areas

pre-cedes the calculation of centroids by integration, thus making it possible

to establish the concept of moment of area firmly before introducing the

use of integration The presentation of numerical solutions takes into

account the universal use of calculators by engineering students, and

instructions on the proper use of calculators for the solution of typical

statics problems have been included in Chap 2.

Each chapter begins with an introductory section setting the purpose

and goals of the chapter and describing in simple terms the material to be

covered and its application to the solution of engineering problems The

body of the text is divided into units, each consisting of one or several

theory sections, one or several sample problems, and a large number of

homework problems Each unit corresponds to a well-defined topic and

generally can be covered in one lesson In a number of cases, however,

the instructor will find it desirable to devote more than one lesson to a

given topic Each chapter ends with a review and summary of the

mate-rial covered in that chapter Marginal notes are included in these sections

to help students organize their review work, and cross-references are

used to help them find the portions of material requiring their special

attention.

The sample problems are set up in much the same form that

stu-dents will use when solving the assigned problems They thus serve the

double purpose of amplifying the text and demonstrating the type of neat

and orderly work that students should cultivate in their own solutions.

A section entitled Solving Problems on Your Own has been added to

each lesson, between the sample problems and the problems to be

as-signed The purpose of these new sections is to help students organize in

their own minds the preceding theory of the text and the solution

meth-ods of the sample problems so that they may more successfully solve the

homework problems Also included in these sections are specific

sugges-tions and strategies which will enable the students to more efficiently

attack any assigned problems.

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 text and to help students understand the

basic principles of mechanics The problems have been grouped

accord-ing to the portions of material they illustrate and have been arranged in

order of increasing difficulty Problems requiring special attention have

been indicated by asterisks Answers to 70% of the problems are given at

the end of the book Problems for which no answer is given are indicated

by a number set in italic.

Preface XV

xvi Preface The inclusion in the engineering cmrimIum of instruction in

com-puter programming and the widespread avaiJability of personal ers or mainframe terminals on most campuses make it possible for engi- neering students to solve a number of challenging mechanics problems.

comput-At one time these problems would have been considered inappropriate for an undergraduate course because of the large number of computa-

tions their solutions require In this new edition of Vector Mechanics for

Engineers: Statics, a group of problems designed to be solved with a computer follow the review problems at the end of each chapter Many

of these problems are relevant to the design process; they may involve the analysis of a structure for various configurations and loadings of the structure, or the determination of the equilibrium positions of a given mechanism which may require an iterative method of solution Develop- ing the algorithm required to solve a given mechanics problem will bene- fit the students in two different ways: (1) it will help them gain a better understanding of the mechanics principles involved; (2) it will provide them with an opportunity to apply the skills acquired in their computer programming course to the solution of a meaningful engineering prob- lem.

The authors wish to acknowledge the helpful collaboration of sor Elliot Eisenberg to this sixth edition of Vector Mechanics for Engi-

Profes-neers and thank him especially for contributing many new and ing problems The authors also gratefully acknowledge the many helpful comments and suggestions offered by the users of the previous editions

challeng-of Mechanics for Engineers and of Vector Mechanics for Engineers.

Ferdinand P Beer

E Russell Johnston, Jr.

List of Symbols

Constant; radius; distance

Reactions at supports and connections

Base of natural logarithms

Force; friction force

Acceleration of gravity

Center of gravity; constant of gravitation

Height; sag of cable

U nit vectors along coordinate axes

Moment about point 0

Moment resultant about point 0

Magnitude of couple or moment; mass of earth

Moment about axis OL

Normal component of reaction

xviii List of Symbols Position vector

Radius; distance; polar coordinate Resultant force; resultant vector; reaction Radius of earth

Position vector Length of arc; length of cable Force; vector

Thickness Force Tension Work Vector product; shearing force Volume; potential energy; shear Load per unit length

Weight; load Rectangular coordinates; distances Rectangular coordinates of centroid or center

of gravity Angles Elongation Virtual displacement Virtual work Unit vector along a line Efficiency

