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Vector Mechanics
For Engineers
Dynamics
Eleventh Edition in SI Units

Eleventh Edition in SI Units
Vector Mechanics
For Engineers
Ferdinand P. Beer
Late of Lehigh University
E. Russell Johnston, Jr.
Late of University of Connecticut
Phillip J. Cornwell
Rose-Hulman Institute of Technology
Brian P. Self
California Polytechnic State University—San Luis Obispo
Sanjeev Sanghi
Indian Institute of Technology, Delhi
Dynamics
McGraw Hill Education (India) Private Limited
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v
About the Authors
Ferdinand P. Beer. Born in France and educated in France and Switzer-
land, Ferd received an M.S. degree from the Sorbonne and an Sc. D. degree
in theoretical mechanics from the University of Geneva. He came to the
United States after serving in the French army during the early part of
World War II and taught for four years at Williams College in the Williams-
MIT joint arts and engineering program. Following his service at Williams
College, Ferd joined the faculty of Lehigh University where he taught for
thirty-seven years. He held several positions, including University Distin-
guished Professor and chairman of the Department of Mechanical Engi-
neering and Mechanics, and in 1995 Ferd was awarded an honorary Doctor
of Engineering degree by Lehigh University.
E. Russell Johnston, Jr. Born in Philadelphia, Russ received a B.S. degree
in civil engineering from the University of Delaware and an Sc. D. degree
in the field of structural engineering from the Massachusetts Institute of
Technology. He taught at Lehigh University and Worcester Polytechnic
Institute before joining the faculty of the University of Connecticut where
he held the position of chairman of the Civil Engineering Department and
taught for twenty-six years. In 1991 Russ received the Outstanding Civil
Engineer Award from the Connecticut Section of the American Society of
Civil Engineers.
Phillip J. Cornwell. Phil holds a B.S. degree in mechanical engineering
from Texas Tech University and M.A. and Ph.D. degrees in mechanical
and aerospace engineering from Princeton University. He is currently a
professor of mechanical engineering and Vice President of Academic
Affairs at Rose-Hulman Institute of Technology where he has taught since
1989. Phil received an SAE Ralph R. Teetor Educational Award in 1992,
the Dean’s Outstanding Teacher Award at Rose-Hulman in 2000, and the
Board of Trustees’ Outstanding Scholar Award at Rose-Hulman in 2001.
Phil was one of the developers of the Dynamics Concept Inventory.
Brian P. Self. Brian obtained his B.S. and M.S. degrees in Engineering
Mechanics from Virginia Tech, and his Ph.D. in Bioengineering from the
University of Utah. He worked in the Air Force Research Laboratories
before teaching at the U.S. Air Force Academy for seven years. Brian has
taught in the Mechanical Engineering Department at Cal Poly, San Luis
Obispo since 2006. He has been very active in the American Society of
Engineering Education, serving on its Board from 2008–2010. With a
team of five, Brian developed the Dynamics Concept Inventory to help
assess student conceptual understanding. His professional interests include
educational research, aviation physiology, and biomechanics.
Sanjeev Sanghi. obtained his bachelor’s degree in Mechanical Engineer-
ing from the Indian Institute of Technology, Kanpur. He continued his
education further at Cornell University and The Levich Institute of City
University of New York from where he obtained his MS and PhD respec-

vi About the Authors
tively. He is now Professor in the Applied Mechanics Department at the
Indian Institute of Technology, Delhi. He joined the department in 1992
and since then has been teaching courses in Engineering Mechanics, Fluid
Mechanics at the undergraduate and graduate levels. He has been the Head
of Educational Technology and Service Centre, IIT Delhi and was Visiting
Professor at The University of Sussex, UK in 2007–2008. At Cornell Uni-
versity, he won the Distinguished Teaching Assistant Award in 1988. In
2002, he was given the Distinguished Teacher Award by the C V Kapoor
Foundation. He has published papers in leading journals like Journal of
Fluid Mechanics, Journal of Computational Physics, Physics of Fluids,
Chaos, AIAA Journal, ASME Journal of Heat Transfer, Journal of Sound
and Vibration, Computers and Structures and Computers and Fluids.

