CONVERSIONS BETWEEN U.S. CUSTOMARY UNITS AND SI UNITS (Continued)
Times conversion factor
U.S. Customary unit
Equals SI unit
Moment of inertia (area)
inch to fourth power
in.4
inch to fourth power
in.4
Accurate
Practical
416,231
416,000
0.416231 3 1026
millimeter to fourth
power
meter to fourth power
mm4
m4
kilogram meter squared
kg·m2
watt (J/s or N·m/s)
watt
watt
W
W
W
47.9
6890
47.9
6.89
pascal (N/m2)
pascal
kilopascal
megapascal
Pa
Pa
kPa
MPa
16,400
16.4 3 1026
millimeter to third power
meter to third power
mm3
m3
meter per second
meter per second
meter per second
kilometer per hour
m/s
m/s
m/s
km/h
cubic meter
cubic meter
cubic centimeter (cc)
liter
cubic meter
m3
m3
cm3
L
m3
0.416 3 1026
Moment of inertia (mass)
slug foot squared
slug-ft2
1.35582
1.36
Power
foot-pound per second
foot-pound per minute
horsepower (550 ft-lb/s)
ft-lb/s
ft-lb/min
hp
1.35582
0.0225970
745.701
1.36
0.0226
746
Pressure; stress
pound per square foot
pound per square inch
kip per square foot
kip per square inch
psf
psi
ksf
ksi
Section modulus
inch to third power
inch to third power
in.3
in.3
Velocity (linear)
foot per second
inch per second
mile per hour
mile per hour
ft/s
in./s
mph
mph
Volume
cubic foot
cubic inch
cubic inch
gallon (231 in.3)
gallon (231 in.3)
ft3
in.3
in.3
gal.
gal.
47.8803
6894.76
47.8803
6.89476
16,387.1
16.3871 3 1026
0.3048*
0.0254*
0.44704*
1.609344*
0.0283168
16.3871 3 1026
16.3871
3.78541
0.00378541
0.305
0.0254
0.447
1.61
0.0283
16.4 3 1026
16.4
3.79
0.00379
*An asterisk denotes an exact conversion factor
Note: To convert from SI units to USCS units, divide by the conversion factor
Temperature Conversion Formulas
5
T(°C) 5 } }[T(°F) 2 32] 5 T(K) 2 273.15
9
5
T(K) 5 } }[T(°F) 2 32] 1 273.15 5 T(°C) 1 273.15
9
9
9
T(°F) 5 } }T(°C) 1 32 5 } }T(K) 2 459.67
5
5
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Engineering Mechanics
Dynamics
Third Edition
SI Edition
Andrew Pytel
The Pennsylvania State University
Jaan Kiusalaas
The Pennsylvania State University
SI Edition prepared by Ishan Sharma
Indian Institute of Technology, Kanpur
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Engineering Mechanics:
Dynamics, Third Edition
Andrew Pytel and Jaan Kiusalaas
SI Edition prepared by Ishan Sharma
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Contents
Preface to the SI Edition
xi
Preface
xiii
Chapter 11
11.1
11.2
11.3
11.4
Introduction to Dynamics
Introduction 1
Derivatives of Vector Functions 3
Position, Velocity, and Acceleration of a Particle
Newtonian Mechanics 5
1
4
Chapter 12 Dynamics of a Particle: Rectangular
Coordinates
12.1
12.2
12.3
12.4
12.5
*12.6
Chapter 13 Dynamics of a Particle: Curvilinear
Coordinates
13.1
13.2
13.3
13.4
69
Introduction 69
Kinematics—Path (Normal-Tangential) Coordinates 70
Kinematics—Polar and Cylindrical Coordinates 82
Kinetics: Force-Mass-Acceleration Method 95
Chapter 14 Work-Energy and Impulse-Momentum
Principles for a Particle
14.1
14.2
14.3
14.4
15
Introduction 15
Kinematics 16
Kinetics: Force-Mass-Acceleration Method 27
Dynamics of Rectilinear Motion 29
Curvilinear Motion 44
Analysis of Motion by the Area Method 56
117
Introduction 117
Work of a Force 118
Principle of Work and Kinetic Energy 122
Conservative Forces and the Conservation of Mechanical
Energy 133
* Indicates optional articles
vii
viii
Contents
14.5
14.6
14.7
*14.8
Power and Efficiency 144
Principle of Impulse and Momentum 150
Principle of Angular Impulse and Momentum 158
Space Motion under a Gravitational Force 168
Chapter 15
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
15.10
*15.11
Chapter 16
16.1
16.2
16.3
16.4
16.5
16.6
16.7
16.8
16.9
*16.10
Dynamics of Particle Systems
Planar Kinematics of Rigid Bodies
273
Introduction 273
Plane Angular Motion 275
Rotation about a Fixed Axis 278
Relative Motion of Two Points in a Rigid Body 287
Method of Relative Velocity 288
Instant Center for Velocities 301
Method of Relative Acceleration 312
Absolute and Relative Derivatives of Vectors 326
Motion Relative to a Rotating Reference Frame 329
Method of Constraints 344
Chapter 17 Planar Kinetics of Rigid Bodies:
