• a variety of problems for practice. Sections end with Fundamental Problems and Conceptual
Problems, and chapters conclude with Review Problems.
• an emphasis on free-body diagrams. Specific sections, examples, and homework problems
are devoted to the drawing of free-body diagrams, essential to solving problems.
• the Procedure for Analysis feature. Introduced in the first chapter and customized in later
ones, this procedure is a versatile approach to analyzing a variety of mechanics problems.
NEW TO THIS EDITION
• New Fundamental Problems have been added, with partial solutions at the back of the book.
Engineering Mechanics
STATICS
Engineering Mechanics: Statics features Russell Hibbeler’s hallmark approach to teaching the
subject—a clear, thorough, and student-friendly presentation of theory alongside examples and
practice problems for the application of concepts. Now in its fifteenth edition, the text continues
to empower students with
GLOBAL
EDITION
GLOB AL
EDITION
GLOBAL
EDITION
This is a special edition of an established title widely used by colleges and
universities throughout the world. Pearson published this exclusive edition
for the benefit of students outside the United States. If you purchased this
book within the United States, you should be aware that it has been imported
without the approval of the Publisher or Author.
Engineering Mechanics
STATICS
Fifteenth Edition in SI Units
• Over 300 new problems involve applications to many different fields of engineering.
Available separately for purchase is Mastering Engineering for Engineering Mechanics: Statics, the
teaching and learning platform that empowers instructors to personalize learning for every student.
This optional suite helps deliver the desired learning outcomes when combined with Pearson’s
trusted educational content and features like the following:
• Video Solutions: Developed by the author, they summarize key concepts discussed in the text,
demonstrate how to solve problems, and model the best way to reach a solution.
• GeoGebra 3D Interactive Figures: These figures enable students to interact directly with
the graph in a manner that replicates how they would graph on paper.
CVR_HIBB4048_15_GE_CVR_Vivar.indd All Pages
R. C. Hibbeler
Hibbeler
• Enhanced feedback: Tutorials and many end-of-section problems provide enhanced feedback,
specific to student errors, and optional hints, which break problems down into simpler steps.
Fifteenth Edition
in SI Units
• New or updated photos throughout the book illustrate how principles apply to real-world
situations and how materials behave under load.
07/07/22 2:43 PM
SI Prefixes
Multiple
Exponential Form
Prefix
SI Symbol
1 000 000 000
109
giga
G
1 000 000
106
mega
M
1 000
103
kilo
k
10-3
milli
m
0.000 001
-6
10
micro
μ
0.000 000 001
10-9
nano
n
Submultiple
0.001
Conversion Factors (SI) to (FPS)
Quantity
Unit of
Measurement (SI)
Force
N
Mass
kg
0.06852 slug
Length
m
3.281 ft
CVR_HIBB4048_15_GE_CVR_Vivar_IFC.indd All Pages
Equals
Unit of
Measurement (FPS)
0.2248 lb
16/06/22 2:24 PM
Fundamental Equations of Statics
Cartesian Vector
A = Axi + Ayj + Azk
Equilibrium
Particle
Magnitude
ΣFx = 0, ΣFy = 0, ΣFz = 0
A = 2A2x + A2y + A2z
Directions
Rigid Body-Two Dimensions
Ay
Az
Ax
A
uA =
=
i +
j +
k
A
A
A
A
= cos ai + cos bj + cos gk
cos2 a + cos2 b + cos2 g = 1
ΣFx = 0, ΣFy = 0, ΣMO = 0
Rigid Body-Three Dimensions
ΣFx = 0, ΣFy = 0, ΣFz = 0
ΣMx = 0, ΣMy = 0, ΣMz = 0
Dot Product
A · B = AB cos u
= AxBx + AyBy + AzBz
Cross Product
i
3
C = A : B = Ax
Bx
j
Ay
By
k
Az 3
Bz
Cartesian Position Vector
r = (x2 - x1)i + (y2 - y1)j + (z2 - z1)k
Cartesian Force Vector
Friction
Static (maximum)
Fs = μsN
Kinetic
Fk = μkN
Center of Gravity
Particles or Discrete Parts
r =
Body
r
F = Fu = Fa b
r
r =
Moment of a Force
MO = Fd
i
MO = r : F = 3 rx
Fx
j
ry
Fy
uy
ry
Fy
I =
uz
rz 3
Fz
Simplification of a Force and Couple System
FR = ΣF
( MR ) O = ΣM + ΣMO
CVR_HIBB4971_15_SE_EP.indd
A01_HIBB4048_15_GE_FM.indd 13
L
∼
r dW
L
dW
Area and Mass Moments of Inertia
k
rz 3
Fz
Moment of a Force about a Specified Axis
ux
M = u # r : F = 3 rx
Fx
Σ∼
r W
ΣW
L
r 2dA
I =
L
r 2dm
Parallel-Axis Theorem
I = I + Ad 2
I = I + md 2
Radius of Gyration
k =
Virtual Work
I
AA
k =
I
Am
dU = 0
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20:12
Geometric Properties of Line and Area Elements
Centroid Location
Centroid Location
y
y
L 5 2ur
r
u
C
u
A 5 ur 2
r
u
C
x
u
x
r sin u
u
2 r sin u
3
u
Circular arc segment
Circular sector area
y
L 5 p–2 r
r
Area Moment of Inertia
A 5 14 pr 2
L 5 pr
C
2r
—
p
C
y9
4r
—
3p
r
C
r
x9
x
C
h
y
x
–1 2 a 1 b
3 a 1b
b
Trapezoidal area
pr
A5—
2
1 4
1
r (u + sin 2u)
4
2
Ix =
1 4
πr
16
Iy =
1 4
πr
16
4
π
- b r4
16 9 π
4
π
- b r4
16 9 π
2
4r
—
Ix =
1 4
πr
8
Iy =
1 4
πr
8
3p
r
h
Iy =
Iy′ = a
Quarter circle area
A 5 –12 h (a 1 b)
a
1 4
1
r (u - sin 2u)
4
2
Ix′ = a
4r
—
3p
Quarter and semicircle arcs
Ix =
C
x
Semicircular area
y
b
a C
A 5 pr2
A5 23– ab
1
Ix = πr4
4
r
3–
5a
x
C
1
Iy = πr4
4
3–
8b
Semiparabolic area
Circular area
y
1
A5 —
ab
3
b
C
3
—
b
10
–3 a
4
h
b
a
Ix =
1 3
bh
12
Iy =
1 3
hb
12
Rectangular area
a
y9
1
A 5—
bh
2
h
C
4 ab
A 5—
3
h
—
3
x9
C
b
—
3
b
2
—
a
5
Parabolic area
CVR_HIBB4971_15_SE_EP.indd
4
A01_HIBB4048_15_GE_FM.indd 2
x
C
Exparabolic area
b
A 5 bh
1 3
bh
36
1
Iy′ = hb3
36
Ix′ =
Triangular area
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20:12
CVR_HIBB
Center of Gravity and Mass Moment of Inertia of Homogeneous Solids
z
z
r
V5 4–3 pr 3
V 5 pr 2 h
r
h
–
2
G
G
x
x
Cylinder
1
m(3r2 + h2)
Ixx = Iyy =
12
Sphere
Ixx = Iyy = Izz =
2
mr2
5
y
h
–
2
y
Izz =
1
mr2
2
z
z
1
V 5 – r 2h
3
V 5 –23 pr 3
G
r
y
Cone
3
3
Ixx = Iyy =
m(4r2 + h2) Izz =
mr2
80
10
Hemisphere
Ixx = Iyy = 0.259 mr2 Izz =
z
h
y
r
x
3– r
8
x
h
–
4
G
2
mr2
5
z
z9
G
G
r
y
y
a
b
x
x
Thin plate
Thin circular disk
1
1
3
Ixx = Iyy = mr2 Izz = mr2 Iz′z′ = mr2
4
2
2
1
Ixx =
mb2
12
Iyy =
1
ma2
12
Izz =
1
m(a2 + b2)
12
z
z
l
2
G
r
G
y
y
l
2
y9
x9
x
Thin ring
1
Ixx = Iyy = mr2 Izz = mr2
2
CVR_HIBB4971_15_SE_EP.indd
A01_HIBB4048_15_GE_FM.indd 35
x
Slender rod
1
Ixx = Iyy =
ml 2
12
Ix′x′ = Iy′y′ =
1
ml 2
3
Iz′z′ = 0
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20:12
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ENGINEERING MECHANICS
STATICS
FIFTEENTH EDITION IN SI UNITS
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ENGINEERING MECHANICS
STATICS
FIFTEENTH EDITION IN SI UNITS
R. C. HIBBELER
SI Conversion by
Jun Hwa Lee
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Content Strategy: Shabnam Dohutia and Aurko Mitra
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Cover Image: Orla/Shutterstock
Please contact https://support.pearson.com/getsupport/s/contactsupport with any queries on this content.
© 2023 by R. C. Hibbeler
The right of R. C. Hibbeler to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Authorized adaptation from the United States edition, entitled Engineering Mechanics: Statics, Fifteenth Edition, ISBN 978-0-13-481497-1, by Russell C. Hibbeler,
published by Pearson Education, Inc. © 2022.
Microsoft and/or its respective suppliers make no representations about the suitability of the information contained in the documents and related graphics published
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To the Student
With the hope that this work will stimulate
an interest in Engineering Mechanics
and provide an acceptable guide to its understanding.
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PREFACE
The main purpose of this book is to provide the student with a clear and thorough
presentation of the theory and application of engineering mechanics. To achieve this
objective, this work has been shaped by the comments and suggestions of hundreds
of reviewers in the teaching profession, as well as many of the author’s students.
New to this Edition
Expanded Answer Section. The answer section in the back of the book now
includes additional information related to the solution of select Fundamental
Problems in order to offer the student some guidance in solving the problems.
Re-writing of Text Material. Some concepts have been clarified further in this
edition, and throughout the book, the accuracy has been enhanced, and important
definitions are now in boldface throughout the text to highlight their importance.
Additional Fundamental Problems. Some new fundamental problems
have been added along with their partial solutions which are given in the back of
the book.
New Photos. The relevance of knowing the subject matter is reflected by the
real-world applications depicted in the over 15 new or updated photos placed
throughout the book. These photos generally are used to explain how the relevant
principles apply to real-world situations and how materials behave under load.
New Problems. There are approximately 30% new problems that have been
added to this edition, which involve applications to many different fields of engineering.
New Videos. Three types of videos are available that are designed to enhance
the most important material in the book. Lecture videos serve to test the student’s
ability to understand concepts, example problem videos are intended to review
these problems, and fundamental problem videos guide the student in solving these
problems that are in the book. They are available for select sections in the chapters
and marked with a video icon. The videos appear in the Pearson eText and on a
companion website available for purchase at www.pearsonglobaleditions.com.
11
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12
Preface
Hallmark Features
Besides the new features mentioned, other outstanding features that define the
contents of the book include the following:
Organization and Approach. Each chapter is organized into well-defined
sections that contain an explanation of specific topics, illustrative example problems,
and a set of homework problems. The topics within each section are placed into
subgroups defined by boldface titles. The purpose of this is to present a structured
method for introducing each new definition or concept and to make the book
convenient for later reference and review.
Chapter Contents. Each chapter begins with an illustration demonstrating a
broad-range application of the material within the chapter. A bulleted list of the
chapter contents is provided to give a general overview of the material that will
be covered.
Emphasis on Free-Body Diagrams. Drawing a free-body diagram is
particularly important when solving problems, and for this reason this step is strongly
emphasized throughout the book. In particular, special sections and examples are
devoted to show how to draw free-body diagrams. Specific homework problems
have also been added to develop this practice.