Angular coordinate; angle; polar coordinate Coefficient of friction

Density Angle of friction; angle

THIRD 51 METRIC EDITION

for Engineers

2 Introduction 1.1 WHAT IS MECHANICS?

Mechanics can be defined as that science which describes and dicts the conditions of rest or motion of bodies under the action of

pre-forces It is divided into three parts: mechanics of rigid bodies, chanics of deformable bodies, and mechanics of fluids.

me-The mechanics of rigid bodies is subdivided into statics and

dy-namics, the former dealing with bodies at rest, the latter with bodies

in motion In this part of the study of mechanics, bodies are assumed

to 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 ap-preciably affect the conditions of equilibrium or motion of the struc-ture under consideration They are important, though, as far as theresistance 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, is

subdivided into the study of incompressible fluids and of

compress-ible fluids An important subdivision of the study of incompresscompress-ible

fluids is 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 math-ematics, while many consider it as an engineering subject Both theseviews are justified in part Mechanics is the foundation of most engi-neering sciences and is an indispensable prerequisite to their study

However, it does not have the empiricism found in some engineering

sciences, i.e., it does not rely on experience or observation alone; byits rigor and the emphasis it places on deductive reasoning it resem-

bles mathematics But, again, it is not an abstract or even a pure ence; mechanics is an applied science The purpose of mechanics is

sci-to explain and predict physical phenomena and thus sci-to lay the dations for engineering applications

foun-1.2 FUNDAMENTAL CONCEPTS AND PRINCIPLES

Although the study of mechanics goes back to the time of Aristotle(384-322 B.C.) and Archimedes (287-212 B.C.), one has to wait untilNewton (1642-1727) to find a satisfactory formulation of its funda-mental principles These principles were later expressed in a modi-fied form by d'Alembert, Lagrange, and Hamilton Their validity re-

mained unchallenged, however, until Einstein formulated his theory

of relativity (1905) While its limitations have now been recognized, newtonian mechanics still remains the basis of today's engineering

sciences

The basic concepts used in mechanics are space, time, mass, and

force These concepts cannot be truly defined; they should be

ac-cepted on the basis of our intuition and experience and used as a tal frame of reference for our study of mechanics

men-The concept of space is associated with the notion of the position

of a point P The position of P can be defined by three lengths

mea-sured from a certain reference point, or origin, in three given tions These lengths are known as the coordinates of P.

direc-To define an event, it is not sufficient to indicate its position in

space The time of the event should also be given.

The concept of mass is used to characterize and compare bodies

on the basis of certain fundamental mechanical experiments Two

bod-ies of the same mass, for example, will be attracted by the earth in

the same manner; they will also offer the same resistance to a change

in translational motion

A force represents the action of one body on another It can be

exerted by actual contact or at a distance, as in the case of

gravita-tional forces and magnetic forces A force is characterized by its point

of application, its magnitude, and its direction; a force is represented

by a vector (Sec 2.3).

In newtonian mechanics, space, time, and mass are absolute

con-cepts, independent of each other (This is not true in relativistic

me-chanics, where the time of an event depends upon its position, and

where 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,

one of the fundamental principles of newtonian mechanics listed

be-low indicates that the resultant force acting on a body is related to

the mass of the body and to the manner in which its velocity varies

with time

You will study the conditions of rest or motion of particles and

rigid bodies in terms of the four basic concepts we have introduced

By particle we mean a very small amount of matter which may be

as-sumed to occupy a single point in space A rigid body is a

combina-tion of a large number of particles occupying fixed posicombina-tions with

re-spect to each other The study of the mechanics of particles is

obviously a prerequisite to that of rigid bodies Besides, the results

obtained for a particle can be used directly in a large number of

prob-lems dealing with the conditions of rest or motion of actual bodies

The study of elementary mechanics rests on six fundamental

prin-ciples based on experimental evidence

The Parallelogram Law for the Addition of Forces. This

states that two forces acting on a particle may be replaced by a

sin-gle 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

condi-tions of equilibrium or of motion of a rigid body will remain

un-changed if a force acting at a given point of the rigid body is replaced

by a force of the same magnitude and same direction, but acting at a

different point, provided that the two forces have the same line of

ac-tion (Sec 3.3)