Force-Mass-Acceleration Method
17.1
17.2
17.3
17.4
17.5
*17.6
185
Introduction 185
Kinematics of Relative Motion 186
Kinematics of Constrained Motion 192
Kinetics: Force-Mass-Acceleration Method 198
Work-Energy Principles 214
Principle of Impulse and Momentum 217
Principle of Angular Impulse and Momentum 218
Plastic Impact 234
Impulsive Motion 236
Elastic Impact 248
Mass Flow 257
357
Introduction 357
Mass Moment of Inertia; Composite Bodies 358
Angular Momentum of a Rigid Body 368
Equations of Motion 371
Force-Mass-Acceleration Method: Plane Motion 373
Differential Equations of Motion 398
Chapter 18 Planar Kinetics of Rigid Bodies:
Work-Energy and Impulse-Momentum Methods
18.1 Introduction 415
Part A: Work-Energy Method 416
18.2 Work and Power of a Couple 416
18.3 Kinetic Energy of a Rigid Body 418
18.4 Work-Energy Principle and Conservation of Mechanical
Energy 429
415
Contents
Part B: Impulse-Momentum Method
18.5 Momentum Diagrams 442
18.6 Impulse-Momentum Principles
18.7 Rigid-Body Impact 459
442
444
Chapter 19 Rigid-Body Dynamics in
Three Dimensions
*19.1
*19.2
*19.3
*19.4
*19.5
*19.6
Introduction 475
Kinematics 476
Impulse-Momentum Method 491
Work-Energy Method 497
Force-Mass-Acceleration Method 511
Motion of an Axisymmetric Body 527
Chapter 20
20.1
20.2
20.3
20.4
*20.5
475
Vibrations
Introduction 547
Free Vibrations of Particles 548
Forced Vibrations of Particles 565
Rigid-Body Vibrations 578
Methods Based on Conservation of Energy
547
587
Appendix D Proof of the Relative Velocity Equation
for Rigid-Body Motion
599
Appendix E
Equations
601
E.1
E.2
E.3
E.4
Introduction 601
Numerical Methods 601
Application of MATLAB 602
Linear Interpolation 605
Appendix F
F.1
F.2
F.3
F.4
F.5
F.6
F.7
F.8
Numerical Solution of Differential
Mass Moments and Products of Inertia
607
Introduction 607
Review of Mass Moment of Inertia 607
Moments of Inertia of Thin Plates 608
Mass Moment of Inertia by Integration 609
Mass Products of Inertia; Parallel-Axis Theorems 616
Products of Inertia by Integration; Thin Plates 617
Inertia Tensor; Moment of Inertia about an Arbitrary
Axis 618
Principal Moments and Principal Axes of Inertia 619
Answers to Even-Numbered Problems
633
Index
641
ix
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Preface to the SI Edition
This edition of Engineering Mechanics: Dynamics has been adapted to
incorporate the International System of Units (Le Système International
d’Unités or SI) throughout the book.
Le Système International d’Unités
The United States Customary System (USCS) of units uses FPS (footpound−second) units (also called English or Imperial units). SI units are
primarily the units of the MKS (meter-kilogram-second) system. However,
CGS (centimeter-gram-second) units are often accepted as SI units, especially
in textbooks.
Using SI Units in this Book
In this book, we have used both MKS and CGS units. USCS units or FPS units
used in the US Edition of the book have been converted to SI units throughout
the text and problems. However, in case of data sourced from handbooks,
government standards, and product manuals, it is not only extremely difficult
to convert all values to SI, it also encroaches upon the intellectual property
of the source. Also, some quantities such as the ASTM grain size number
and Jominy distances are generally computed in FPS units and would lose
their relevance if converted to SI. Some data in figures, tables, examples, and
references, therefore, remains in FPS units. For readers unfamiliar with the
relationship between the FPS and the SI systems, conversion tables have been
provided inside the front and back covers of the book.
To solve problems that require the use of sourced data, the sourced values can be converted from FPS units to SI units just before they are to be used
in a calculation. To obtain standardized quantities and manufacturers’ data
in SI units, the readers may contact the appropriate government agencies or
authorities in their countries/regions.
xi
xii
Preface to the SI Edition
Instructor Resources
A Printed Instructor’s Solution Manual in SI units is available on request.