vii
Brief Contents
11 Kinematics of Particles 615
12 Kinetics of Particles: Newton’s Second Law 718
13 Kinetics of Particles: Energy and Momentum
Methods 795
14 Systems of Particles 915
15 Kinematics of Rigid Bodies 977
16 Plane Motion of Rigid Bodies: Forces and
Accelerations 1107
17 Plane Motion of Rigid Bodies: Energy and Momentum
Methods 1181
18 Kinetics of Rigid Bodies in Three Dimensions 1264
19 Mechanical Vibrations 1332
Appendix A: Some Useful Definitions and Properties
of Vector Algebra A1
Appendix B: Mass Moment of Inertia A7
Appendix C: Fundamentals of Engineering
Examination A45
Answers to Problems AN1
Photo Credits C1
Index I1

viii
Contents
Preface x
Guided Tour xiv
Digital Resources xvii
Acknowledgments xix
List of Symbols xx
11 Kinematics of Particles 615
11.1 Rectilinear Motion of Particles 617
11.2 Special Cases and Relative Motion 635
*11.3 Graphical Solutions 652
11.4 Curvilinear Motion of Particles 663
11.5 Non-Rectangular Components 690
Review and Summary 711
Review Problems 715
12 Kinetics of Particles:
Newton’s Second Law 718
12.1 Newton’s Second Law and Linear Momentum 720
12.2 Angular Momentum and Orbital Motion 763
*12.3 Applications of Central-Force Motion 774
Review Problems 792
13 Kinetics of Particles: Energy and
Momentum Methods 795
13.1 Work and Energy 797
13.2 Conservation of Energy 827
13.3 Impulse and Momentum 855
13.4 Impacts 877
Review Problems 911
*Advanced or specialty topics

ix
Contents
14 Systems of Particles 915
14.1 Applying Newton’s Second Law and Momentum
Principles to Systems of Particles 917
14.2 Energy and Momentum Methods for a System of
Particles 936
*14.3 Variable Systems of Particles 950
Review Problems 974
15 Kinematics of Rigid Bodies 977
15.1 Translation and Fixed Axis Rotation 980
15.2 General Plane Motion: Velocity 997
15.3 Instantaneous Center of Rotation 1015
15.4 General Plane Motion: Acceleration 1029
15.5 Analyzing Motion with Respect to a Rotating
Frame 1048
*15.6 Motion of a Rigid Body in Space 1065
*15.7 Motion Relative to a Moving Reference Frame 1082
Review Problems 1104
16 Plane Motion of Rigid Bodies:
Forces and Accelerations 1107
16.1 Kinetics of a Rigid Body 1109
16.2 Constrained Plane Motion 1144
17 Plane Motion of Rigid Bodies: Energy
and Momentum Methods 1181
17.1 Energy Methods for a Rigid Body 1183
17.2 Momentum Methods for a Rigid Body 1211
17.3 Eccentric Impact 1234

x Contents
18 Kinetics of Rigid Bodies
in Three Dimensions 1264
18.1 Energy and Momentum of a Rigid Body 1266
*18.2 Motion of a Rigid Body in Three Dimensions 1285
*18.3 Motion of a Gyroscope 1305
19 Mechanical Vibrations 1332
19.1 Vibrations without Damping 1334
19.2 Free Vibrations of Rigid Bodies 1350
19.3 Applying the Principle of Conservation
of Energy 1364
19.4 Forced Vibrations 1375
19.5 Damped Vibrations 1389
Appendix A: Some Useful Definitions and Properties
of Vector Algebra A1
Appendix B: Mass Moment of Inertia A7
Appendix C: Fundamentals of Engineering
Examination A45
Answers to Problems AN1
Photo Credits C1
Index I1