Newton's Three Fundamental Laws. Formulated by Sir Isaac

Newton in the latter part of the seventeenth century, these laws can

be stated as follows:

FIRST LAW If the resultant force acting on a particle is zero,

the particle 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

The principles we have just listed will be introduced in the course

of our study of mechanics as they are needed The study of the

stat-ics of particles carried out in Chap 2, will be based on the

parallelo-gram law of addition and on Newton's first law alone The principle

of transmissibility will be introduced in Chap 3 as we begin the study

of the statics of rigid bodies, and Newton's third law in Chap 6 as we

analyze the forces exerted on each other by the various members

form-ing a structure In the study of dynamics, Newton's second law and

Newton's law of gravitation will be introduced It will then be shown

that Newton's first law is a particular case of Newton's second law

(Sec 12.2) and that the principle of transmissibility could be derived

from the other principles and thus eliminated (Sec 16.5) In the

mean-time, however, Newton's first and third laws, the parallelogram law of

addition, and the principle of transmissibility will provide us with the

necessary and sufficient 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 are

based on experimental evidence Except for Newton's first law and

the principle of transmissibility, they are independent principles which

cannot be derived mathematically from each other or from any other

elementary physical principle On these principles rests most of the

intricate structure of newtonian mechanics For more than two

cen-turies a tremendous number of problems dealing with the conditions

of rest and motion of rigid bodies, deformable bodies, and fluids have

been solved by applying these fundamental principles Many of the

solutions obtained could be checked experimentally, thus providing a

further verification of the principles from which they were derived

It is only in this century that Newton's mechanics was found at fault,

in the study of the motion of atoms and in the study of the motion of

certain planets, where it must be supplemented by the theory of

rel-ativity But on the human or engineering scale, where velocities are

small compared with the speed of light, Newton's mechanics has yet

to be disproved

1.3 SYSTEMS OF UNITS

With the four fundamental concepts introduced in the preceding

sec-tion are associated the so-called kinetic units, Le., the units of length,

time, mass, and force 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 base 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

aconsistent system of units.

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

base units are the units of length, mass, and time, and they are called,

respectively, the metre (m), the kilogram (kg), and the second (s) All

three are arbitrarily defined The second, which was originally

cho-sen to reprecho-sent 1/86 400 of the mean solar day, is now defined as the

duration of 9 192 631 770 periods of the radiation corresponding to

t

1.3 Systems of Units 5

t The first syllable of every prefix is accented so that the prefix will retain its identity Thus,

the preferred pronunciation of kilometre places the accent on the first syllable, not the

second.

t The use of these prefixes should be avoided, except for the measurement of areas and

"olumes and for the nontechnical use of centimetre, as for body and clothing

measure-ments.

The SI unit of time is the second (s), and multiples and

submul-tiples of this unit are created according to the prefixes listed in Table

1.1 However, the minute (min), hour (h), day (d), and year (a) are

units that are permitted for use along with the SI

The SI unit of plane angle is the radian (rad), and multiples and

submultiples of this unit are again created according to the prefixes

listed in Table 1.1 However, the degree (0), minute ('), second ("), and

revolution (r) are units that are permitted for use along with the SI

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 mm

rather than 0.002 16 m.t

Units of Area and Volume The unit of area is the square

me-tre (m2), which represents the area of a square of side 1 m; the unit

of volume is the cubic metre (m3), equal to the volume of a cube of

side 1 m In order to avoid exceedingly small numerical values in the

computation of areas and volumes, one uses submultiples of the

me-tre, namely, the decimetre (dm), the centimetre (cm) and the

milli-metre (mm) Since, by definition,

1 dm = 0.1 m = 10-1 m

1 cm = 0.01 m = 10-2 ill

1 mm = 0.001 m = 10-3 m

t It should be noted that when more than four digits are used on either side of the

decimal point to express a quantity in 51 units-as in 427 200 m or 0.002 16 m-spaces,

never commas, should be used to separate the digits into groups of three This is to avoid

confusion with the comma used in place of a decimal point, which is the convention in