Procedures for Analysis. A general procedure for analyzing any mechanics
problem is presented at the end of the first chapter. Then this procedure is customized
to relate to specific types of problems that are covered throughout the book. This
unique feature provides the student with a logical and orderly method to follow when
applying the theory. The example problems are solved using this outlined method in
order to clarify its numerical application. Realize, however, that once the relevant
principles have been mastered and enough confidence and judgment have been
obtained, the student can then develop his or her own procedures for solving problems.
Important Points. This feature provides a review or summary of the most
important concepts in a section and highlights the most significant points that should
be known when applying the theory to solve problems.
Fundamental Problems. These problem sets are selectively located just after
most of the example problems. They provide students with simple applications of
the concepts, and therefore, the chance to develop their problem-solving skills
before attempting to solve any of the standard problems that follow. In addition,
they can be used for preparing for exams, and they can be used at a later time when
preparing for the Fundamentals of Engineering Exam. The partial solutions are
given in the back of the book.
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Preface
13
Conceptual Understanding. Through the use of photographs placed
throughout the book, the theory is applied in a simplified way in order to illustrate
some of its more important conceptual features and instill the physical meaning of
many of the terms used in the equations.
Homework Problems. Apart from the Fundamental and Conceptual type
problems mentioned previously, other types of problems contained in the book
include the following:
• Free-Body Diagram Problems. Some sections of the book contain introductory
problems that only require drawing the free-body diagram for the specific
problems within a problem set. These assignments will impress upon the student
the importance of mastering this skill as a requirement for a complete solution of
any equilibrium problem.
• General Analysis and Design Problems. The majority of problems in the book
depict realistic situations encountered in engineering practice. Some of these
problems come from actual products used in industry. It is hoped that this realism
will both stimulate the student’s interest in engineering mechanics and provide a
means for developing the skill to reduce any such problem from its physical
description to a model or symbolic representation to which the principles of
mechanics may be applied.
Throughout the book, in any set of problems, an attempt has been made to a­ rrange
them in order of increasing difficulty except for the end of chapter ­review problems, which are presented in random order.
• Computer Problems. An effort has been made to include a few problems that
may be solved using a numerical procedure executed on either a desktop computer
or a programmable pocket calculator. The intent here is to broaden the student’s
capacity for using other forms of mathematical analysis without sacrificing the
time needed to focus on the application of the principles of mechanics. Problems
of this type, which either can or must be solved using numerical procedures, are
identified by a “square” symbol (j) preceding the problem number.
The many homework problems in this edition, have been placed into two different
categories. Problems that are simply indicated by a problem number have an
answer and in some cases an additional numerical result given in the back of the
book. An asterisk (*) before every fourth problem number indicates a problem
without an answer.
Accuracy. As with the previous editions, apart from the author, the accuracy of
the text and problem solutions has been thoroughly checked by Kai Beng Yap, who
was a practicing engineer, and a team of specialists at EPAM, including Georgii
Kolobov, Ekaterina Radchenko, and Artur Akberov. Thanks are also due to Keith
Steuer from Snow College and Mike Freeman, Professor Emeritus at the University
of Alabama.
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14
Preface
Contents
The book is divided into 11 chapters, in which the principles are first applied to
simple, then to more complicated situations. In a general sense, each principle is
applied first to a particle, then a rigid body subjected to a coplanar system of forces,
and finally to three-dimensional force systems acting on a rigid body.
Chapter 1 begins with an introduction to mechanics and a discussion of units.
The vector properties of a concurrent force system are introduced in Chapter 2.
This theory is then applied to the equilibrium of a particle in Chapter 3. Chapter 4
contains a general discussion of both concentrated and distributed force systems
and the methods used to simplify them. The principles of rigid-body equilibrium
are developed in Chapter 5 and then applied to specific problems involving the
equilibrium of trusses, frames, and machines in Chapter 6, and to the analysis of
internal forces in beams and cables in Chapter 7. Applications to problems involving
frictional forces are discussed in Chapter 8, and topics related to the center of
gravity and centroid are treated in Chapter 9. If time permits, sections involving
more advanced topics, indicated by stars (★), may be covered. Most of these topics
are included in Chapter 10 (area and mass moments of inertia) and Chapter 11
(virtual work and potential energy). Note that this material also provides a suitable
reference for basic principles when it is discussed in more advanced courses. Finally,
Appendix A provides a review and list of mathematical formulas needed to solve the
problems in the book.
Alternative Coverage. At the discretion of the instructor, some of the material
may be presented in a different sequence with no loss of continuity. For example, it is
possible to introduce the concept of a force and all the necessary methods of vector
analysis by first covering Chapter 2 and Section 4 (the cross product). Then after
covering the rest of Chapter 4 (force and moment systems), the equilibrium methods
of Chapters 3 and 5 can be discussed.
Acknowledgments
The author has endeavored to write this book so that it will appeal to both the student
and instructor. Through the years, many people have helped in its development, and
I will always be grateful for their valued suggestions and comments. Specifically, I
wish to thank all the individuals who have sent comments to me. These include J.
Aurand, D. Boyajian, J. Callahan, D. Dikin, I. Elishakoff, R. Hendricks, F. Herrera,
J. Hilton, H. Kuhlman, K. Leipold, C. Roche, M. Rosengren, R. Scott, and J. Tashbar.
A long-time friend and associate, Kai Beng Yap, was of great help to me in
preparing and checking problem solutions. During the production process I am
thankful for the assistance of Rose Kernan, my production editor, and Marta Samsel,
who worked on the cover of the book. And finally, to my wife, Conny, who helped in
the proofreading of the manuscript for publication.
Lastly, many thanks are extended to all my students and to members of the teaching
profession who have freely taken the time to offer their suggestions and comments.
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Preface
15
Since this list is too long to mention, it is hoped that those who have given help in
this manner will accept this anonymous recognition.
I would greatly appreciate hearing from you if at any time you have any comments,
suggestions, or issues related to any matters regarding this edition.
Russell Charles Hibbeler
hibbeler@bellsouth.net
Acknowledgments for the Global Edition
Pearson would like to thank and acknowledge the following for their work on the
Global Edition.
Contributor
Jun Hwa Lee
Jun has a PhD in Mechanical Engineering from the Korea Advanced Institute of
Science and Technology.
Reviewers
Imad Abou-Hayt, Aalborg University
Fred Afagh, Carleton University
Rishad Irani, Carleton University
Akbar Afaghi Khatibi, RMIT University
Payam Khazaeinejad, Kingston University
Murat Saribay, Istanbul Bilgi University
Pearson would also like to thank Kai Beng Yap for his contributions to the previous
Global Edition. Kai was a registered professional engineer working in Malaysia.
He had BS and MS degrees in Civil Engineering from the University of Louisiana,
Lafayette, Louisiana.
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Resources
• Mastering Engineering This online tutorial and assessment program allows you to integrate dynamic
homework and practice problems with automated grading of exercises from the textbook. Tutorials and many
end-of-section problems provide enhanced student feedback and optional hints. Mastering Engineering™
allows you to easily track the performance of your entire class on an assignment-by-assignment basis, or the
detailed work of an individual student. For more information visit www.masteringengineering. com.
• Videos Developed by the author, three different types of videos are now available to reinforce learning the
basic theory and applying the principles. The first set provides a lecture review and a self-test of the material
related to the theory and concepts presented in the book, the second set provides a self-test of the example
problems and the basic procedures used for their solution, and the third set provides an engagement for solving
the Fundamental Problems throughout the book. They are available for select sections in the chapters and marked
with a video icon. The videos appear in the Pearson eText and on a companion website available for purchase at
www.pearsonglobaleditions.com.
• Instructor’s Solutions Manual This supplement provides complete solutions supported by problem
statements and problem figures. The Instructor’s Solutions Manual is available in the Instructor Resource
Center at www.pearsonglobaleditions.com.
• PowerPoint Slides A complete set of all the figures and tables from the textbook are available in PowerPoint
format in the Instructor Resource Center at www.pearsonglobaleditions.com.
16
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CONTENTS
1
General Principles
25
25
Chapter Objectives
1.1
Mechanics
25
1.2
Fundamental Concepts
1.3
The International System of Units
1.4
Numerical Calculations
1.5
General Procedure for Analysis
26
29
32
34
2
Force Vectors
39
Chapter Objectives
39
2.1
Scalars and Vectors
39
2.2
Vector Operations
40
2.3
Vector Addition of Forces
2.4
ddition of a System of Coplanar
A
Forces 54
2.5
Cartesian Vectors
2.6
Addition of Cartesian Vectors
2.7
Position Vectors
2.8
Force Vector Directed Along a Line
2.9
Dot Product
42
65
68
76
78
86
17
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18 C o n t e n t s
3
Equilibrium of a
Particle 103
Chapter Objectives
103
3.1
ondition for the Equilibrium
C
of a Particle 103
3.2
The Free-Body Diagram
104
3.3
Coplanar Force Systems
107
3.4
Three-Dimensional Force Systems
120
4
Force System
Resultants 135
Chapter Objectives
A01_HIBB4048_15_GE_FM.indd 18
135
4.1
Moment of a Force—Scalar
Formulation 135
4.2
Principle of Moments
4.3
Cross Product
4.4
Moment of a Force—Vector
Formulation 148
4.5
Moment of a Force about a
Specified Axis 158
4.6
Moment of a Couple
4.7
Simplification of a Force and Couple
System 179
4.8
Further Simplification of a Force and
Couple System 190
4.9
Reduction of a Simple Distributed
Loading 202
137
145
167
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19
Contents
5
Equilibrium of a
Rigid Body 217
Chapter Objectives
217
5.1
onditions for Rigid-Body
C
Equilibrium 217
5.2
Free-Body Diagrams
5.3
Equations of Equilibrium
5.4
Two- and Three-Force Members
5.5
Free-Body Diagrams
5.6
Equations of Equilibrium
5.7
Constraints and Statical Determinacy
219
230
240
253
258
259
6
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Structural Analysis
279
Chapter Objectives
279
6.1
Simple Trusses
279
6.2
The Method of Joints
282
6.3
Zero-Force Members
288
6.4
The Method of Sections
6.5
Space Trusses
6.6
Frames and Machines
296
306
310
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20 C o n t e n t s
7
Internal Forces
347
Chapter Objectives
347
7.1
Internal Loadings
347
7.2
hear and Moment Equations and
S
Diagrams 363
7.3
elations among Distributed Load, Shear,
R
and Moment 372
7.4
Cables
383
8
Friction
A01_HIBB4048_15_GE_FM.indd 20
403
Chapter Objectives
403
8.1
Characteristics of Dry Friction
403
8.2
Problems Involving Dry Friction
8.3
Wedges
8.4
Frictional Forces on Screws
8.5
Frictional Forces on Flat Belts
8.6
rictional Forces on Collar Bearings, Pivot
F
Bearings, and Disks 447
8.7
Frictional Forces on Journal Bearings
8.8
Rolling Resistance
408
430
432
439
450
452
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21
Contents
9
Center of Gravity and
Centroid 465
Chapter Objectives
465
9.1
enter of Gravity, Center of Mass, and the
C
Centroid of a Body 465
9.2
Composite Bodies
9.3
Theorems of Pappus and Guldinus
9.4
Resultant of a General Distributed
Loading 511
9.5
Fluid Pressure
488
502
512
10
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Moments of Inertia
529
Chapter Objectives
529
10.1
D
efinition of Moments of Inertia for
Areas 529
10.2
Parallel-Axis Theorem for an Area
10.3
Radius of Gyration of an Area 531
10.4
M
oments of Inertia for Composite
Areas 540
10.5
Product of Inertia for an Area
10.6
M
oments of Inertia for an Area about
Inclined Axes 552
10.7
Mohr’s Circle for Moments of Inertia
10.8
Mass Moment of Inertia
530
548
555
563
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22 C o n t e n t s
11
Virtual Work
581
Chapter Objectives
581
11.1
Definition of Work
581
11.2
Principle of Virtual Work
11.3
rinciple of Virtual Work for a System of
P
Connected Rigid Bodies 585
11.4
Conservative Forces
11.5
Potential Energy
11.6
otential-Energy Criterion for
P
Equilibrium 600
11.7
Stability of Equilibrium Configuration
583
597
598
601
Appendix
A.