xi
Preface
Objectives
A primary objective in a first course in mechanics is to help develop a
student’s ability first to analyze problems in a simple and logical manner,
and then to apply basic principles to their solutions. A strong conceptual
understanding of these basic mechanics principles is essential for success-
fully solving mechanics problems. We hope that this text, as well as the
preceding volume, Vector Mechanics for Engineers: Statics, will help
instructors achieve these goals.†
General Approach
Vector algebra was introduced at the beginning of the first volume and is
used in the presentation of the basic principles of statics, as well as in the
solution of many problems, particularly three-dimensional problems. Sim-
ilarly, the concept of vector differentiation will be introduced early in this
volume, and vector analysis will be used throughout the presentation of
dynamics. This approach leads to more concise derivations of the funda-
mental principles of mechanics. It also makes it possible to analyze many
problems in kinematics and kinetics which could not be solved by scalar
methods. The emphasis in this text, however, remains on the correct under-
standing of the principles of mechanics and on their application to the
solution of engineering problems, and vector analysis is presented chiefly
as a convenient tool.‡
Practical Applications Are Introduced Early. One of the
characteristics of the approach used in this book is that mechanics of
particles is clearly separated from the mechanics of rigid bodies. This
approach makes it possible to consider simple practical applications at an
early stage and to postpone the introduction of the more difficult concepts.
For example:
• In Dynamics, the same division is observed. The basic concepts of
force, mass, and acceleration, of work and energy, and of impulse
and momentum are introduced and first applied to problems involv-
ing only particles. Thus, students can familiarize themselves with
the three basic methods used in dynamics and learn their respective
advantages before facing the difficulties associated with the motion of
rigid bodies.
†
Both texts also are available in a single volume, Vector Mechanics for Engineers: Statics
and Dynamics, eleventh edition.
‡
In a parallel text, Mechanics for Engineers: Dynamics, fifth edition, the use of vector algebra
is limited to the addition and subtraction of vectors, and vector differentiation is omitted.
11.4 CURVILINEAR MOTION
OF PARTICLES
When a particle moves along a curve other than a straight line, we say
that the particle is in curvilinear motion. We can use position, velocity,
and acceleration to describe the motion, but now we must treat these
quantities as vectors because they can have directions in two or three
dimensions.
11.4A Position, Velocity, and
Acceleration Vectors
To define the position P occupied by a particle in curvilinear motion at a
given time t, we select a fixed reference system, such as the x, y, z axes
shown in Fig. 11.12a, and draw the vector r joining the origin O and
point P. The vector r is characterized by its magnitude r and its direction
with respect to the reference axes, so it completely defines the position of
the particle with respect to those axes. We refer to vector r as the position
vector of the particle at time t.
Consider now the vector r9 defining the position P9 occupied by the
same particle at a later time t 1 Dt. The vector Dr joining P and P9
represents the change in the position vector during the time interval Dt
and is called the displacement vector. We can check this directly from
Fig. 11.12a, where we obtain the vector r9 by adding the vectors r and
Dr according to the triangle rule. Note that Dr represents a change in
direction as well as a change in magnitude of the position vector r.
We define the average velocity of the particle over the time interval
Dt as the quotient of Dr and Dt. Since Dr is a vector and Dt is a scalar,
the quotient Dr/Dt is a vector attached at P with the same direction as Dr
and a magnitude equal to the magnitude of Dr divided by Dt (Fig. 11.12b).
We obtain the instantaneous velocity of the particle at time t by
taking the limit as the time interval Dt approaches zero. The instantaneous
velocity is thus represented by the vector
x
y
z
x
y
z
x
y
z
O
O
P
P'
P
P
P'
r
r
r'
r
r' ∆r ∆s
(a)
(b)
(c)
∆r
∆t
P0
v
s
O
The 11th
edition has undergone a complete
rewrite to modernize and streamline the
language throughout the text.
NEW!

xii Preface
New Concepts Are Introduced in Simple Terms. Since this
text is designed for the first course in dynamics, new concepts are pre-
sented in simple terms and every step is explained in detail. On the other
hand, by discussing the broader aspects of the problems considered, and
by stressing methods of general applicability, a definite maturity of
approach has been achieved. For example, the concept of potential energy
is discussed in the general case of a conservative force. Also, the study of
the plane motion of rigid bodies is designed to lead naturally to the study
of their general motion in space. This is true in kinematics as well as in
kinetics, where the principle of equivalence of external and effective forces
is applied directly to the analysis of plane motion, thus facilitating the
transition to the study of three-dimensional motion.
Fundamental Principles Are Placed in the Context of Simple
Applications. The fact that mechanics is essentially a deductive sci-
ence based on a few fundamental principles is stressed. Derivations have
been presented in their logical sequence and with all the rigor warranted
at this level. However, the learning process being largely inductive, simple
applications are considered first. For example:
• The kinematics of particles (Chap. 11) precedes the kinematics of
rigid bodies (Chap. 15).
• The fundamental principles of the kinetics of rigid bodies are first
applied to the solution of two-dimensional problems (Chaps. 16
and 17), which can be more easily visualized by the student, while
three-dimensional problems are postponed until Chap. 18.
The Presentation of the Principles of Kinetics Is Unified.
The eleventh edition of Vector Mechanics for Engineers retains the unified
presentation of the principles of kinetics which characterized the previous
ten editions. The concepts of linear and angular momentum are introduced
in Chap. 12 so that Newton’s second law of motion can be presented not
only in its conventional form F 5 ma, but also as a law relating, respectively,
the sum of the forces acting on a particle and the sum of their moments to
the rates of change of the linear and angular momentum of the particle. This
makes possible an earlier introduction of the principle of conservation of
angular momentum and a more meaningful discussion of the motion of a
particle under a central force (Sec. 12.3A). More importantly, this approach
can be readily extended to the study of the motion of a system of particles
(Chap. 14) and leads to a more concise and unified treatment of the kinetics
of rigid bodies in two and three dimensions (Chaps. 16 through 18).
Systematic Problem-Solving Approach. New to this edition of
the text, all the sample problems are solved using the steps of Strategy,
Modeling, Analysis, and Reflect & Think, or the “SMART” approach.
This methodology is intended to give students confidence when approach-
ing new problems, and students are encouraged to apply this approach in
the solution of all assigned problems.
17.1 ENERGY METHODS FOR A
RIGID BODY
We now use the principle of work and energy to analyze the plane motion
of rigid bodies. As we pointed out in Chap. 13, the method of work and
energy is particularly well adapted to solving problems involving veloci-
ties and displacements. Its main advantage is that the work of forces and
the kinetic energy of particles are scalar quantities.
17.1A Principle of Work and Energy
To apply the principle of work and energy to the motion of a rigid body,
we again assume that the rigid body is made up of a large number n of
particles of mass Dmi. From Eq. (14.30) of Sec. 14.2B, we have
Principle of work
and energy, rigid body
T1 1 U1y2 5 T2 (17.1)
where T1, T2 5 the initial and final values of total kinetic energy of
particles forming the rigid body
U1y2 5 work of all forces acting on various particles of the body
Just as we did in Chap. 13, we can express the work done by nonconser-
vative forces as UNC
1 y2, and we can define potential energy terms for con-
servative forces. Then we can express Eq. (17.1) as
T1 1 Vg1
1 Ve1
1 UNC
1 y2 5 T2 1 Vg2
1 Ve2
(17.19)
where Vg1
and Vg2
are the initial and final gravitational potential energy of
the center of mass of the rigid body with respect to a reference point or
datum, and Ve1
and Ve2
are the initial and final values of the elastic energy
associated with springs in the system.
We obtain the total kinetic energy
T 5
1
2
O
n
i51
Dmi v2
i (17.2)
by adding positive scalar quantities, so it is itself a positive scalar quantity.
You will see later how to determine T for various types of motion of a
rigid body.
The expression U1y2 in Eq. (17.1) represents the work of all the
forces acting on the various particles of the body whether these forces are
internal or external. However, the total work of the internal forces holding
together the particles of a rigid body is zero. To see this, consider two
particles A and B of a rigid body and the two equal and opposite forces F
and –F they exert on each other (Fig. 17.1). Although, in general, small
displacements dr and dr9 of the two particles are different, the components
of these displacements along AB must be equal; otherwise, the particles
would not remain at the same distance from each other and the body
would not be rigid. Therefore, the work of F is equal in magnitude and
T1 1 U1y2 5 T2
NEW!