1.3 Systems of Units 7

mea-by dividing a base unit mea-by another base unit, a prefix may be used in the numerator of the derived unit but not in its denominator For ex- ample, the constant k of a spring which stretches 20mm under a load

of 100N will be expressed as

Table 1.2. Principal 51 Units Used in Mechanics

Acceleration metre per second squared · mls 2

Angular acceleration radian per second squared rad/s2Angular velocity radian per second · rad/s

Density kilogram per cubic metre · kglm3

1.4 METHOD OF PROBLEM SOLUTION

You should approach a problem in mechanics as you would approach

an actual engineering situation By drawing on your own experience

and intuition, you will find it easier to understand and formulate the

problem Once the problem has been clearly stated, however, there

is no place in its solution for your particular fancy The solution must

be based on the six fundamental principles stated in Sec 1.2 or on

theorems derived from them Every step taken must be justified on

that basis Strict rules must be followed, which lead to the solution in

an almost automatic fashion, leaving no room for your intuition or

"feeling." After an answer has been obtained, it should be checked

Here again, you may call upon your common sense and personal

ex-perience If not completely satisfied with the result obtained, you

should carefully check your formulation of the problem, the validity

of the methods used for its solution, and the accuracy of your

com-putations

The statement of a problem should be clear and precise It should

contain the given data and indicate what information is required A

neat drawing showing all quantities involved should be included

Sep-arate diagrams should be drawn for all bodies involved, indicating

clearly the forces acting on each body These diagrams are known as

free-body diagrams and are described in detail in Sees 2.11 and 4.2.

The fundamental principles of mechanics listed in Sec 1.2 will

be used to write equations expressing the conditions of rest or motion

of the bodies considered Each equation should be clearly related to

one of the free-body diagrams You will then proceed to solve the

problem, observing strictly the usual rules of algebra and recording

neatly 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 For example, to determine the moment of a force of 50 N about

a point 0.60 m from its line of action, we would have written (Sec

3.12)

M = Fd= (50 N)(0.60 m) = 30 N mThe unit N m obtained by multiplying newtons by meters is the cor-

rect 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 the

numerical values obtained into an equation which has not yet been

used and verifYing that the equation is satisfied The importance of

correct computations in engineering cannot be overemphasized

1.5 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

com-putations performed

The solution cannot be more accurate than the less accurate of

these two items For example, if the load supported by a bridge is

known to have a mass of 40 000 kg with a possible error of 50 kg

1,5, Numerical Accuracy 9

1 0 Introduction

In computing the reaction at one of the bridge supports, it would then

be unwarranted accuracy to record it as, say, 253.42 kn The accuracy

of the solution cannot be greater than 0.125 percent, no matter howaccurate the computations are, and the possible error in the answermay be as large as (0.125/100)(253.42 kN)=0.30 kN The answershould be properly recorded as (253.4 ± 0.3) kN

In engineering problems, the data are seldom known with an curacy 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 be-ginning with a "1" and 3 figures in all other cases Unless otherwiseindicated, the data given in a problem should be assumed known with

ac-a compac-arac-able degree of ac-accurac-acy A force of 40 N, for exac-ample, should

be read 40.0 N, and a force of 15 N should be read 15.00 N

Pocket electronic calculators are widely used by practicing neers and engineering students The speed and accuracy of these cal-culators facilitate the numerical computations in the solution of manyproblems However, students should not record more significant fig-ures than can be justified merely because they are easily obtained Asnoted above, an accuracy greater than 0.2 percent is seldom neces-sary or meaningful in the solution of practical engineering problems

engi-12 Statics of Particles

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 original

forces This single equivalent force is the resultant of the original

forces acting on the particle Later the relations which exist among

the various forces acting on a particle in a state of equilibrium will be

derived and used to determine some of the forces acting on the ticle

par-The use of the word "particle" does not imply that our study will

be limited to that of small corpuscles What it means is that the sizeand shape of the bodies under consideration will not significantly af-fect the solution of the problems treated in this chapter and that allthe forces acting on a given body will be assumed to be applied at thesame point Since such an assumption is verified in many practical ap-plications, you will be able to solve a number of engineering prob-lems in this chapter