athematical Review and
M
Formulations 616
Fundamental Problem
Solutions and
Answers 620
Review Problem
Answers 637
Selected Answers
Index
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640
653
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CREDITS
Cover: SHUTTERSTOCK: Orla/Shutterstock
Chapter 1: Image Credits
024 ALAMY IMAGES: Gaertner/Alamy Stock Photo; 029 SHUTTERSTOCK:
NikoNomad/Shutterstock
Chapter 2: Image Credits
038 ALAMY IMAGES: Marcus Hofmann/Alamy Stock Photo
Chapter 3: Image Credits
102 SHUTTERSTOCK: Djgis/Shutterstock
Chapter 4: Image Credits
134 SHUTTERSTOCK: Zbynek Jirousek/Shutterstock
Chapter 5: Image Credits
216 ALAMY IMAGES: Panther Media GmbH/Alamy Stock Photo
Chapter 6: Image Credits
278 ALAMY IMAGES: Ray Hardinge/Alamy Stock Photo
Chapter 7: Image Credits
346 SHUTTERSTOCK: Northlight/Shutterstock
Chapter 8: Image Credits
402 ALAMY IMAGES: Eugene Sergeev/Alamy Stock Photo
Chapter 9: Image Credits
464 SHUTTERSTOCK: Ann Baldwin/Shutterstock
Chapter 10: Image Credits
528 ALAMY IMAGES: Olaf Speier/Alamy Stock Photo
Chapter 11: Image Credits
580 ALAMY IMAGES: APIX/Alamy Stock Photo
23
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CHAPTER
1
Cranes such as this one are required to lift extremely large loads. Their design is
based on the basic principles of statics and dynamics, which form the subject matter
of engineering mechanics.
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GENERAL
PRINCIPLES
Lecture Summary and Quiz,
Example, and Problemsolving videos are available
where this icon appears.
CHAPTER OBJECTIVES
■■ To provide an introduction to the basic quantities and idealizations
of mechanics.
■■ To state Newton’s Laws of Motion and Gravitation.
■■ To review the principles for applying the SI system of units.
■■ To examine the standard procedures for performing numerical
calculations.
■■ To present a general guide for solving problems.
1.1
MECHANICS
Mechanics is a branch of the physical sciences that is concerned with the
state of rest or motion of bodies that are subjected to the action of forces.
In general, this subject can be subdivided into three branches: rigid-body
mechanics, deformable-body mechanics, and fluid mechanics. In this book
we will study rigid-body mechanics since it is a basic requirement for
the study of the mechanics of deformable bodies and the mechanics of
fluids. Furthermore, rigid-body mechanics is essential for the design and
analysis of many types of structural members, mechanical components,
or electrical devices encountered in engineering.
Rigid-body mechanics is divided into two areas: statics and dynamics.
Statics deals with the equilibrium of bodies, that is, those that are either
at rest or move with a constant velocity; whereas dynamics is concerned
with the accelerated motion of bodies. We can consider statics as a
special case of dynamics, in which the acceleration is zero; however,
statics deserves separate treatment in engineering education since many
objects are designed with the intention that they remain in equilibrium.
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25
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26
C h a p t e r 1 G e n e r a l P r i n c i p l e s
Historical Development. The subject of statics developed at
1
a very early time because its principles can be formulated simply from
measurements of geometry and force. For example, the writings of
Archimedes (287–212 B.C.) deal with the principle of the lever. Studies
of the pulley, inclined plane, and wrench are also recorded in ancient
writings—at times when the requirements for engineering were limited
primarily to building construction.
Since the principles of dynamics depend on an accurate measurement
of time, this subject developed much later. Galileo Galilei (1564–1642)
was one of the first major contributors to this field. His work consisted
of experiments using pendulums and falling bodies. The most significant
contributions in dynamics, however, were made by Isaac Newton
(1642–1727), who is noted for his formulation of the three fundamental
laws of motion and the law of universal gravitational attraction. Shortly
after these laws were postulated, important techniques for their
application were developed by other scientists and engineers, some of
whom will be mentioned throughout the book.
1.2
FUNDAMENTAL CONCEPTS
Before we begin our study of engineering mechanics, it is important
to understand the meaning of certain fundamental concepts and
principles.
Basic
Quantities. The following four quantities are used
throughout mechanics.
Length. Length is used to locate the position of a point in space and
thereby describe the size of a physical system. Once a standard unit of
length is defined, one can then use it to define distances and geometric
properties of a body as multiples of this unit.
Time. Time is conceived as a succession of events. Although the
principles of statics are time independent, this quantity plays an important
role in the study of dynamics.
Mass. Mass is a measure of a quantity of matter that is used to compare
the action of one body with that of another. This property manifests itself
as a gravitational attraction between two bodies and provides a measure
of the resistance of matter to a change in velocity.
Force. In general, force is considered as a “push” or “pull” exerted by
one body on another. This interaction can occur when there is direct
contact between the bodies, such as a person pushing on a wall, or it
can occur through a distance when the bodies are physically separated.
Examples of the latter type include gravitational, electrical, and magnetic
forces. In any case, a force is completely characterized by its magnitude,
direction, and point of application.
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1.2
Fundamental Concepts
27
Idealizations. Models or idealizations are used in mechanics in
order to simplify application of the theory. Here we will consider three
important idealizations.
Particle. A particle has a mass, but a size that can be neglected.
For example, the size of the earth is insignificant compared to the size
of its orbit, and therefore the earth can be modeled as a particle when
studying its orbital motion. When a body is idealized as a particle,
the principles of mechanics reduce to a rather simplified form since
the geometry of the body will not be involved in the analysis of the
problem.
Rigid Body. A rigid body can be considered as a combination of a
large number of particles in which all the particles remain at a fixed
distance from one another, both before and after applying a load.
This model is important because the body’s shape does not change
when a load is applied, and so we do not have to consider the type
of material from which the body is made. In most cases the actual
deformations occurring in structures, machines, mechanisms, and the
like are relatively small, and the rigid-body assumption is suitable for
analysis.
1
Three forces act on the ring. Since these
forces all meet at a point, then for any
force analysis, we can assume the ring to be
represented as a particle.
Concentrated Force. A concentrated force represents the effect
of a loading which is assumed to act at a point on a body. We can
represent a load by a concentrated force, provided the area over which
the load is applied is very small compared to the overall size of the
body. An example would be the contact force between a wheel and
the ground.
Steel is a common engineering material that does not deform very much under load.
Therefore, we can consider this railroad wheel to be a rigid body acted upon by the
concentrated force of the rail.
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28
C h a p t e r 1 G e n e r a l P r i n c i p l e s
Newton’s Three Laws of Motion. Engineering mechanics is
formulated on the basis of Newton’s three laws of motion, the validity
of which is based on experimental observation. These laws apply to the
motion of a particle as measured from a nonaccelerating reference frame.
They may be briefly stated as follows.
1
First Law. A particle originally at rest, or moving in a straight line with
constant velocity, tends to remain in this equilibrium state provided the
particle is not subjected to an unbalanced force, Fig. 1–1a.
F1
F2
v
F3
Equilibrium
(a)
Second Law. A particle acted upon by an unbalanced force F
experiences an acceleration a that has the same direction as the force and
a magnitude that is directly proportional to the force, Fig. 1–1b.* If the
particle has a mass m, this law may be expressed mathematically as
F = ma
(1–1)
a
F
Accelerated motion
(b)
Third Law. The mutual forces of action and reaction between two
particles are equal, opposite, and collinear, Fig. 1–1c.
force of A on B
F
F
A
B
force of B on A
Action–reaction
(c)
Fig. 1–1
*Stated another way, the unbalanced force acting on the particle is proportional to the time
rate of change of the particle’s linear momentum.
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1.3 The International System of Units
29
Newton’s Law of Gravitational Attraction. Shortly after
formulating his three laws of motion, Newton postulated a law governing the
gravitational attraction between any two particles. Stated mathematically,
F = G
m1m2
r2
(1–2)
1
where
F = force of gravitation between the two particles
G = universal constant of gravitation; according to experimental
evidence, G = 66.73 ( 10 - 12 ) m3 > ( kg # s2 )
m1, m2 = mass of each of the two particles
r = distance between the two particles
Weight.
According to Eq. 1–2, any two particles or bodies have a
mutual attractive (gravitational) force acting between them. In the case
of a particle located at or near the surface of the earth, however, the
only gravitational force having any sizable magnitude is that between
the earth, because of its very large mass, and the particle. Consequently,
this force, called the weight, will be the only gravitational force we will
consider.
From Eq. 1–2, if the particle has a mass m1 = m, and we assume the
earth is a nonrotating sphere of constant density and having a mass
m2 = Me, then if r is the distance between the earth’s center and the
particle, the weight W of the particle becomes
W = G
If we let g = GMe >r 2, we have
mMe
The astronaut’s weight is diminished since
she is far removed from the gravitational
field of the earth.
r2
W = mg
(1–3)
If we allow the particle to fall downward, then neglecting air resistance,
the only force acting on the particle is its weight, and so Eq. 1–1 becomes
W = ma. Comparing this result with Eq. 1–3, we see that a = g.
In other words, g is the acceleration due to gravity. Since it depends on r,
then the weight of the particle or body is not an absolute quantity. Instead,
its magnitude depends upon the elevation where the measurement was
made. For most engineering calculations, however, g is determined at sea
level and at a latitude of 45°, which is considered the “standard location.”
Refer to the companion website for Lecture
Summary and Quiz videos.
1.3 THE INTERNATIONAL SYSTEM
OF UNITS
The four basic quantities—length, time, mass, and force—are not all
independent from one another; in fact, they are related by Newton’s
second law of motion, F = ma. Because of this, the units used to measure
these quantities cannot all be selected arbitrarily. The equality F = ma is
maintained only if three of the four units, called base units, are defined
and the fourth unit is then derived from the equation.
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30
C h a p t e r 1 G e n e r a l P r i n c i p l e s
1 kg
1
9.81 N
(a)
Fig. 1-2
The International System of units, abbreviated SI after the French
Système International d’Unités, is a modern version of the metric
system which has received worldwide recognition. As shown in Table 1–1,
the system defines length in meters (m), time in seconds (s), and mass in
kilograms (kg).† The unit of force, called a newton (N), is derived from
F = ma. Thus, 1 newton is equal to a force required to give 1 kilogram of
mass an acceleration of 1 m>s2 ( N = kg # m>s2 ) . Think of this force as
the weight of a small apple.
If the weight of a body located at the “standard location” is to
be determined in newtons, then Eq. 1–3 must be applied. Here
measurements give g = 9.806 65 m>s2; however, for calculations, the
value g = 9.81 m>s2 will be used. Thus,
W = mg
( g = 9.81 m>s2 )
(1–4)
Therefore, a body of mass 1 kg has a weight of 9.81 N, a 2-kg body weighs
19.62 N, and so on, Fig. 1–2.
TABLE 1–1 International System of Units
Quantity
Length
Time
Mass
Force
SI Units
meter
second
kilogram
newton*
m
s
kg
N
kg # m
¢
s2
≤
*Derived unit.
Prefixes. When a numerical quantity is either very large or very
small, the SI units used to define its size may be modified by using a
prefix. Some of these prefixes used are shown in Table 1–2. Each
represents a multiple or submultiple of a unit which, if applied
successively, moves the decimal point of a numerical quantity to every
third place.‡ For example, 4 000 000 N = 4 000 kN (kilo-newton) = 4
MN (mega-newton), or 0.005 m = 5 mm (milli-meter). Notice that the SI
system does not include the multiple deca (10) or the submultiple centi
(0.01), which form part of the metric system. Except for some volume
and area measurements, the use of these prefixes is generally avoided in
science and engineering.