xiii
Preface
Free-Body Diagrams Are Used Both to Solve Equilibrium
Problems and to Express the Equivalence of Force
Systems. Free-body diagrams were introduced early in statics, and their
importance was emphasized throughout. They were used not only to solve
equilibrium problems but also to express the equivalence of two systems
of forces or, more generally, of two systems of vectors. In dynamics we
will introduce a kinetic diagram, which is a pictorial representation of
inertia terms. The advantage of this approach becomes apparent in the
study of the dynamics of rigid bodies, where it is used to solve three-
dimensional as well as two-dimensional problems. By placing the empha-
sis on the free-body diagram and kinetic diagram, rather than on the
standard algebraic equations of motion, a more intuitive and more com-
plete understanding of the fundamental principles of dynamics can be
achieved. This approach, which was first introduced in 1962 in the first
edition of Vector Mechanics for Engineers, has now gained wide accep-
tance among mechanics teachers in this country. It is, therefore, used in
preference to the method of dynamic equilibrium and to the equations
of motion in the solution of all sample problems in this book.
Optional Sections Offer Advanced or Specialty Topics. A
large number of optional sections have been included. These sections are
indicated by asterisks and thus are easily distinguished from those which
form the core of the basic dynamics course. They can be omitted without
prejudice to the understanding of the rest of the text.
The topics covered in the optional sections include graphical meth-
ods for the solution of rectilinear-motion problems, the trajectory of a
particle under a central force, the deflection of fluid streams, problems
involving jet and rocket propulsion, the kinematics and kinetics of rigid
bodies in three dimensions, damped mechanical vibrations, and electrical
analogues. These topics will be found of particular interest when dynamics
is taught in the junior year.
The material presented in the text and most of the problems require
no previous mathematical knowledge beyond algebra, trigonometry, elemen-
tary calculus, and the elements of vector algebra presented in Chaps. 2
and 3 of the volume on statics.†
However, special problems are included,
which make use of a more advanced knowledge of calculus, and certain
sections, such as Secs. 19.5A and 19.5B on damped vibrations, should be
assigned only if students possess the proper mathematical background. In
portions of the text using elementary calculus, a greater emphasis is placed
on the correct understanding and application of the concepts of differentia-
tion and integration, than on the nimble manipulation of mathematical
formulas. In this connection, it should be mentioned that the determination
of the centroids of composite areas precedes the calculation of centroids by
integration, thus making it possible to establish the concept of moment of
area firmly before introducing the use of integration.
†
Some useful definitions and properties of vector algebra have been summarized in Appendix
A at the end of this volume for the convenience of the reader. Also, Secs. 9.5 and 9.6 of the
volume on statics, which deal with the moments of inertia of masses.