The first part of the chapter is devoted to the study of forces tained in a single plane, and the second part to the analysis of forces

con-in three-dimensional space

FORCES IN A PLANE 2.2 FORCE ON A PARTICLE RESULTANT OF TWO FORCES

A force represents the action of one body on another and is

gener-ally characterized by its point of application, its magnitude, and its

di-rection Forces acting on a given particle, however, have the same

point of application Each force considered in this chapter will thus

be completely 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 tomeasure the magnitude of a force are the newton (N) and its multi-ple the kilonewton (kN), equal to 1000 N The direction of a force

is defined by the line of action and the sense of the force The line

of action is the infinite straight line along which the force acts; it ischaracterized by the angle it forms with some fixed axis (Fig 2.1).The force itself is represented by a segment of that line; through the

use of an appropriate scale, the length of this segment may be sen to represent the magnitude of the force Finally, the sense of theforce should be indicated by an arrowhead It is important in defin-ing a force to indicate its sense Two forces having the same magni-tude and the same line of action but different sense, such as theforces shown in Fig 2.1a and b, will have directly opposite effects

cho-on a particle

Experimental evidence shows that two forces P and Q acting on

a particle A (Fig.2.2a) can be replaced by a single force R which hasthe same effect on the particle (Fig.2.2c). This force is called there-

sultant of the forces P and Q and can be obtained, as shown in Fig 2.2b, by constructing a parallelogram, using P and Q as two adjacent

sides of the parallelogram The diagonal that passes through A

rep-resents the resultant This method for finding the resultant is known

as the parallelogram law for the addition of two forces This law is

based on experimental evidence; it cannot be proved or derived ematically

math-2.3 VECTORS

It appears from the above that forces do not obey the rules of tion defined in ordinary arithmetic or algebra For example, twoforces acting at a right angle to each other, one of 4 N and the other

addi-of 3 N, add up to a force addi-of 5 N, not to a force addi-of 7 N Forces are

not the only quantities which follow the parallelogram law of

addi-tion As you will see later, displacements, velocities, accelerations, and

momenta are other examples of physical quantities possessing

mag-nitude and direction that are added according to the parallelogram

law All these quantities can be represented mathematically by

vec-tors, while those physical quantities which have magnitude but not

direction, such as volume, mass, or energy, are represented by plain numbers or scalars.

Vectors are defined as mathematical expressions possessing

mag-nitude and direction, which add according to the parallelogram law.

Vectors are represented by arrows in the illustrations and will be tinguished from scalar quantities in this text through the use of bold-face type (P).Inlonghand writing, a vector may be denotedBYdraw-ing a short arrow above the letter used to represent it (P) or byunderlining the letter (f). The last method may be preferred sinceunderlining can also be used on a typewriter or computer The mag-nitude of a vector defines the length of the arrow used to represent

dis-the vector In this text, italic type will be used to denote dis-the

magni-tude of a vector Thus, the magnimagni-tude of the vector P will be denoted

byP.

A vector used to represent a force acting on a given particle has

a well-defined point of application, namely, the particle itself Such a

vector is said to be a fixed, or bound, vector and cannot be moved

without modifying the conditions of the problem Other physicalquantities, however, such as couples (see Chap 3), are represented

by vectors which may be freely moved in space; these vectors are

called free vectors Still other physical quantities, such as forces

act-ing on a rigid body (see Chap 3), are represented by vectors whichcan be moved, or slid, along their lines of action; they are known as

sliding vectors.t

Two vectors which have the same magnitude and the same

di-rection are said to be equal, whether or not they also have the same

point 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 vector

hav-ing the same magnitude as P and a direction opposite to that of P(Fig 2.5); the negative of the vector P is denoted by - P The vec-

tors P and - P are commonly referred to as equal and opposite

vec-tors Clearly, we have

P + (-P) = 0

2.4 ADDITION OF VECTORS

We saw in the preceding section that, by definition, vectors add cording to the parallelogram law Thus, the sum of two vectors P and

ac-Q is obtained by attaching the two vectors to the same point A and

constructing a parallelogram, using P and Q as two sides of the

par-allelogram (Fig 2.6) The diagonal that passes through A represents

the sum of the vectors P and Q, and this sum is denoted by P +Q.The fact that the sign + is used to denote both vector and scalar ad-dition should not cause any confusion if vector and scalar quantitiesare alwayscarefully distinguished Thus, we should note that the mag-nitude of the vector P +Q is not, in general, equal to the sum P +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 con-

clude that the addition of two vectors is commutative, and we write

t Some expressions have magnitude and direction, but do not add according to the parallelogram law While these expressions may be represented by arrows they cannot be

considered as vectors.