†
Historically, the meter was defined as 1/10,000,000 the distance from the Equator to
the North Pole, and the kilogram is 1/1000 of a cubic meter of water.
‡
The kilogram is the only base unit that is defined with a prefix.
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1.3 The International System of Units
TABLE 1–2
31
Prefixes
Exponential Form
Prefix
SI Symbol
Multiple
1 000 000 000
1 000 000
1 000
109
106
103
giga
mega
kilo
G
M
k
Submultiple
0.001
0.000 001
0.000 000 001
10-3
10-6
10-9
milli
micro
nano
m
m
n
1
Rules for Use. Here are a few of the important rules that describe
the proper use of the various SI symbols:
• Quantities defined by several units which are multiples of one
another are separated by a dot to avoid confusion with prefix
notation, as indicated by N = kg # m>s2 = kg # m # s - 2. Also, m # s
(meter-second), whereas ms (milli-second).
• The exponential power on a unit having a prefix refers to both the
unit and its prefix. For example, mN2 = (mN) 2 = mN # mN. Likewise,
mm2 represents (mm) 2 = mm # mm.
• With the exception of the base unit the kilogram, in general avoid
the use of a prefix in the denominator of composite units. For
example, do not write N>mm, but rather kN>m; also, m>mg should
be written as Mm>kg.
• When performing calculations, represent the numbers in terms of
their base or derived units by converting all prefixes to powers of
10. The final result should then be expressed using a single prefix.
Also, after calculation, it is best to keep numerical values between 0.1
and 1000; otherwise, a suitable prefix should be chosen. For example,
(50 kN) (60 nm) = 3 50 ( 103 ) N 4 3 60 ( 10 - 9 ) m 4
= 3000 ( 10 - 6 ) N # m = 3 ( 10 - 3 ) N # m = 3 mN # m
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32
C h a p t e r 1 G e n e r a l P r i n c i p l e s
1.4
1
NUMERICAL CALCULATIONS
Numerical work in engineering practice is most often performed by
using handheld calculators and computers. It is important, however, that
the answers to any problem be reported with justifiable accuracy using
appropriate significant figures. In this section we will discuss these topics
together with some other important aspects involved in all engineering
calculations.
Dimensional Homogeneity. The terms of any equation used to
describe a physical process must be dimensionally homogeneous; that is,
each term must be expressed in the same units. Provided this is the case,
all the terms of an equation can then be combined if numerical values
are substituted for the variables. Consider, for example, the equation
s = vt + 12 at 2, where, in SI units, s is the position in meters, m, t is time in
seconds, s, v is velocity in m>s and a is acceleration in m>s2. Regardless of
how this equation is evaluated, it maintains its dimensional homogeneity.
In the form stated, each of the three terms is expressed in meters
3 m, ( m>s ) s, ( m>s2 ) s2 4 or solving for a, a = 2s>t 2 - 2v>t, the terms are
each expressed in units of m>s2 3 m>s2, m>s2, ( m>s ) >s 4 .
Keep in mind that problems in mechanics always involve the solution
of dimensionally homogeneous equations, and so this fact can then be
used as a partial check for algebraic manipulations of an equation.
Significant Figures. The number of significant figures contained
in any number determines the accuracy of the number. For instance,
the number 4981 contains four significant figures. However, if zeros
occur at the end of a whole number, it may be unclear as to how many
significant figures the number represents. For example, 23 400 might
have three (234), four (2340), or five (23 400) significant figures. To avoid
these ambiguities, we will use engineering notation to report a result.
This requires that numbers be rounded off to the appropriate number
of significant digits and then expressed in multiples of (103), such as
(103), (106), or (10–9). For instance, if 23 400 has five significant figures,
it is written as 23.400(103), but if it has only three significant figures, it is
written as 23.4(103).
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1.4 Numerical Calculations
If zeros occur at the beginning of a number that is less than one,
then the zeros are not significant. For example, 0.008 21 has three
significant figures. Using engineering notation, this number is expressed
as 8.21 1 10-3 2 . Likewise, 0.000 582 can be expressed as 0.582 1 10-3 2 or
582 1 10-6 2 .
33
1
Rounding Off Numbers. Rounding off a number is necessary
so that the accuracy of the result will be the same as that of the problem
data. As a general rule, any numerical figure ending in a number greater
than five is rounded up and a number less than five is not rounded up. The
rules for rounding off numbers are best illustrated by examples. Suppose
the number 3.5587 is to be rounded off to three significant figures.
Because the fourth digit (8) is greater than 5, the third number is rounded
up to 3.56. Likewise 0.5896 becomes 0.590 and 9.3866 becomes 9.39. If we
round off 1.341 to three significant figures, because the fourth digit (1)
is less than 5, then we get 1.34. Likewise 0.3762 becomes 0.376 and 9.871
becomes 9.87. There is a special case for any number that ends in a 5. As a
general rule, if the digit preceding the 5 is an even number, then this digit
is not rounded up. If the digit preceding the 5 is an odd number, then it
is rounded up. For example, 75.25 rounded off to three significant digits
becomes 75.2, 0.1275 becomes 0.128, and 0.2555 becomes 0.256.
Calculations. When a sequence of calculations is performed, it is
best to store the intermediate results in the calculator. In other words, do
not round off calculations until expressing the final result. This procedure
maintains precision throughout the series of steps to the final solution.
In this book we will generally round off the answers to three significant
figures since most of the data in engineering mechanics, such as geometry
and loads, may be reliably measured to this accuracy.
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34
C h a p t e r 1 G e n e r a l P r i n c i p l e s
1.5 GENERAL PROCEDURE FOR
ANALYSIS
Attending a lecture, reading this book, and studying the example
problems helps, but the most effective way of learning the principles of
engineering mechanics is to solve problems. To be successful at this, it is
important to always present the work in a logical and orderly manner, as
suggested by the following sequence of steps:
1
• Read the problem carefully and try to correlate the actual physical
situation with the theory studied.
• Tabulate the problem data and draw to a large scale any necessary
When solving problems, do the work as
neatly as possible. Being neat will stimulate
clear and orderly thinking, and vice versa.
diagrams.
• Apply the relevant principles, generally in mathematical form. When
writing any equations, be sure they are dimensionally homogeneous.
• Solve the necessary equations, and report the answer with no more
than three significant figures.
• Study the answer with technical judgment and common sense to
determine whether or not it seems reasonable.
I MPO RTA N T PO I N T S
• Statics is the study of bodies that are at rest or move with constant
Refer to the companion website for Lecture
Summary and Quiz videos.
M01_HIBB4048_15_GE_C01.indd 34
velocity.
• A particle has a mass but a size that can be neglected, and a rigid
body does not deform under load.
• A force is considered as a “push” or “pull” of one body on another.
• Concentrated forces are assumed to act at a point on a body.
• Newton’s three laws of motion should be memorized.
• Mass is measure of a quantity of matter that does not change
from one location to another. Weight refers to the gravitational
attraction of the earth on a body or quantity of mass. Its magnitude
depends upon the elevation at which the mass is located.
• In the SI system the unit of force, the newton, is a derived unit.
The meter, second, and kilogram are base units.
• Prefixes G, M, k, m, m, and n are used to represent large and small
numerical quantities. Their exponential size should be known,
along with the rules for using the SI units.
• Perform numerical calculations with several significant figures,
and then report the final answer to three significant figures.
• Algebraic manipulations of an equation can be checked in part by
verifying that the equation remains dimensionally homogeneous.
• Know the rules for rounding off numbers.
07/07/22 2:26 PM
1.5 General Procedure for Analysis
EXAMPLE
35
1.1
Convert 100 km>h to m>s and 24 m>s to km>h.
SOLUTION
Since 1 km = 1000 m and 1 h = 3600 s, the factors of conversion are
arranged in the following order, so that a cancellation of the units can
be applied:
100 km 1000 m
1h
¢
≤¢
≤
h
km
3600 s
100 km>h =
100(103) m
= 27.8 m>s
3600 s
=
24 m>s = a
=
1
24 m
1 km
3600 s
ba
ba
b
s
1000 m
1h
86.4 (103) km
= 86.4 km/h
1000 h
Ans.
Ans.
Note: Remember to round off the final answer to three significant
figures.
EXAMPLE
1.2
Convert the density of steel 7.85 g/cm3 to kg/m3.
SOLUTION
Using 1 kg = 1000 g and 1 m = 100 cm, and arrange the conversion
factor in such a way that g and cm3 can be canceled out.
7.85 g/cm3 = a
= a
7.85 g
3
cm
ba
1 kg
100 cm 3
ba
b
1000 g
1m
1 kg
1003 cm3
b
a
b
a
b
1000 g
cm3
1 m3
7.85 g
= 7.85(103) kg/m3
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Ans.
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36
C h a p t e r 1 G e n e r a l P r i n c i p l e s
EXAMPLE
1
1.3
Evaluate each of the following and express with SI units having an
appropriate prefix: (a) (50 mN)(6 GN), (b) (400 mm)(0.6 MN)2,
(c) 45 MN3 >900 Gg.
SOLUTION
First convert each number to base units, perform the indicated
operations, then choose an appropriate prefix.
Part (a)
(50 mN)(6 GN) = 3 50 ( 10-3 ) N 4 3 6 ( 109 ) N 4
= 300 ( 106 ) N2
= 300 ( 106 ) N2 a
= 300 kN2
1 kN
1 kN
ba
b
103 N 103 N
Ans.
Note: Keep in mind the convention kN2 = (kN) 2 = 106 N2.
Part (b)
(400 mm)(0.6 MN)2 = 3 400 ( 10-3 ) m 4 3 0.6 ( 106 ) N 4 2
= 3 400 ( 10-3 ) m 4 3 0.36 ( 1012 ) N2 4
= 144 ( 109 ) m # N2
= 144 Gm # N2
Ans.
We can also write
144 ( 109 ) m # N2 = 144 ( 109 ) m # N2 a
= 0.144 m # MN2
1 MN 1 MN
ba 6 b
106 N
10 N
Ans.
Part (c)
45 ( 106 N ) 3
45 MN3
=
900 Gg
900 ( 106 ) kg
= 50 ( 109 ) N3 >kg
= 50 ( 109 ) N3 a
= 50 kN3 >kg
M01_HIBB4048_15_GE_C01.indd 36
1 kN 3 1
b
103 N kg
Ans.
07/07/22 2:26 PM
Problems
37
PROBLEMS
The answers to all but every fourth problem (asterisk)
are given in the back of the book.
1–1. Round off the following numbers to three significant
figures: (a) 58 342 m, (b) 68.534 s, (c) 2553 N, (d) 7555 kg.
1–2. Represent each of the following combinations of
units in the correct SI form using an appropriate prefix:
(a) Mg>mm, (b) mN >ms, (c) mm # Mg.
1–13. Using the SI system of units, show that Eq. 1–2 is
a dimensionally homogeneous equation which gives F
in ­
newtons. Determine to three significant figures the
1
gravitational force acting between two spheres that are
­
touching each other. The mass of each sphere is 200 kg and
the radius is 300 mm.
1–14. Evaluate each of the following and express with
an appropriate prefix: (a) (430 kg)2, (b) (0.002 mg)2, and
(c) (230 m)3.
1–3. Represent each of the following combinations of
units in the correct SI form using an appropriate prefix:
(a) kN>ms, (b) Mg>mN, (c) MN>(kg # ms).
1–15. Evaluate each of the following to three significant
­figures and express each answer in SI units using an a­ ppropriate
prefix: (a) (200 kN)2, (b) (0.005 mm)2, (c) (400 m)3.