xiv
Guided Tour
Chapter Introduction. Each chapter begins with a list of learning
objectives and an outline that previews chapter topics. An introductory
section describes the material to be covered in simple terms, and how it
will be applied to the solution of engineering problems.
Chapter Lessons. The body of the text is divided into sections, each
consisting of one or more sub-sections, several sample problems, and a large
number of end-of-section problems for students to solve. Each section cor-
responds to a well-defined topic and generally can be covered in one lesson.
In a number of cases, however, the instructor will find it desirable to devote
more than one lesson to a given topic.
Sample Problems. The Sample Problems are set up in much the
same form that students will use when solving assigned problems, and
they employ the SMART problem-solving methodology that students are
encouraged to use in the solution of their assigned problems. They thus
serve the double purpose of reinforcing the text and demonstrating the
type of neat and orderly work that students should cultivate in their own
solutions. In addition, in-problem references and captions have been added
to the sample problem figures for contextual linkage to the step-by-step
solution.
Solving Problems on Your Own. A section entitled Solving Prob-
lems on Your Own is included for each lesson, between the sample prob-
lems and the problems to be assigned. The purpose of these sections is to
help students organize in their own minds the preceding theory of the text
and the solution methods of the sample problems so that they can more
successfully solve the homework problems. Also included in these sec-
tions are specific suggestions and strategies that will enable the students
to more efficiently attack any assigned problems.
Homework Problem Sets. Most of the problems are of a practical
nature and should appeal to engineering students. They are primarily designed,
however, to illustrate the material presented in the text and to help students
understand the principles of mechanics. The problems are grouped according
to the portions of material they illustrate and, in general, are arranged in
order of increasing difficulty. Problems requiring special attention are indi-
cated by asterisks. Answers to 70 percent of the problems are given at the
end of the book.
Sample Problem 11.4
An uncontrolled automobile traveling at 72 km/h strikes a highway crash
barrier square on. After initially hitting the barrier, the automobile deceler-
ates at a rate proportional to the distance x the automobile has moved into
the barrier; specifically, a 5 2302x, where a and x are expressed in m/s2
and m, respectively. Determine the distance the automobile will move into
the barrier before it comes to rest.
v0
y
x
z
–a (m/s2)
x (m)
STRATEGY: Since you are given the deceleration as a function of
displacement, you should start with the basic kinematic relationship
a 5 v dv/dx.
MODELING and ANALYSIS: Model the car as a particle. First find
the initial speed in ft/s,
v0 5 a72
km
hr
ba
1 hr
3600 s
ba
1000 m
km
b 5 20
m
s
Substituting a 5 2302x into a 5 v dv/dx gives
a 5 2302x 5
v dv
dx
Separating variables and integrating gives
v dv 5 2302x dx y #
0
v0
v dv 5 2 #
x
0
302x dx
1
2
v2
2
1
2
v2
0 5 220x3/2
y x 5 a
1
40
(v2
0 2 v2
)b
2/3
(1)
Substituting v 5 0, v0 5 20 m/s gives
d 5 4.64 m b
REFLECT and THINK: A distance of 4.64 m seems reasonable for a
barrier of this type. If you substitute d into the equation for a, you find a
maximum deceleration of about 7 g’s. Note that this problem would have
been much harder to solve if you had been asked to find the time for the
automobile to stop. In this case, you would need to determine v(t) from
Eq. (1). This gives v 5 2v2
0 2 40x3/2
. Using the basic kinematic relation-
ship v 5 dx/dt, you can easily show that
#
t
0
dt 5 #
x
0
dx
2v2
0 2 40x3/2
Unfortunately, there is no closed-form solution to this integral, so you
would need to solve it numerically.
The motion of the paraglider can be described in terms of its
position, velocity, and acceleration. When landing, the pilot of the
paraglider needs to consider the wind velocity and the relative
motion of the glider with respect to the wind. The study of
motion is known as kinematics and is the subject of this chapter.
Kinematics of Particles
11
More than 40 new sample problems have
been added to this volume.
NEW!
Over 300 of the homework problems in
the text are new or revised.
NEW!