A group of such expressions is the finite rotations of a rigid body Place a closed book

on a table in front of you, so that it lies in the usual fashion, with its front cover up and its binding to the left Now rotate it through 180 0 about an axis parallel to the binding (Fig 2.3a); this rotation may be represented by an arrow of length equal to 180 units and ori-

From the parallelogram law,we can derive an alternative methodfor determining the sum of two vectors This method, known as the

triangle rule, is derived as follows Consider Fig 2.6, where the sum

of the vectors P andQhas been determined by the parallelogram law.Since the side of the parallelogram opposite Qis equal to Q in mag-nitude and direction, we could draw only half of the parallelogram

(Fig 2.7a) The sum of the two vectors can thus be found by arranging

P and Q in tip-to-tail fashion and then connecting the tail of P with

the tip ofQ In Fig 2.7b, the other half of the parallelogram is

con-sidered, and the same result is obtained This confirms the fact thatvector addition is commutative

The subtraction of a vector is defined as the addition of the

cor-responding negative vector Thus, the vector P - Q representing thedifference between the vectors P and Q is obtained by adding to Pthe negative vector -Q (Fig 2.8) We write

Here again we should observe that, while the same sign is used to note both vector and scalar subtraction, confusion will be avoided ifcare is taken to distinguish between vector and scalar quantities

de-We will now consider the sum of three or more vectors The sum

of three vectorsP, Q, and S will, by definition, be obtained by first

adding the vectors P and Q and then adding the vector S to the tor P +Q We thus write

vec-P +Q +S= (P +Q)+S (2.3)Similarly,the sum of four vectors will be obtained by adding the fourthvector to the sum of the first three It follows that the sum of anynumber of vectors can be obtained by applying repeatedly the paral-lelogram law to successive pairs of vectors until all the given vectorsare replaced by a single vector

180 0 about a horizontal axis perpendicular to the binding (Fig 2.3b); this second rotation

may be represented by an arrow 180 units long and oriented as shown But the book could have been placed in this final position through a single 180 0 rotation about a vertical axis

(Fig 2.3c) We conclude that the sum of the two 180 0 rotations represented by arrows rected respectively along the z and x axes is a 1800 rotation represented by an arrow di- rected along the y axis (Fig 2.3d) Clearly, the finite rotations of a rigid body do not obey

di-If the given vectors are coplanar, i.e., if they are contained in the

same plane, their sum can be easily obtained graphically For this case,the repeated application of the triangle rule is preferred to the ap-plication of the parallelogram law In Fig 2.9 the sum of three vec-tors P, Q, and S was obtained in that manner The triangle rule was

first applied to obtain the sum P +Qof the vectors P and Q; it wasapplied again to obtain the sum of the vectors P +Q and S The de-termination of the vector P +Q, however, could have been omittedand the sum of the three vectors could have been obtained directly,

as shown in Fig 2.10, by arranging the given vectors in tip-to-tail

fashion and connecting the tail of the first vector with the tip of the last one This is known as the polygon rule for the addition of vectors.