*1–4. Determine the mass of an object that has a weight
of (a) 20 mN, (b) 150 kN, (c) 60 MN. Express the answer to
three significant figures.
*1–16. Evaluate each of the following to three significant
figures and express each answer in SI units using an
appropriate prefix: (a) (212 mN)2, (b) (52 800 ms)2,
(c) [548(106)]1>2 ms.
1–5. Round off the following numbers to three significant
figures: (a) 3.455 55 m, (b) 45.556 s, (c) 5555 N, (d) 4525 kg.
1–6. Represent each of the following as a number between
0.1 and 1000 using an appropriate prefix: (a) 45 320 kN,
(b) 568(105) mm, (c) 0.005 63 mg.
1–7. Represent each of the following combinations
of units in the correct SI form: (a) Mg>ms, (b) N>mm,
(c) mN>(kg # ms).
*1–8. Represent each of the following quantities in the correct
SI form using an appropriate prefix: (a) 0.000 431 kg,
(b) 35.3 ( 103 ) N, (c) 0.005 32 km.
1–9. Represent each of the following combinations of
units in the correct SI form using an appropriate prefix:
(a) mMN, (b) N>mm, (c) MN>ks2, (d) kN>ms.
1–10. Represent each of the following combinations of
units in the correct SI form using an appropriate prefix:
(a) m>ms, (b) mkm, (c) ks>mg, (d) km # mN.
1–17. Evaluate (204 mm)(0.004 57 kg)>(34.6 N) to three
significant figures and express the answer in SI units using
an appropriate prefix.
1–18. What is the weight in newtons of an object that has a
mass of (a) 8 kg, (b) 0.04 g, (c) 760 Mg?
1–19. A concrete column has a diameter of 350 mm and
a length of 2 m. If the density (mass>volume) of concrete is
2.45 Mg>m3, determine the weight of the column.
*1–20. Two particles have a mass of 8 kg and 12 kg,
respectively. If they are 800 mm apart, determine the force
of gravity acting between them. Compare this result with
the weight of each particle.
1–21. If a man weighs 690 newtons on earth, specify
(a) his mass in kilograms. If the man is on the moon, where
the acceleration due to gravity is gm = 1.61 m>s2, determine
(b) his weight in newtons, and (c) his mass in kilograms.
1–11. Represent each of the following combinations of
units in the correct SI form using an appropriate prefix:
(a) GN # mm, (b) kg>mm, (c) N>ks2, (d) kN>ms.
*1–12. A rocket has a mass 3.529(106) kg on earth. Specify
(a) its mass in SI units, and (b) its weight in SI units. If the
rocket is on the moon, where the acceleration due to gravity is gm = 1.61 m>s2, determine to three significant figures
(c) its weight in SI units, and (d) its mass in SI units.
M01_HIBB4048_15_GE_C01.indd 37
07/07/22 2:26 PM
CHAPTER
2
This electric transmission tower is stabilized by cables that exert forces on the tower
at their points of connection. In this chapter we will show how to express these
forces as Cartesian vectors, and then determine their resultant.
M02_HIBB4048_15_GE_C02.indd 38
07/07/2022 16:33
FORCE
VECTORS
Lecture Summary and Quiz,
Example, and Problemsolving videos are available
where this icon appears.
CHAPTER OBJECTIVES
■■ To show how to add forces and resolve them into components
using the parallelogram law.
■■ To
express force and position in Cartesian vector form and
explain how to determine the vector’s magnitude and direction.
■■ To introduce the dot product in order to use it to find the angle
between two vectors or the projection of one vector onto another.
2.1
SCALARS AND VECTORS
Many physical quantities in engineering mechanics are measured using
either scalars or vectors.
Scalar. A scalar is any positive or negative physical quantity that can
be completely specified by its magnitude. Examples of scalar quantities
include length, mass, and time.
Vector. A vector is any physical quantity that requires both a
magnitude and a direction for its complete description. Examples of
vectors encountered in statics are force, position, and moment. A vector
is shown graphically by an arrow, Fig. 2–1. The length of the arrow
represents the magnitude of the vector, and the angle u between the
vector and a fixed axis defines the direction of its line of action. The head
or tip of the arrow indicates the sense of direction of the vector.
In print, vector quantities are represented by boldface letters such as
A, and the magnitude of a vector is italicized, A. For handwritten work,
it is often convenient to denote a vector quantity by simply drawing an
S
arrow above it, A .
M02_HIBB4048_15_GE_C02.indd 39
Line of action
A
Magnitude
u Direction
Sense
Fig. 2–1
39
07/07/2022 16:33
40
C h a p t e r 2 F o r c e V e c t o r s
2.2
VECTOR OPERATIONS
Multiplication and Division of a Vector by a Scalar. If
a vector is multiplied or divided by a positive scalar, its magnitude is
changed by that amount. Multiplying or dividing by a negative scalar
will also change the directional sense of the vector. Graphic examples of
these operations are shown in Fig. 2–2.
2A
2
A
2A
2 0.5 A
Scalar multiplication and division
Fig. 2–2
Vector Addition.
When adding two vectors together it is important
to account for both their magnitudes and their directions. To do this
we must use the parallelogram law of addition. To illustrate, the two
component vectors A and B in Fig. 2–3a are added to form a resultant
vector R = A + B using the following procedure:
• First join the tails of the components at a point to make them
concurrent, Fig. 2–3b.
• From the head of B, draw a line parallel to A. Draw another line
from the head of A that is parallel to B. These two lines intersect at
point P to form the adjacent sides of a parallelogram.
• The diagonal of this parallelogram that extends to P forms R, which
then represents the resultant vector R = A + B, Fig. 2–3c.
A
A
A
R
P
B
B
B
R5A1B
Parallelogram law
(a)
(b)
(c)
Fig. 2–3
M02_HIBB4048_15_GE_C02.indd 40
07/07/2022 16:33
2.2 Vector Operations
41
We can also add B to A, Fig. 2–4a, using the triangle rule, which is a special
case of the parallelogram law, whereby vector B is added to vector A in
a “head-to-tail” fashion, i.e., by connecting the tail of B to the head of A,
Fig. 2–4b. The resultant R extends from the tail of A to the head of B. In
a similar manner, R can also be obtained by adding A to B, Fig. 2–4c. By
comparison, it is seen that vector addition is commutative; in other words,
the vectors can be added in either order, i.e., R = A + B = B + A.
A
B
A
2
R
R
B
A
B
(a)
R5A1B
R5B1A
Triangle rule
Triangle rule
(b)
(c)
Fig. 2–4
As a special case, if the two vectors A and B are collinear, i.e.,
both have the same line of action, the parallelogram law reduces to an
algebraic or scalar addition R = A + B, as shown in Fig. 2–5.
R
A
B
R5A1B
Vector Subtraction. The resultant of the difference between two
vectors A and B of the same type may be expressed as
Addition of collinear vectors
Fig. 2–5
R′ = A - B = A + 1 -B2
This vector sum is shown graphically in Fig. 2–6. Subtraction is therefore
defined as a special case of addition, so the rules of vector addition also
apply to vector subtraction.
B
A
R9
A
or
R9
A
B
B
Parallelogram law
Triangle construction
Vector subtraction
Fig. 2–6
M02_HIBB4048_15_GE_C02.indd 41
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42
C h a p t e r 2 F o r c e V e c t o r s
2.3
F2
F1
FR
2
The parallelogram law must be used to
determine the resultant of the two forces
acting on the hook.
VECTOR ADDITION OF FORCES
Experimental evidence has shown that a force is a vector quantity since
it has a specified magnitude, direction, and sense and it adds according
to the parallelogram law. Two common problems in statics involve
either finding the resultant force, knowing its components, or resolving
a known force into two components. We will now describe how each of
these problems is solved using the parallelogram law.
Finding a Resultant Force. The two component forces F1 and
F2 acting on the pin in Fig. 2–7a are added together to form the resultant
force FR = F1 + F2, using the parallelogram law as shown in Fig. 2–7b.
From this construction, or using the triangle rule, Fig. 2–7c, we can apply
the law of cosines or the law of sines to the triangle in order to obtain the
magnitude of the resultant force and its direction.
F1
F1
F1
FR
F2
F2
FR
F2
FR 5 F1 1 F2
y
Fy
F
(a)
u
(b)
(c)
Fig. 2–7
Fu
Finding the Components of a Force. Sometimes it is necessary
Using the parallelogram law the supporting
force F can be resolved into components
acting along the u and v axes.
M02_HIBB4048_15_GE_C02.indd 42
to resolve a force into two components in order to study its pulling or
pushing effect in two specific directions. For example, in Fig. 2–8a, F is to
be resolved into two components along the two members, defined by the
u and v axes. In order to determine the magnitude of each component, a
parallelogram is constructed first, by drawing lines starting from the tip
of F, one line parallel to u, and the other line parallel to v. These lines intersect
the v and u axes, forming a parallelogram. The force components Fu and Fv are
established by simply joining them to the tail of F, to the intersection points
on the u and v axes, Fig. 2–8b. This parallelogram can be reduced to a triangle,
which represents the triangle rule, Fig. 2–8c. From this, the law of sines can be
applied to determine the unknown magnitudes of the components.
07/07/2022 16:33
43
2.3 Vector Addition of Forces
v
v
F
F
Fv
F
Fv
u
u
Fu
(a)
(b)
Fu
(c)
Fig. 2–8
2
Addition of Several Forces. If more than two forces are to be
added, successive applications of the parallelogram law can be carried out
in order to obtain the resultant force. For example, if three forces F1, F2,
F3 act at a point O, Fig. 2–9, the resultant of any two of the forces is found,
say, F1 + F2, and then this resultant is added to the third force, yielding
the resultant of all three forces; i.e., FR = 1F1 + F2 2 + F3. Using the
parallelogram law to add more than two forces, as shown here, generally
requires extensive geometric and trigonometric calculation to determine
the magnitude and direction of the resultant. Instead, problems of this
type are easily solved by using the “rectangular-component method,”
which is explained in the next section.
F1 1 F2
FR
F2
F1
F3
O
Fig. 2–9
FR
F1 1 F2
F2
F1
F3
The resultant force FR on the hook requires
the addition of F1 + F2, then this resultant is
added to F3.
M02_HIBB4048_15_GE_C02.indd 43
07/07/2022 16:33
44
C h a p t e r 2 F o r c e V e c t o r s
I MPO RTA N T PO I N T S
• A scalar is a positive or negative number.
• A vector is a quantity that has a magnitude, direction, and sense.
• Multiplication or division of a vector by a scalar will change
the magnitude of the vector. The sense of the vector will
change if the scalar is negative.
• Vectors are added or subtracted using the parallelogram law or
the triangle rule.
• As a special case, if the vectors are collinear, the resultant is
2
formed by an algebraic or scalar addition.
F1
FR
F2
(a)
PROCEDURE FOR ANALYSIS
Problems that involve the addition of two forces can be solved as
follows:
v
F
u
Fv
Fu
(b)
A
c
b
• If a force F is to be resolved into components along two axes
B
a
C
Cosine law:
C 5 A2 1 B2 2 2AB cos c
Sine law:
A 5 B 5 C
sin a sin b sin c
(c)
Fig. 2–10
Refer to the companion website for Lecture
Summary and Quiz videos.
M02_HIBB4048_15_GE_C02.indd 44
Parallelogram Law.
• Sketch the two “component” forces F1 and F2 added together
according to the parallelogram law, yielding the resultant force
FR that forms the diagonal of the parallelogram, Fig. 2–10a.
u and v, then start at the head of force F and construct lines
parallel to the axes, thereby forming the parallelogram, Fig. 2–10b.
The sides of the parallelogram represent the components, Fu and Fv.
• Label all the known and unknown force magnitudes and the angles
on the sketch and identify the two unknowns as the magnitude
and direction of FR, or the magnitudes of its components.
Trigonometry.