xv
Guided Tour
Chapter Review and Summary. Each chapter ends
with a review and summary of the material covered in that
chapter. Marginal notes are used to help students organize
their review work, and cross-references have been included to
help them find the portions of material requiring their special
attention.
Review Problems. A set of review problems is included
at the end of each chapter. These problems provide students
further opportunity to apply the most important concepts intro-
duced in the chapter.
Concept Questions. Educational research has shown that students can
often choose appropriate equations and solve algorithmic problems without
having a strong conceptual understanding of mechanics principles.†
To help
assess and develop student conceptual understanding, we have included Con-
cept Questions, which are multiple choice problems that require few, if any,
calculations. Each possible incorrect answer typically represents a common
misconception (e.g., students often think that a vehicle moving in a curved
path at constant speed has zero acceleration). Students are encouraged to
solve these problems using the principles and techniques discussed in the
text and to use these principles to help them develop their intuition. Mastery
and discussion of these Concept Questions will deepen students’ conceptual
understanding and help them to solve dynamics problems.
905
Review and Summary
This chapter was devoted to presenting the method of work and energy and
the method of impulse and momentum. In the first half of the chapter, we
studied the method of work and energy and its application to the analysis of
the motion of particles.
Work of a Force
We first considered a force F acting on a particle A and defined the work of
F corresponding to the small displacement dr [Sec. 13.2] as the quantity
dU 5 F?dr (13.1)
or recalling the definition of the scalar product of two vectors, as
dU 5 F ds cos α (13.19)
where α is the angle between F and dr (Fig. 13.30). We obtained the work
of F during a finite displacement from A1 to A2, denoted by U1y2, by integrating
Eq. (13.1) along the path described by the particle as
U1y2 5 #
A2
A1
F?dr (13.2)
For a force defined by its rectangular components, we wrote
U1y2 5 #
A2
A1
(Fx dx 1 Fy dy 1 Fz dz) (13.20)
Work of a Weight
We obtain the work of the weight W of a body as its center of gravity moves
from the elevation y1 to y2 (Fig. 13.31) by substituting Fx 5 Fz 5 0 and
Fy 5 2W into Eq. (13.20) and integrating. We found
U1y2 5 2 #
y2
y1
W dy 5 Wy1 2 Wy2 (13.4)
A2
A
A1
y2
y1
dy
y
W
Fig. 13.31
A1
s1
s2
s
A2
F
O
A
dr
ds
a
Fig. 13.30
1104
15.248 A straight rack rests on a gear of radius r and is attached to a block
B as shown. Denoting by vD the clockwise angular velocity of gear
D and by u the angle formed by the rack and the horizontal, derive
expressions for the velocity of block B and the angular velocity of
the rack in terms of r, u, and vD.
15.249 A carriage C is supported by a caster A and a cylinder B, each of
50-mm diameter. Knowing that at the instant shown the carriage has
an acceleration of 2.4 m/s2
and a velocity of 1.5 m/s, both directed
to the left, determine (a) the angular accelerations of the caster and
of the cylinder, (b) the accelerations of the centers of the caster and
of the cylinder.
A B
C
Fig. P15.249
15.250 A baseball pitching machine is designed to deliver a baseball with
a ball speed of 108 kmph and a ball rotation of 300 rpm clockwise.
Knowing that there is no slipping between the wheels and the base-
ball during the ball launch, determine the angular velocities of
wheels A and B.
175 mm
75 mm
B
175 mm
A
Fig. P15.250
15.251 Knowing that inner gear A is stationary and outer gear C starts from
rest and has a constant angular acceleration of 4 rad/s2
clockwise,
determine at t 5 5 s (a) the angular velocity of arm AB,
(b) the angular velocity of gear B, (c) the acceleration of the point
on gear B that is in contact with gear A.
Review Problems
A
D
B
q
r
Fig. P15.248
80 mm
C
40 mm 80 mm
B
A
Fig. P15.251
†
Hestenes, D., Wells, M., and Swakhamer, G (1992). The force concept inventory. The Physics
Teacher, 30: 141–158.
Streveler, R. A., Litzinger, T. A., Miller, R. L., and Steif, P. S. (2008). Learning conceptual knowl-
edge in the engineering sciences: Overview and future research directions, JEE, 279–294.

xvi Guided Tour
Free Body and Impulse-Momentum Diagram Practice
Problems. Drawing diagrams correctly is a critical step in solving
kinetics problems in dynamics. A new type of problem has been added to
the text to emphasize the importance of drawing these diagrams. In
Chaps. 12 and 16 the Free Body Practice Problems require students to
draw a free-body diagram (FBD) showing the applied forces and an
equivalent diagram called a “kinetic diagram” (KD) showing ma or its
components and Ia. These diagrams provide students with a pictorial
representation of Newton’s second law and are critical in helping students
to correctly solve kinetic problems. In Chaps. 13 and 17 the Impulse-
Momentum Diagram Practice Problems require students to draw diagrams
showing the momenta of the bodies before impact, the impulses exerted
on the body during impact, and the final momenta of the bodies. The
answers to all of these questions can be accessed through Connect.
1039
FREE-BODY PRACTICE PROBLEMS
16.F1 A 6-ft board is placed in a truck with one end resting against a block
secured to the floor and the other leaning against a vertical partition.
Draw the FBD and KD necessary to determine the maximum
allowable acceleration of the truck if the board is to remain in the
position shown.
16.F2 A uniform circular plate of mass 3 kg is attached to two links AC
and BD of the same length. Knowing that the plate is released
from rest in the position shown, in which lines joining G to A and
B are, respectively, horizontal and vertical, draw the FBD and KD
for the plate.
75°
75°
C
A
D
B
G
Fig. P16.F2
16.F3 Two uniform disks and two cylinders are assembled as indicated.
Disk A weighs 20 lb and disk B weighs 12 lb. Knowing that the
system is released from rest, draw the FBD and KD for the whole
system.
18 lb
15 lb
6 in.
8 in.
B
C D
A
Fig. P16.F3
16.F4 The 400-lb crate shown is lowered by means of two overhead cranes.
Knowing the tension in each cable, draw the FBD and KD that can
be used to determine the angular acceleration of the crate and the
acceleration of the center of gravity.
A
B
78°
Fig. P16.F1
TA TB
6.6 ft
3.6 ft
3.3 ft
1.8 ft
A
G
B
Fig. P16.F4