An electronic version of the Instructor’s Solutions Manual, and PowerPoint
slides of the figures from the SI text are available through www.cengage.com/
engineering.
The readers’ feedback on this SI Edition will be highly appreciated and
will help us improve subsequent editions.
The Publishers
Preface
Statics and dynamics are the foundation subjects in the branch of engineering
known as engineering mechanics. Engineering mechanics is, in turn, the basis
of many of the traditional fields of engineering, such as aerospace engineering, civil engineering, and mechanical engineering. In addition, engineering
mechanics often plays a fundamental role in such diverse fields as medicine
and biology. Applying the principles of statics and dynamics to such a wide
range of applications requires reasoning and practice rather than memorization. Although the principles of statics and dynamics are relatively few, they
can only be truly mastered by studying and analyzing problems. Therefore, all
modern textbooks, including ours, contain a large number of problems to be
solved by the student. Learning the engineering approach to problem solving
is one of the more valuable lessons to be learned from the study of statics and
dynamics.
In this, our third edition of Statics and Dynamics, we have made
every effort to improve our presentation without compromising the following
principles that formed the basis of the previous editions.
• Each sample problem is carefully chosen to help students master the
intricacies of engineering problem analysis.
• The selection of homework problems is balanced between “textbook”
problems that illustrate the principles of engineering mechanics in a
straightforward manner, and practical engineering problems that are
applicable to engineering design.
• The number of problems using U.S. Customary Units and SI Units are
approximately equal.
• The importance of correctly drawn free-body diagrams is emphasized
throughout.
• Whenever applicable, the number of independent equations is compared
to the number of unknowns before the governing equations are written.
• Numerical methods for solving problems are seamlessly integrated into
the text, the emphasis being on computer applications, not on computer
programming.
• Review Problems appear at the end of each chapter to encourage
students to synthesize the topics covered in the chapter.
Both Statics and Dynamics contain several optional topics, which are
marked with an asterisk (*). Topics so indicated can be omitted without
jeopardizing the presentation of other subjects. An asterisk is also used to
xiii
xiv
Preface
indicate problems that require advanced reasoning. Articles, sample problems, and problems associated with numerical methods are preceded by an
icon representing a compact disk.
In this third edition of Dynamics, we have made what we consider to
be a number of significant improvements based upon the feedback received
from students and faculty who have used the previous editions. In addition,
we have incorporated many of the suggestions provided by the reviewers of
the second edition.
A number of articles have been reorganized, or rewritten, to make the
topics easier for the student to understand. For example, the discussion of the
work-energy method in Chapter 18 has been streamlined. Also, Chapter 20
(Vibrations) has been reorganized to provide a more concise presentation of
the material. In addition, sections entitled Review of Equations have been
added at the end of each chapter as an aid to problem solving.
The total numbers of sample problems and problems remain about the
same as in the previous edition; however, the introduction of two colors
improves the overall readability of the text and artwork. Compared with the
previous edition, approximately one-third of the problems are new, or have
been modified.
New to this edition, the Sample Problems that require numerical solutions have been solved using MATLAB© , the software program that is
familiar to many engineering students.
Study Guide to Accompany Pytel and Kiusalaas Engineering
Mechanics, Dynamics, Third Edition, J.L. Pytel and A. Pytel, 2009. The goals
of this study guide are two-fold. First, self-tests are included to help the student focus on the salient features of the assigned reading. Second, the study
guide uses “guided” problems which give the student an opportunity to work
through representative problems, before attempting to solve the problems in
the text.
Ancillary
We are grateful to the following reviewers for their
valuable suggestions:
Acknowledgments
Hamid R. Hamidzadeh, Tennessee State University
Aiman S. Kuzmar, The Pennsylvania State University—Fayette,
The Eberly Campus
Gary K. Matthew, University of Florida
Noel Perkins, University of Michigan
Corrado Poli, University of Massachusetts, Amherst
ANDREW PYTEL
JAAN KIUSALAAS
11
Introduction to Dynamics
11.1
Introduction
Sir Isaac Newton (1643–1727) in
his treatise Philosophiae Naturalis
Principia Mathematica established
the groundwork for dynamics with
his three laws of motion and the
universal theory of gravitation,
which are discussed in this chapter.
(Time & Life Pictures/Getty Images)
Classical dynamics studies the motion of bodies using the principles established
by Newton and Euler.* The organization of this text is based on the subdivisions
of classical dynamics shown in Fig. 11.1.