• Redraw a half portion of the parallelogram to illustrate the
triangular head-to-tail addition of the components.
• From this triangle, the magnitude of the resultant force can
be determined using the law of cosines, and its direction is
determined from the law of sines. The magnitudes of two
force components are determined from the law of sines. The
formulas are given in Fig. 2–10c.
07/07/2022 16:34
2.3 Vector Addition of Forces
EXAMPLE
45
2.1
The screw eye in Fig. 2–11a is subjected to two forces, F1 and F2.
Determine the magnitude and direction of the resultant force.
108
F2 5 150 N
A
150 N
658
1158
F1 5 100 N
108
158
2
FR
3608 2 2(658)
5 1158
2
100 N
u
158
908 2 258 5 658
(a)
(b)
SOLUTION
Parallelogram Law. The parallelogram is formed by drawing a
line from the head of F1 that is parallel to F2, and another line from
the head of F2 that is parallel to F1. The resultant force FR extends to
where these lines intersect at point A, Fig. 2–11b. The two unknowns
are the magnitude of FR and the angle u (theta).
FR
Trigonometry. From the parallelogram, the vector triangle is
constructed, Fig. 2–11c. Using the law of cosines
FR = 21100 N2 2 + 1150 N2 2 - 21100 N21150 N2 cos 115°
= 210 000 + 22 500 - 30 0001 -0.42262 = 212.6 N
= 213 N
Applying the law of sines to determine u,
150 N
212.6 N
=
sin u
sin 115°
sin u =
Ans.
150 N
1158
u
f
158
100 N
(c)
Fig. 2–11
150 N
(sin 115°)
212.6 N
u = 39.8°
Thus, the direction f (phi) of FR, measured from the horizontal, is
f = 39.8° + 15.0° = 54.8°
Ans.
note: The results seem reasonable, since Fig. 2–11b shows FR to have a
magnitude larger than its components and a direction that is between them.
M02_HIBB4048_15_GE_C02.indd 45
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46
C h a p t e r 2 F o r c e V e c t o r s
EXAMPLE
2.2
Resolve the horizontal 600-N force in Fig. 2–12a into components
acting along the u and v axes and determine the magnitudes of these
components.
u
2
u
B
Fu
30
30
30
30
A
600 N
Fv
120
600 N
120
Fu
30
30
120
30
Fv
600 N
C
v
(a)
v
(b)
(c)
Fig. 2–12
SOLUTION
The parallelogram is constructed by extending a line from the head
of the 600-N force parallel to the v axis until it intersects the u axis at
point B, Fig. 2–12b. The arrow from A to B represents Fu. Similarly, the
line extended from the head of the 600-N force drawn parallel to the
u axis intersects the v axis at point C, which gives Fv.
The vector addition using the triangle rule is shown in Fig. 2–12c. The two
unknowns are the magnitudes of Fu and Fv. Applying the law of sines,
Fu
600 N
=
sin 120°
sin 30°
Fu = 1039 N
Fv
600 N
=
sin 30°
sin 30°
Fv = 600 N
Ans.
Ans.
note: The result for Fu shows that sometimes a component can have a
greater magnitude than the resultant.
M02_HIBB4048_15_GE_C02.indd 46
07/07/2022 16:34
2.3 Vector Addition of Forces
EXAMPLE
47
2.3
Determine the magnitude of the component force F in Fig. 2–13a and
the magnitude of the resultant force FR if FR is directed along the
positive y axis.
y
2
y
45
F
FR
200 N
45
F
45
30
45
60
75
30
30
(a)
F
45
FR
60
200 N
200 N
(b)
(c)
Fig. 2–13
SOLUTION
The parallelogram law of addition is shown in Fig. 2–13b, and the
triangle rule is shown in Fig. 2–13c. The magnitudes of FR and F are
the two unknowns. They can be determined by applying the law of
sines.
F
200 N
=
sin 60°
sin 45°
F = 245 N
Ans.
FR
200 N
=
sin 75°
sin 45°
FR = 273 N
M02_HIBB4048_15_GE_C02.indd 47
Ans.
07/07/2022 16:34
48
C h a p t e r 2 F o r c e V e c t o r s
EXAMPLE
2.4
It is required that the resultant force acting on the eyebolt in Fig. 2–14a
be directed along the positive x axis and that F2 have a minimum
magnitude. Determine this magnitude, the angle u, and the
corresponding resultant force.
F1 5 800 N
2
608
x
F1 5 800 N
F2
F1 5 800 N
u
F2
608
u
FR
608
x
FR
F2
(a)
(b)
x
u 5 908
(c)
Fig. 2–14
SOLUTION
The triangle rule for FR = F1 + F2 is shown in Fig. 2–14b. Since the
magnitudes (lengths) of FR and F2 are not specified, then F2 can
actually be any vector that has its head touching the line of action of
FR, Fig. 2–14c. However, as shown, the magnitude of F2 is a minimum
or the shortest length when its line of action is perpendicular to the
line of action of FR, that is, when
u = 90°
Ans.
Since the vector addition now forms the shaded right triangle, the two
unknown magnitudes can be obtained by trigonometry.
Refer to the companion website for a self quiz of these
Example problems.
M02_HIBB4048_15_GE_C02.indd 48
FR = 1800 N2cos 60° = 400 N
F2 = 1800 N2sin 60° = 693 N
Ans.
Ans.
13/07/2022 17:40
49
Fundamental Problems
FUN DAMEN TA L PR O B L EM S
Partial solutions and answers to all Fundamental Problems are given in the back of the book. Video solutions are also available
for select problems on the companion website.
F2–1. Determine the magnitude of the resultant force and
its direction measured clockwise from the positive x axis.
F2–4. Resolve the 30-N force into components along the
u and v axes, and determine the magnitude of each of these
components.
v
x
608
30 N
15
458
30
2
u
2 kN
6 kN
Prob. F2–1
F2–2. Two forces act on the hook. Determine the magnitude
of the resultant force.
Prob. F2–4
F2–5. The force F = 450 N acts on the frame. Resolve this
force into components acting along members AB and AC,
and determine the magnitude of each component.
A
30
C
45
308
408
450 N
200 N
500 N
Prob. F2–2
B
F2–3. Determine the magnitude of the resultant force and
its direction measured counterclockwise from the positive
x axis.
y
Prob. F2–5
F2–6. If force F is to have a component along the u axis of
Fu = 6 kN, determine the magnitude of F and the magnitude
of its component Fv along the v axis.
u
800 N
F
458
1058
x
308
v
600 N
Prob. F2–3
M02_HIBB4048_15_GE_C02.indd 49
Prob. F2–6
07/07/2022 16:34
50
C h a p t e r 2 F o r c e V e c t o r s
P ROBLEMS
2–1. If u = 60° and F = 450 N, determine the magnitude
of the resultant force and its direction, measured
counterclockwise from the positive x axis.
2–2. If the magnitude of the resultant force is to be 500 N,
directed along the positive y axis, determine the magnitude
of force F and its direction u.
2–6. If FB = 2 kN and the resultant force acts along the
positive u axis, determine the magnitude of the resultant
force and the angle u.
2–7. If the resultant force is required to act along the
positive u axis and have a magnitude of 5 kN, determine the
required magnitude of FB and its direction u.
y
2
y
F
FA
u
x
15
u 30
3 kN
x
A
700 N
B
u
FB
Probs. 2–1/2
Probs. 2–6/7
2–3. Determine the magnitude of the resultant force
FR = F1 + F2 and its direction, measured clockwise from
the positive u axis.
*2–8. Two forces are applied at the end of a screw
eye in order to remove the post. Determine the angle
u10° … u … 90°2 and the magnitude of force F so that
the resultant force acting on the post is directed vertically
upward and has a magnitude of 750 N.
*2–4. Resolve the force F1 into components along the u
and v axes and determine the magnitudes of the components.
2–5. Resolve the force F2 into components along the u and
v axes and determine the magnitudes of the components.
y
F
500 N
u
308
x
v
308
758
F1 5 4 kN
308
u
F2 5 6 kN
Probs. 2–3/4/5
M02_HIBB4048_15_GE_C02.indd 50
Prob. 2–8
07/07/2022 16:34
Problems
2–9. If u = 60°, determine the magnitude of the resultant
force and its direction measured clockwise from the horizontal.
2–10. Determine the angle u for connecting member A to
the plate so that the resultant force of FA and FB is directed
horizontally to the right. Also, what is the magnitude of the
resultant force?
2–13. If u = 30° and T = 6 kN, determine the magnitude
of the resultant force acting on the eyebolt and its direction
measured clockwise from the positive x axis.
2–14. If u = 60° and T = 5 kN, determine the magnitude
of the resultant force acting on the eyebolt and its direction
measured clockwise from the positive x axis.
2–15. If the magnitude of the resultant force is to be 9 kN
directed along the positive x axis, determine the magnitude
of force T acting on the eyebolt and its angle u.
FA 5 8 kN
u
51
A
2
y
T
u
408
B
FB 5 6 kN
x
45
Probs. 2–9/10
8 kN
Probs. 2–13/14/15
2–11. Determine the magnitude of the resultant force
FR = F1 + F2 and its orientation u, measured clockwise
from the positive x axis.
*2–12. Determine the magnitude of the resultant
force FR = F1 + F3 and its orientation u, measured
counter­clockwise from the positive x axis.
*2–16. The pelvis P is connected to the femur F at A
using three different muscles, which exert the forces shown
on the femur. Determine the resultant force and specify its
orientation u, measured counterclockwise from the positive x
axis.
y
F3 5 250 N
y
308
120 N
x
60 N
13 12
5
80 N
308
458
F1 5 400 N
F2 5 360 N
A
308
x
F
Probs. 2–11/12
M02_HIBB4048_15_GE_C02.indd 51
P
Prob. 2–16
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52
C h a p t e r 2 F o r c e V e c t o r s
2–17. Determine the magnitude and direction of the
resultant force, FR measured counterclockwise from
the positive x axis. Solve the problem by first finding the
resultant F′ = F1 + F2 and then forming FR = F′ + F3.
2–18. Determine the magnitude and direction of the
resultant force, FR measured counterclockwise from
the positive x axis. Solve the problem by first finding the
resultant F′ = F2 + F3 and then forming FR = F′ + F1.
2–21. Determine the magnitude of the two towing forces
FB and FC if the resultant force has a magnitude FR = 10 kN
and is directed along the positive x axis. Set u = 15°.
2–22. If the resultant FR of the two forces acting on the
jet aircraft is to be directed along the positive x axis and
have a magnitude of 10 kN, determine the angle u of the
cable attached to the truck at B so that FB is a minimum.
What is the magnitude of force in each cable when this
occurs?
y
2
C
F1 400 N
FC
F2 200 N
90º
208
x
x
150º
u
A
FB
F3 300 N
B
Prob. 2–17/18
Probs. 2–21/22
2–19. Determine the magnitude and direction of the
resultant FR = F1 + F2 + F3 of the three forces by first
finding the resultant F′ = F1 + F2 and then finding
FR = F′ + F3.
2–23. Two forces act on the screw eye. If F1 = 400 N
and F2 = 600 N, determine the angle u (0° … u … 180°)
between them, so that the resultant force has a magnitude
of FR = 800 N.
*2–20. Determine the magnitude and direction of the
resultant FR = F1 + F2 + F3 of the three forces by first
finding the resultant F′ = F2 + F3 and then finding
FR = F′ + F1.
*2–24. Two forces F1 and F2 act on the screw eye. If their
lines of action are at an angle u apart and the magnitude of
each force is F1 = F2 = F, determine the magnitude of the
resultant force FR and the angle between FR and F1.
F1
y
F1 5 30 N
5
3
F3 5 50 N
4
x
u
208
F2 5 20 N
F2
Probs. 2–19/20
M02_HIBB4048_15_GE_C02.indd 52
Probs. 2–23/24
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53
Problems
2–25. Determine the magnitude and direction u of FA so
that the resultant force is directed along the positive x axis
and has a magnitude of 1250 N.