xvii
Online Learning Centre
Find the following instructor resources available through
(http://www.mhhe.com/beer/vme/11e/dynamics):
• Instructor’s and Solutions Manual. The Instructor’s and Solutions
Manual that accompanies the eleventh edition features solutions to all
end of chapter problems.
• Lecture PowerPoint Slides for each chapter that can be modified.
These generally have an introductory application slide, animated
worked-out problems that you can do in class with your students,
concept questions, and “what-if?” questions at the end of the units.
• Textbook images
LearnSmart is available as
an integrated feature of McGraw-Hill Connect. It is an adaptive learning
system designed to help students learn faster, study more efficiently, and
retain more knowledge for greater success. LearnSmart assesses a student’s
knowledge of course content through a series of adaptive questions. It
pinpoints concepts the student does not understand and maps out a
personalized study plan for success. This innovative study tool also has
features that allow instructors to see exactly what students have
accomplished and a built-in assessment tool for graded assignments.
SmartBook™ is the first and
only adaptive reading experience available for the higher education mar-
ket. Powered by an intelligent diagnostic and adaptive engine, SmartBook
facilitates the reading process by identifying what content a student knows
and doesn’t know through adaptive assessments. As the student reads, the
reading material constantly adapts to ensure the student is focused on
the content he or she needs the most to close any knowledge gaps.
Visit the following site for a demonstration of LearnSmart or Smart-
Book: www.learnsmartadvantage.com
Digital Resources
NEW!
NEW!

xviii
A special thanks to Jim Widmann of California Polytechnic State Univer-
sity, who thoroughly checked the solutions and answers of all problems
in this edition and then prepared the solutions for the accompanying
Instructor’s and Solutions Manual. The authors would also like to thank
Baheej Saoud, who helped develop and solve several of the new problems
in this edition.
We are pleased to acknowledge David Chelton, who carefully reviewed
the entire text and provided many helpful suggestions for revising this edition.
The authors thank the many companies that provided photographs
for this edition. We also wish to recognize the determined efforts and
patience of our photo researcher Danny Meldung.
The authors also thank the members of the staff at McGraw-Hill for
their support and dedication during the preparation of this new edition.
Phillip J. Cornwell
Brian P. Self
The authors gratefully acknowledge the many helpful comments and
suggestions offered by focus group attendees and by users of the previous
editions of Vector Mechanics for Engineers:
Acknowledgments
George Adams
Northeastern University
William Altenhof
University of Windsor
Sean B. Anderson
Boston University
Manohar Arora
Colorado School of Mines
Gilbert Baladi
Michigan State University
Francois Barthelat
McGill University
Oscar Barton, Jr.
U.S. Naval Academy
M. Asghar Bhatti
University of Iowa
Shaohong Cheng
University of Windsor
Philip Datseris
University of Rhode Island
Timothy A. Doughty
University of Portland
Howard Epstein
University of Connecticut
Asad Esmaeily
Kansas State University,
Civil Engineering Department
David Fleming
Florida Institute of Technology
Jeff Hanson
Texas Tech University
David A. Jenkins
University of Florida
Shaofan Li
University of California, Berkeley
William R. Murray
Cal Poly State University
Eric Musslman
University of Minnesota, Duluth
Masoud Olia
Wentworth Institute of
Technology
Renee K. B. Petersen
Washington State University
Amir G Rezaei
California State Polytechnic
University, Pomona
Martin Sadd
University of Rhode Island
Stefan Seelecke
North Carolina State University
Yixin Shao
McGill University
Muhammad Sharif
The University of Alabama
Anthony Sinclair
University of Toronto
Lizhi Sun
University of California, lrvine
Jeffrey Thomas
Northwestern University
Jiashi Yang
University of Nebraska
Xiangwa Zeng
Case Western Reserve University

xix
Guided Tour
The publishers would like to acknowledge the suggestions and
praise received from the reviewers of Dynamics, Special Indian Edition
(SIE). There names are as follows:
Sandeep Chabbra
Deptt of Mech Engg, KIET,
Ghaziabad
Sri Hari Prasad Anne
Jawahar Lal Nehru Technological
University, Andhra Pradesh
A Nelson
Jawahar Lal Nehru Technological
University, Hyderabad
Sujit Majumder
Maulana Abdul Kalam Azad
University of Technology, Kolkata
Kanchan Chatterjee
Dr. B. C. Roy Engineering College
Durgapur, West Bengal
Gandhirajan Balaji
PSNA College of Engineering and
Technology, Dindigul, Tamilnadu
Goutam Paul
Department of Mechanical
Engineering, MCKV Institute of
Engineering, Howrah, West Bengal
Sunil Pansare
St. Franics Institute of Technology,
Mumbai
Manish Joglekar
IIT Roorkee
Abhishek Mishra
NIT Delhi
Arun Jalan
BITS Pilani, Pilani
Dr. Cherian Samuel
BHU
Dr. J. S. Rathore
BITS Pilani, Pilani
Dr. Nilanjan Mallik
IIT (BHU) Varanasi
Rajnesh Tyagi
IIT (BHU) Varanasi