We observe that the result obtained would have been unchanged

if, as shown in Fig 2.11, the vectors Q and S had been replaced bytheir sum Q + S We may thus write

P +Q +S = (P +Q) +S = P +(Q + S) (2.4)

which expresses the fact that vector addition is associative Recalling

that vector addition has also been shown, in the case of two vectors,

(Fig 2.13)

2.5 RESULTANT OF SEVERAL CONCURRENT FORCES

Consider a particle A acted upon by several coplanar forces, i.e., by

several forces contained in the same plane (Fig 2.14a) Since the forces considered here all pass through A, they are also said to be con-

current The vectors representing the forces acting on A may be added

by the polygon rule (Fig 2.14b) Since the use of the polygon rule is

equivalent to the repeated application of the parallelogram law, thevector R thus obtained represents the resultant of the given concur-rent forces, i.e., the single force which has the same effect on the par-ticle A as the given forces As indicated above, the order in which thevectors P,Q, and S representing the given forces are added together

is immaterial

2.6 RESOLUTION OF A FORCE INTO COMPONENTS

We have seen that two or more forces acting on a particle may be placed by a single force which has the same effect on the particle.Conversely, a single force F acting on a particle may be replaced bytwo or more forces which, together, have the same effect on the par-

re-ticle These forces are called the components of the original force F, and the process of substituting them for F is called resolving the force

Finto components.

Clearly, for each force F there exist an infinite number of

possi-ble sets of components Sets of two components P and Q are the most

important as far as practical applications are concerned But, eventhen, the number of ways in which a given force F may be resolvedinto two components is unlimited (Fig 2.15) Two cases are of par-ticular interest:

1 One of the Two Components, P, Is Known The second

com-ponent, Q, is obtained by applying the triangle rule and ing the tip of P to the tip of F (Fig 2.16); the magnitude anddirection of Q are determined graphically or by trigonometry.Once Q has been determined, both components P and Qshould be applied at A

join-2 The Line of Action of Each Component Is Known The

mag-nitude and sense of the components are obtained by applyingthe parallelogram law and drawing lines, through the tip of F,parallel to the given lines of action (Fig 2.17) This processleads to two well-defined components, P andQ,which can bedetermined graphically or computed trigonometrically by ap-plying the law of sines

Many other cases can be encountered; for example, the direction

of one component may be known, while the magnitude of the othercomponent is to be as small as possible (see Sample Prob 2.2) In allcases the appropriate triangle or parallelogram which satisfies thegiven conditions is drawn

of known direction.

You will now be asked to solve problems on your own.Somemayresemble one ofthe sample problems; others may not What all problemsand.sample problems inthis section have in common is that they can be solved by the direct application ofthe parallelogram law

Your solution of a given prohlem should consist of the following steps:

1 Identify whichoftheforces are the applied forces and which is the sultant. It is often helpful to write the vector equation which shows how theforces are related For example, in Sample Prob 2.1 we would have

re-R=·P +QYou may want to keep that relation in.mindas you formulate the next part of yoursolution

2 Draw a parallelogram with the applied forces as two adjacent sides and the resultant as the included diagonal (Fig 2.2) Alternatively, you can use

the triangle rule, with the applied forces drawn in tip-to-tail fashion and the

resul-tant extending from the tail of the first vector to the tip of the second (Fig 2.7)

3 Indicate all dimensions. Using one of the triangles of the parallelogram, orthe triangle constructed according to the triangle rule, indicate all dimensions-whether sides or angles and determine the unknown dimensions either graphi-cally or by trigonometry If you use trigonometry, remember that the law of cosinessho1.l1dbeapplied first if two sides and the included angle are known [Sample Prob.2.1], and the law of sines should be applied first if one side and all angles are known[Sample Prob 2.2]

If you have had prior exposure to mechanics, you might be tempted to ignore thesolution techniques of this lesson in favor of resolving the forces into rectangularcomponents While this latter method is important and will be considered in thenext section, use of the parallelogram law simplifies the solution of many problemsand should be mastered at this time

2.1 Two forces are applied at point B of beam AB. Determine

graph-~- the magnitude and direction of their resultant using (a) the

parallelo-~ law,(b) the triangle rule.

2.2 Two forces P and Q are applied as shown at point A of a hook support Knowing that P = 75 Nand Q = 125 N, determine graphically the magnitude and direction of their resultant using(a) the parallelogram law,

ib) the triangle rule.

2.3 Two forces P and Q are applied as shown at point A of a hook support Knowing that P = 60 Nand Q= 25 N, determine graphically the magnitude and direction of their resultant using (a) the parallelogram law,

(b) the triangle rule.