*2–28. If the resultant force of the two tugboats is 3 kN,
directed along the positive x axis, determine the required
magnitude of force FB and its direction u.
2–26. Determine the magnitude of the resultant force acting
on the ring at O, if FA = 750 N and u = 45°. What is its
direction, measured counterclockwise from the positive x axis?
2–29. If FB = 3 kN and u = 45°, determine the magnitude
of the resultant force and its direction measured clockwise
from the positive x axis.
2–30. If the resultant force of the two tugboats is required to be
directed toward the positive x axis, and FB is to be a minimum,
determine the magnitude of FR and FB and the angle u.
y
FA
A
2
y
u
x
308
O
A
FA 5 2 kN
B
308
FB 5 800 N
x
u
C
FB
Probs. 2–25/26
B
2–27. Two forces act on the screw eye. If F = 600 N,
determine the magnitude of the resultant force and the
angle u if the resultant force is directed vertically upward.
2–31. Determine the magnitude of force F so that the
resultant FR of the three forces is as small as possible. What
is the minimum magnitude of FR?
y
F
500 N
Probs. 2–28/29/30
30
u
8 kN
x
F
30
6 kN
Probs. 2–27
M02_HIBB4048_15_GE_C02.indd 53
Probs. 2–31
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C h a p t e r 2 F o r c e V e c t o r s
2.4 ADDITION OF A SYSTEM
OF COPLANAR FORCES
When a force is resolved into two components along the x and y axes,
the components are then called rectangular components. For analytical
work we can represent these components in one of two ways, using either
scalar or Cartesian vector notation.
Scalar Notation.
2
The rectangular components of force F shown
in Fig. 2–15a are found using the parallelogram law, so that F = Fx + Fy.
Because these components form a right triangle, they can be
determined from
y
Fx = F cos u
F
Fy
u
x
and
Fy = F sin u
Instead of using the angle u, however, the direction of F can also be
defined using a small “slope” triangle, as in the example shown in Fig. 2–15b.
Since this triangle and the larger shaded triangle are similar, the proportional
length of the sides gives
Fx
Fx
a
=
c
F
(a)
or
y
Fx
Fy
b
x
and
c
Fy
a
F
(b)
Fig. 2–15
a
Fx = F a b
c
F
=
b
c
or
b
Fy = -F a b
c
Here the y component is a negative scalar since Fy is directed along the
negative y axis.
It is important to keep in mind that this positive and negative
scalar notation is to be used only for calculations, not for graphical
representations in figures. Throughout the text, the head of a vector
arrow in any figure indicates the sense of the vector graphically; algebraic
signs are not used for this purpose. Thus, the vectors in Figs. 2–15a and
2–15b are designated by using boldface (vector) notation.* Whenever
italic symbols are written near vector arrows in figures, they indicate the
magnitude of the vector, which is always a positive quantity.
*Negative signs are used only in figures with boldface notation when showing equal but
opposite pairs of vectors, as in Fig. 2–2.
M02_HIBB4048_15_GE_C02.indd 54
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2.4 Addition of a System of Coplanar Forces
Cartesian Vector Notation. It is also possible to represent
the x and y components of a force in terms of Cartesian unit vectors
i and j. They are called unit vectors because they have a dimensionless
magnitude of 1, and so they can be used to designate the directions of the
x and y axes, respectively, Fig. 2–16.*
Since the magnitude of each component of F is always a positive
quantity, which is represented by the (positive) scalars Fx and Fy, then we
can express F as a Cartesian vector,
y
j
F
Fy
x
Fx
F = Fx i + Fy j
i
Fig. 2–16
Coplanar Force Resultants. We can use either of the two
methods just described to determine the resultant of several coplanar
forces. To do this, each force is first resolved into its x and y components,
and then the respective components are added using scalar algebra since
they are collinear. The resultant force is then formed by adding the
resultant components using the parallelogram law. For example, consider
the three concurrent forces in Fig. 2–17a, which have x and y components
shown in Fig. 2–17b. Using Cartesian vector notation, each force is first
represented as a Cartesian vector, i.e.,
y
F2
F1
x
F3
(a)
F1 = F1x i + F1y j
F2 = -F2x i + F2y j
F3 = F3x i - F3y j
y
F2y
The vector resultant, Fig. 2–17c, is therefore
FR = F1 + F2 + F3
F1x
F3x
(b)
= 1FR 2 x i + 1FR 2 y j
1FR 2 x = F1x - F2x + F3x
1FR 2 y = F1y + F2y - F3y
Notice that these are the same results as the i and j components of FR
determined above.
*For handwritten work, unit vectors are usually indicated using a circumflex, e.g., î and ĵ.
Also, realize that Fx and Fy in Fig. 2–16 represent the magnitudes of the components, which
are always positive scalars. The directions are defined by i and j. If instead we used scalar
notation, then Fx and Fy could be positive or negative scalars, since they would account for
both the magnitude and direction of the components.
M02_HIBB4048_15_GE_C02.indd 55
x
F3y
= 1F1x - F2x + F3x 2 i + 1F1y + F2y - F3y 2 j
+
¡
+c
F1y
F2x
= F1x i + F1y j - F2x i + F2y j + F3x i - F3y j
If scalar notation is used, then indicating the positive directions of
components along the x and y axes with symbolic arrows, we have
2
Fig. 2–17
y
F 4 F3
F2
F1
x
The resultant force of the four cable forces
acting on the post can be determined by
adding algebraically the separate x and y
components of each cable force. This
resultant FR produces the same pulling effect
on the post as all four cables.
07/07/2022 16:34
56
C h a p t e r 2 F o r c e V e c t o r s
In general then, the components of the resultant force of any number
of coplanar forces can be represented by the algebraic sum of the x and y
components of all the forces, i.e.,
1FR 2 x = ΣFx
1FR 2 y = ΣFy
Once these components are determined, they may be sketched
along the x and y axes with their proper sense of direction, and the
resultant force can be determined from vector addition, Fig. 2–17c. From
this sketch, the magnitude of FR is then found from the Pythagorean
theorem; that is,
y
2
FR
(FR)y
u
(FR)x
(2–1)
x
FR = 21FR 2 2x + 1FR 2 2y
Also, the angle u, which specifies the direction of the resultant force, is
determined from trigonometry:
(c)
Fig. 2–17 (cont.)
u = tan-1 2
1FR 2 y
1FR 2 x
2
The above concepts are illustrated numerically in the examples which
follow.
I MPO RTA N T PO I N T S
• The resultant of several coplanar forces can easily be
determined if an x, y coordinate system is established and the
forces are resolved into components along the axes.
• The direction of each force is specified by the angle its line of
action makes with one of the axes, or by a slope triangle.
• The orientation of the x and y axes is arbitrary, and their
positive direction can be specified by the Cartesian unit vectors
i and j.
• The x and y components of the resultant force are simply the
algebraic addition of the components of all the coplanar forces.
• The magnitude of the resultant force is determined from the
Refer to the companion website for Lecture
Summary and Quiz videos.
M02_HIBB4048_15_GE_C02.indd 56
Pythagorean theorem, and when the resultant components are
sketched on the x and y axes, Fig. 2–17c, the direction u of the
resultant can be determined from trigonometry.
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2.4 Addition of a System of Coplanar Forces
EXAMPLE
2.5
Determine the x and y components of F1 and F2 acting on the boom
shown in Fig. 2–18a. Express each force as a Cartesian vector.
y
F1 5 200 N
SOLUTION
308
Scalar Notation. By the parallelogram law, F1 is resolved into x and y
components, Fig. 2–18b. Since F1x acts in the -x direction, and F1y acts
in the + y direction, we have
F1x = -200 sin 30° N = -100 N = 100 N d Ans.
F1y = 200 cos 30° N = 173 N = 173 N c Ans.
The force F2 is resolved into its x and y components, as shown in
Fig. 2–18c. From this “slope triangle” we could obtain the angle u,
5
e.g., u = tan-1 1 12
2 , and then proceed to determine the magnitudes
of the components in the same manner as for F1. The easier method,
however, consists of using proportional parts of similar triangles, i.e.,
F2x
12
=
260 N
13
Similarly,
F2y = 260 N a
12
F2x = 260 N a b = 240 N
13
F2 5 260 N
(a)
y
F1 5 200 N
F1y 5 200 cos 308 N
F2 = 5240i - 100j6N
M02_HIBB4048_15_GE_C02.indd 57
x
F1x 5 200 sin 308 N
(b)
y
( ( x
12
F2x 5 260 —
— N
13
Ans.
Ans.
Cartesian Vector Notation. Having determined the magnitudes
and directions of the components of each force, we can express each
force as a Cartesian vector.
F1 = 5 -100i + 173j6N
2
12
Notice how the magnitude of the horizontal component, F2x, was
obtained by multiplying the force magnitude by the ratio of the
horizontal leg of the slope triangle divided by the hypotenuse;
whereas the magnitude of the vertical component, F2y, was obtained
by multiplying the force magnitude by the ratio of the vertical leg
divided by the hypotenuse. Using scalar notation to represent the
components, we have
F2y = -100 N = 100 N T 13
308
5
b = 100 N
13
F2x = 240 N = 240 N S x
u
5
( (
5
5 N
F2y 5 260 —
—
13
13
12
F2 5 260 N
(c)
Fig. 2–18
Ans.
Ans.
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58
C h a p t e r 2 F o r c e V e c t o r s
EXAMPLE
2.6
y
F1 5 600 N
F2 5 400 N
458
The link in Fig. 2–19a is subjected to two forces F1 and F2. Determine
the magnitude and direction of the resultant force.
SOLUTION I
308
x
(a)
2
Scalar Notation. First we resolve each force into its x and y
components, Fig. 2–19b, then we sum these components algebraically.
+ 1F 2 = ΣF ;
S
1F 2 = 600 cos 30° N - 400 sin 45° N
R x
x
R x
= 236.8 N S
y
F1 5 600 N
F2 5 400 N
458
308
+ c 1FR 2 y = ΣFy;
1FR 2 y = 600 sin 30° N + 400 cos 45° N
= 582.8 N c
The resultant force, shown in Fig. 2–19c, has a magnitude of
FR = 21236.8 N2 2 + 1582.8 N2 2
x
= 629 N
(b)
and a direction,
y
SOLUTION II
FR
582.8 N
Ans.
u = tan-1 a
582.8 N
b = 67.9°
236.8 N
Ans.
Cartesian Vector Notation. From Fig. 2–19b, each force is first
expressed as a Cartesian vector.
u
236.8 N
(c)
x
Then,
F1 = 5600 cos 30°i + 600 sin 30°j6N
F2 = 5 -400 sin 45°i + 400 cos 45°j6N
FR = F1 + F2 = 1600 cos 30° N - 400 sin 45° N2i
Fig. 2–19
+ 1600 sin 30° N + 400 cos 45° N2j
= 5236.8i + 582.8j6N
The magnitude and direction of FR are determined in the same
manner as before.
note:
Comparing the two methods of solution, notice that the
use of scalar notation is more efficient since the components can
be found directly, without first having to express each force as a
Cartesian vector before adding the components. Later, however, we
will show that Cartesian vector analysis is very beneficial for solving
three-dimensional problems.
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2.4 Addition of a System of Coplanar Forces
EXAMPLE
2.7
The end of the boom O in Fig. 2–20a is subjected to three concurrent
and coplanar forces. Determine the magnitude and direction of the
resultant force.
y
3
F2 5 250 N
458
F3 5 200 N
2
5
4
F1 5 400 N
O
x
y
(a)
250 N
200 N
3
SOLUTION
Each force is resolved into its x and y components, Fig. 2–20b.