* Sir
Isaac Newton is credited with laying the foundation of classical mechanics with the publication
of Principia in 1687. However, the laws of motion as we use them today were developed by Leonhard
Euler and his contemporaries more than sixty years later. In particular, the laws for the motion of finite
bodies are attributable to Euler.
1
2
CHAPTER 11
Introduction to Dynamics
Absolute motion
Kinematics
Relative motion
Particles
Classical dynamics
Force-mass-acceleration method
Rigid bodies
Kinetics
Work-energy method
Impulse-momentum method
Fig. 11.1
The first part of this text deals with dynamics of particles. A particle is a mass
point; it possesses a mass but has no size. The particle is an approximate model of
a body whose dimensions are negligible in comparison with all other dimensions
that appear in the formulation of the problem. For example, in studying the motion
of the earth around the sun, it is permissible to consider the earth as a particle,
because its diameter is much smaller than the dimensions of its orbit.
The second part of this text is devoted mainly to dynamics of rigid bodies.
A body is said to be rigid if the distance between any two material points of
the body remains constant, that is, if the body does not deform. Because any
body undergoes some deformation when loads are applied to it, a truly rigid body
does not exist. However, in many applications the deformation is so small (relative to the dimensions of the body) that the rigid-body idealization is a good
approximation.
As seen in Fig. 11.1, the main branches of dynamics are kinematics and kinetics. Kinematics is the study of the geometry of motion. It is not concerned with the
causes of motion. Kinetics, on the other hand, deals with the relationships between
the forces acting on the body and the resulting motion. Kinematics is not only an
important topic in its own right but is also a prerequisite to kinetics. Therefore,
the study of dynamics always begins with the fundamentals of kinematics.
Kinematics can be divided into two parts as shown in Fig. 11.1: absolute
motion and relative motion. The term absolute motion is used when the motion is
described with respect to a fixed reference frame (coordinate system). Relative
motion, on the other hand, describes the motion with respect to a moving
coordinate system.
Figure 11.1 also lists the three main methods of kinetic analysis. The
force-mass-acceleration (FMA) method is a straightforward application of the
Newton-Euler laws of motion, which relate the forces acting on the body to its
mass and acceleration. These relationships, called the equations of motion, must
be integrated twice in order to obtain the velocity and the position as functions of
time.
The work-energy and impulse-momentum methods are integral forms of
Newton-Euler laws of motion (the equations of motion are integrated with
respect to position or time). In both methods the acceleration is eliminated by
11.2
Derivatives of Vector Functions
the integration. These methods can be very efficient in the solution of problems
concerned with velocity-position or velocity-time relationships.
The purpose of this chapter is to review the basic concepts of Newtonian
mechanics: displacement, velocity, acceleration, Newton’s laws, and units of
measurement.
11.2
Derivatives of Vector Functions
A knowledge of vector calculus is a prerequisite for the study of dynamics. Here
we discuss the derivatives of vectors; integration is introduced throughout the text
as needed.
The vector A is said to be a vector function of a scalar parameter u if the magnitude and direction of A depend on u. (In dynamics, time is frequently chosen
to be the scalar parameter.) This functional relationship is denoted by A(u). If the
scalar variable changes from the value u to (u + u), the vector A will change
from A(u) to A(u + u). Therefore, the change in the vector A can be written as
A = A(u + u) − A(u)
(11.1)
As seen in Fig. 11.2, A is due to a change in both the magnitude and the direction
of the vector A.
The derivative of A with respect to the scalar u is defined as
A
A(u + u) − A(u)
dA
= lim
= lim
u→0 u
u→0
du
u
ΔA
(11.2)
A(u + Δu)
assuming that the limit exists. This definition resembles the derivative of the scalar
function y(u), which is defined as
y
y(u + u) − y(u)
dy
= lim
= lim
u→0 u
u→0
du
u
A(u)
(11.3)
Caution In dealing with a vector function, the magnitude of the derivative
|dA/du| must not be confused with the derivative of the magnitude d|A|/du.
In general, these two derivatives will not be equal. For example, if the magnitude of a vector A is constant, then d|A|/du = 0. However, |dA/du| will not
equal zero unless the direction of A is also constant.
The following useful identities can be derived from the definitions of derivatives (A and B are assumed to be vector functions of the scalar u, and m is also a
scalar):
d(mA)
du
d(A + B)
du
d(A · B)
du
d(A × B)
du
dA dm
+
A
du
du
dA dB
=
+
du
du
dB dA
= A·
+
·B
du
du
dB dA
=A×
+
×B
du
du
=m
(11.4)
(11.5)
(11.6)
(11.7)
O
Fig. 11.2
3