2.4 The cable staysAB and AD help support poleAC. Knowing that the tension is 1200 N inAB and 400 N inAD, determine graphically the mag- nitude and direction of the resultant of the forces exerted by the stays atA

using(a) the parallelogram law,(b) the triangle rule.

2.5 The 200-N force is to be resolved into components along lines

a-a' and b-b' (a) Determine the angle ex by trigonometry knowing that the

component along a-a' is to be l.50 N (b) What is the corresponding value of

the component along b-b'?

2.6 The 200-N force is to be resolved into components along lines

a-a' and b-b' (a) Determine the angleex by trigonometry knowing that the

component along b-b' is to be 120 N (b) What is the corresponding value

of the component along a-a'?

tAnswers to all problems set in straight type (such as 2.1) are given at the end of the

2.7 A telephone cable is clamped at A to the pole AB Knowing that

the tension in the left-hand portion of the cable isTl = 800 N, determine

by trigonometry (a) the required tension Tz in the right-hand portion if the

resultant R of the forces exerted by the cable atA is to be vertical, (b) the corresponding magnitude of R.

2.8 A telephone cable is clamped at A to the pole AB Knowing that the tension in the right-hand portion of the cable is Tz = 1000 N, determine

by trigonometry (a) the required tension Tl in the left-hand portion if the resultant R of the forces exerted by the cable atA is to be vertical, (b) the corresponding magnitude of R.

2.9 Two forces are applied as shown to a hook support Knowing that the magnitude of Pis 35 N, determine by trigonometry (a) the required an- gle CI' if the resultant R of the two forces applied to the support is to be hor- izontal,(b) the corresponding magnitude of R.

2.10 For the hook support of Prob 2.2, knowing that the magnitude

of Pis 75 N, determine by trigonometry (a) the required magnitude of the forceQif the resultant R of the two forces applied at A is to be vertical, (b)

the corresponding magnitude of R.

2.11 A steel tank is to be positioned in an excavation Knowing that CI' = 20°, determine by trigonometry (a) the required magnitude of the force

P if the resultant R of the two forces applied atA is to be vertical, (b) the corresponding magnitude of R.

2.12 A steel tank is to be positioned in an excavation Knowing that the magnitude of P is 500 N, determine by trigonometry (a) the required angle CI' if the resultant R of the two forces applied atA is to be vertical,

(b) the corresponding magnitude of R.

2.13 A steel tank is to be positioned in an excavation Determine by trigonometry (a) the magnitude and direction of the smallest force P for which the resultant R of the two forces applied atA is vertical, (b) the cor- responding magnitude of R.

2.14 For the hook support of Prob 2.9, determine by trigonometry

(a) the magnitude and direction of the smallest force P for which the tant R of the two forces applied to the support is horizontal, (b) the corre- sponding magnitude of R.

resul-2.15 Solve Prob 2.3 by trigonometry.

2.16 Solve Prob 2.4 by trigonometry.

2.17 For the hook support of prob 2.9, knowing that P= 75 Nand

a=50°, determine by trigonometry the magnitude and direction of the sultant of the two forces applied to the support.

re-2.18 Solve Prob 2.1 by trigonometry.

2.19 Two structural members A and B are bolted to a bracket as shown.

Knowing that both members are in compression and that the force is 15 kN

in member A and 10 kN in member B, determine by trigonometry the

mag-nitude and direction of the resultant of the forces applied to the bracket by

members A and B.

2.20 Two structural members A and B are bolted to a bracket as shown.

Knowing that both members are in compression and that the force is 10 kN

in member A and 15 kN in member B, determine by trigonometry the

mag-nitude and direction of the resultant of the forces applied to the bracket by

The x and y axes are usually chosen horizontal and vertical, spectively, as in Fig 2.18; they may, however, be chosen in any two perpendicular directions, as shown in Fig 2.19 In determining the rectangular components of a force, the student should think of the construction lines shown in Figs 2.18 and 2.19 as being parallel to the x and y axes, rather than perpendicular to these axes This prac- tice will help avoid mistakes in determining oblique components as in Sec 2.6.

re-t The properties established in Secs 2.7 and 2.8 may be readily extended to the