Summing the x components, we have
+ (F ) = ΣF ;
S
R x
x
458
5
4
(FR)x = -400 N + 250 sin 45° N - 200 1 45 2 N
x
400 N
O
(b)
= -383.2 N = 383.2 N d
Summing the y components yields
+ c (FR)y = ΣFy;
y
(FR)y = 250 cos 45° N + 200 1 35 2 N
FR
296.8 N
= 296.8 N c
The resultant force, shown in Fig. 2–20c, has a magnitude of
2
FR = 21 -383.2 N2 + 1296.8 N2
u
= 485 N
(c)
Fig. 2–20
Ans.
note: Application of this method is more convenient, compared to
using two applications of the parallelogram law, first to add F1 and F2,
then adding F3 to this resultant.
M02_HIBB4048_15_GE_C02.indd 59
x
Ans.
From the vector addition in Fig. 2–20c, the direction angle u is
296.8
u = tan-1 a
b = 37.8°
383.2
O
383.2 N
2
Refer to the companion website for a self quiz of these
Example problems.
13/07/2022 17:42
60
C h a p t e r 2 F o r c e V e c t o r s
F UN DAMEN TAL PR O B L EM S
F2–7. Resolve each force acting on the post into its x and y
components.
y
F2–10. If the resultant force acting on the bracket is to
be 750 N directed along the positive x axis, determine the
magnitude of F and its direction u.
y
F1 5 300 N
F2 5 450 N
5
F3 5 600 N
325 N
13
4
12
F
5
3
458
x
2
u
x
458
Prob. F2–7
600 N
F2–8. Determine the magnitude and direction of the
resultant force.
3
Prob. F2–10
y
250 N
5
F2–11. If the magnitude of the resultant force acting on
the bracket is to be 80 N directed along the u axis, determine
the magnitude of F and its direction u.
400 N
4
308
y
x
F
300 N
u
5
90 N
Prob. F2–8
F2–9. Determine the magnitude of the resultant force
acting on the corbel and its direction u measured
counterclockwise from the x axis.
y
x
50 N
45
4
u
3
Prob. F2–11
F2–12. Determine the magnitude of the resultant force
and its direction u, measured counterclockwise from the
positive x axis.
y
F3 600 N
4
F2 400 N
5
3
F1 700 N
30
F1 5 15 kN
5
3
x
F2 5 20 kN
5
4
3
F3 5 15 kN
4
x
Prob. F2–9
M02_HIBB4048_15_GE_C02.indd 60
Prob. F2–12
07/07/2022 16:34
61
Problems
PROBLEMS
*2–32. Determine the magnitude of the resultant force and
its direction, measured clockwise from the positive x axis.
y
F3 5 650 N
3
F2 5 750 N
5
4
y
458
400 N
x
F1 5 900 N
B
2
30
x
Probs. 2–34/35
*2–36. Resolve F1 and F2 into their x and y components.
45
800 N
2–37. Determine the magnitude of the resultant force and its
direction measured counterclockwise from the positive x axis.
y
60
30
Prob. 2–32
F1 400 N
2–33. Express each of the three forces acting on the
support in Cartesian vector form and determine the
magnitude of the resultant force and its direction, measured
clockwise from positive x axis.
x
45
y
F1 50 N
5
4
F2 250 N
Probs. 2–36/37
4
3
x
F3 30 N
2–38. Determine the magnitude of the resultant force and its
direction measured counterclockwise from the positive x axis.
15
y
F3
8 kN
F2
5 kN
F1
4 kN
F2 80 N
Prob. 2–33
60
45
2–34. Resolve each force acting on the gusset plate into its
x and y components, and express each force as a Cartesian
vector.
2–35. Determine the magnitude of the resultant force
acting on the gusset plate and its direction, measured
counterclockwise from the positive x axis.
M02_HIBB4048_15_GE_C02.indd 61
x
Prob. 2–38
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62
C h a p t e r 2 F o r c e V e c t o r s
2–39. The three forces are applied to the bracket. Determine
the range of values for the magnitude of force P so that the
resultant of the three forces does not exceed 2400 N.
y
2–43. Express F1, F2, and F3 as Cartesian vectors.
*2–44. Determine the magnitude of the resultant force
and its direction measured counterclockwise from the
positive x axis.
800 N
3000 N
908
608
P
x
y
F3 5 750 N
2
458
3
x
5
4
Prob. 2–39
*2–40. Determine the x and y components of F1 and F2.
2–41. Determine the magnitude of the resultant force and its
direction measured counterclockwise from the positive x axis.
308
F1 5 850 N
F2 5 625 N
Probs. 2–43/44
y
458
F1 5 200 N
2–45. The three concurrent forces acting on the post
produce a zero resultant force FR = 0. If F2 = 12 F1, and
F1 is to be 90° from F2 as shown, determine the required
magnitude of F3 expressed in terms of F1 and the angle u.
308
F2 5 150 N
x
Probs. 2–40/41
2–42. Three forces act on the ring. Determine the range of
values for the magnitude of P so that the magnitude of the
resultant force does not exceed 2500 N. Force P is always
directed to the right.
y
y
1500 N
F2
u
F3
x
600 N
608
458
P
Prob. 2–42
M02_HIBB4048_15_GE_C02.indd 62
x
F1
Prob. 2–45
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63
Problems
2–46. Three forces act on the bracket. Determine the
magnitude and direction u of F1 so that the resultant force
is directed along the positive x′ axis and has a magnitude
of 800 N.
2–50. The four concentric forces act on the post.
Determine the resultant force and its direction, measured
counterclockwise from the positive x axis.
2–47. If F1 = 300 N and u = 10°, determine the magnitude of
the resultant force and its direction measured counterclockwise
from the positive x′ axis.
y
F2 5 600 N
y
F1 5 300 N
F2 5 200 N
2
308
x9
F3 5 180 N
5
608
F1
u
13
x
12
x
608
F4 5 250 N
13
12
5
Probs. 2–46/47
F3 5 450 N
Prob. 2–50
*2–48. Determine the magnitude and orientation u of FB
so that the resultant force is directed along the positive y
axis and has a magnitude of 1500 N.
2–49. If FB = 600 N and u = 20°, determine the
magni­tude of the resultant force and its direction measured
counterclockwise from the positive y axis.
2–51. Express F1, F2 and F3 as Cartesian vectors.
*2–52. Determine the magnitude of the resultant force
and its direction, measured counterclockwise from the
positive x axis.
y
FB
B
y
FA 5 700 N
308
A
u
x
F2 26 kN
5
40
F1 15 kN
13
12
x
30
F3 36 kN
Probs. 2–48/49
M02_HIBB4048_15_GE_C02.indd 63
Probs. 2–51/52
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64
C h a p t e r 2 F o r c e V e c t o r s
2–53. Determine the resultant force acting on the hook, and
its direction measured clockwise from the positive x axis.
*2–56. Three forces act on the bracket. Determine the
magnitude and direction u of F so that the resultant force
is directed along the positive x′ axis and has a magnitude
of 8 kN.
2–57. If F = 5 kN and u = 30°, determine the magnitude
of the resultant force and its direction measured
counterclockwise from the positive x axis.
y
y
4 kN
2
F
158
x
5
3
608
4
F2 5 800 N
u
x9
308
F1 5 500 N
x
6 kN
Prob. 2–53
Probs. 2–56/57
2–54. Express F1 and F2 as Cartesian vectors.
2–55. Determine the magnitude of the resultant force and
its direction measured counterclockwise from the positive
x axis.
2–59. If the resultant force acting on the bracket is
required to be a minimum, determine the magnitudes of F1
and the resultant force. Set f = 30°.
y
F2 5 26 kN
y
13
12
2–58. If the magnitude of the resultant force acting on
the bracket is to be 450 N directed along the positive u axis,
determine the magnitude of F1 and its direction f.
F1
5
u
f
308
x
F2 5 200 N
12
308
x
13
5
F3 5 260 N
F1 5 30 kN
Probs. 2–54/55
M02_HIBB4048_15_GE_C02.indd 64
Probs. 2–58/59
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65
2.5 Cartesian Vectors
2.5
CARTESIAN VECTORS
z
The operations of vector algebra, when applied to solving problems in
three dimensions, are greatly simplified if the vectors are first represented
in Cartesian vector form. In this section we will present a general method
for doing this; then in the next section we will use this method for finding
the resultant force of a system of concurrent forces.
y
x
Right-Handed Coordinate System.
We will use a righthanded coordinate system to describe the theory of vector algebra
that follows. Specifically, a rectangular coordinate system is said to be
right-handed if the thumb of the right hand points in the direction
of the positive z axis when the right-hand fingers are curled about
this axis and directed from the positive x towards the positive y axis,
Fig. 2–21.
Fig. 2–21
Rectangular Components of a Vector.
z
In general, a vector
A may have one, two, or three rectangular components along the x,
y, z coordinate axes, Fig. 2–22. These components are determined
using two successive applications of the parallelogram law; that is,
A = A′ + Az and then A′ = Ax + Ay. Combining these equations
to eliminate A′, A is represented by the vector sum of its three
rectangular components,
A = Ax + Ay + Az
2
Az
A
(2–2)
Ay
y
Cartesian Vector Representation. In three dimensions, the
set of Cartesian unit vectors, i, j, k, is used to designate the directions
of the x, y, z axes, respectively, Fig. 2–23. Using these vectors the
three components of A in Fig. 2–24 can be written in Cartesian vector
form as
Ax
A9
x
Fig. 2–22
z
Az k
z
A
k
k
Ax i
j Ay j
x
Fig. 2–24
M02_HIBB4048_15_GE_C02.indd 65
j
i
x
y
i
y
Fig. 2–23
07/07/2022 16:35
66
C h a p t e r 2 F o r c e V e c t o r s
z
A = A xi + A y j + Azk
(2–3)
A zk
There is a distinct advantage to writing vectors in this manner.
Separating the magnitude and direction of each component vector
will simplify the operations of vector algebra, particularly in three
dimensions.
A
Az
A
Ay j
A xi
2
Magnitude of a Cartesian Vector. If A is expressed as a
Cartesian vector, then its magnitude can be determined. As shown in
Fig. 2–25, from the blue right triangle, A = 2A′2 + A2z, and from the
gray right triangle, A′ = 2A2x + A2y. Combining these equations to
eliminate A′ yields
y
Ax
A9
Ay
x
Fig. 2–25
A = 2A2x + A2y + A2z
z
Hence, the magnitude of A is equal to the positive square root of the sum
of the squares of the magnitudes of its components.
A zk
A
Coordinate Direction Angles. We will define the direction of
A by the coordinate direction angles a (alpha), b (beta), and g (gamma),
measured between the tail of A and the positive x, y, z axes, Fig. 2–26.
Note that regardless of where A is directed, each of these angles will be
between 0° and 180°.
To determine a, b, and g, consider the projection of A onto the x, y, z
axes, Fig. 2–27. Referring to the shaded right triangles shown in the figure,
we have
uA
g
b
a
Ay j
y
Axi
x
Fig. 2–26
cos a =
z
Az
(2–4)
Ax
A
cos b =
Ay
A
cos g =
Az
A
(2–5)
These numbers are known as the direction cosines of A. Once they
have been obtained, the coordinate direction angles a, b, g can then be
determined from the inverse cosines.
An easy way of obtaining these direction cosines is to form a unit
vector uA in the direction of A, Fig. 2–26. To do this, divide A by its
magnitude A, so that
g
A
b
908
a
Ax
Ay
y
uA =
Az
Ay
Ax
A
=
i +
j +
k
A
A
A
A
(2–6)
By comparison with Eqs. 2–5, it is seen that the i, j, k components of uA
represent the direction cosines of A, i.e.,
x
Fig. 2–27
M02_HIBB4048_15_GE_C02.indd 66
u A = cos a i + cos b j + cos g k
(2–7)
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