xxi
a, a Acceleration
a Constant; radius; distance; semimajor
axis of ellipse
a, a Acceleration of mass center
aB/A Acceleration of B relative to frame in
translation with A
aP/^ Acceleration of P relative to rotating
frame ^
ac Coriolis acceleration
A, B, C, . . . Reactions at supports and connections
A, B, C, . . . Points
A Area
b Width; distance; semiminor axis of
ellipse
c Constant; coefficient of viscous damping
C Centroid; instantaneous center of rotation;
capacitance
d Distance
en, et Unit vectors along normal and tangent
er, eθ Unit vectors in radial and transverse
directions
e Coefficient of restitution; base of natural
logarithms
E Total mechanical energy; voltage
f Scalar function
ff Frequency of forced vibration
fn Natural frequency
F Force; friction force
g Acceleration of gravity
G Center of gravity; mass center; constant
of gravitation
h Angular momentum per unit mass
HO Angular momentum about point O
H
#
G Rate of change of angular momentum HG
with respect to frame of fixed orientation
(H
#
G)Gxyz Rate of change of angular momentum HG
with respect to rotating frame Gxyz
i, j, k Unit vectors along coordinate axes
i Current
I, Ix, . . . Moments of inertia
I Centroidal moment of inertia
Ixy, . . . Products of inertia
J Polar moment of inertia
k Spring constant
kx, ky, kO Radii of gyration
k Centroidal radius of gyration
l Length
L Linear momentum
L Length; inductance
m Mass
m9 Mass per unit length
M Couple; moment
MO Moment about point O
MR
O Moment resultant about point O
M Magnitude of couple or moment; mass of
earth
MOL Moment about axis OL
n Normal direction
N Normal component of reaction
O Origin of coordinates
P Force; vector
P
#
Rate of change of vector P with respect
to frame of fixed orientation
q Mass rate of flow; electric charge
Q Force; vector
Q
#
Rate of change of vector Q with respect
to frame of fixed orientation
(Q
#
)Oxyz Rate of change of vector Q with respect to
frame Oxyz
r Position vector
rB/A Position vector of B relative to A
r Radius; distance; polar coordinate
R Resultant force; resultant vector; reaction
R Radius of earth; resistance
s Position vector
s Length of arc
t Time; thickness; tangential direction
T Force
T Tension; kinetic energy
u Velocity
u Variable
U Work
UNC
122 work done by non-conservative forces
v, v Velocity
v Speed
v, v Velocity of mass center
vB/A Velocity of B relative to frame in
translation with A
vP/^ Velocity of P relative to rotating frame ^
List of Symbols

xxii List of Symbols
V Vector product
V Volume; potential energy
w Load per unit length
W, W Weight; load
x, y, z Rectangular coordinates; distances
x
#
, y
#
, z
#
Time derivatives of coordinates x, y, z
x, y, z Rectangular coordinates of centroid,
center of gravity, or mass center
α, α Angular acceleration
α, β, g Angles
g Specific weight
δ Elongation
e Eccentricity of conic section or of orbit
l Unit vector along a line
η Efficiency
θ Angular coordinate; Eulerian angle;
angle; polar coordinate
µ Coefficient of friction
ρ Density; radius of curvature
τ Periodic time
τn Period of free vibration
f Angle of friction; Eulerian angle;
phase angle; angle
w Phase difference
c Eulerian angle
v, v Angular velocity
vf Circular frequency of forced vibration
vn Natural circular frequency
V Angular velocity of frame of reference

The motion of the paraglider can be described in terms of its
position, velocity, and acceleration. When landing, the pilot of the
paraglider needs to consider the wind velocity and the relative
motion of the glider with respect to the wind. The study of
motion is known as kinematics and is the subject of this chapter.
Kinematics of Particles
11

616 Kinematics of Particles
Introduction
11.1 RECTILINEAR MOTION OF
PARTICLES
Acceleration
11.1B Determining the Motion of a
Particle
11.2 SPECIAL CASES AND
RELATIVE MOTION
11.2A Uniform Rectilinear Motion
11.2B Uniformly Accelerated
Rectilinear Motion
11.2C Motion of Several Particles
*11.3 GRAPHICAL SOLUTIONS
11.4 CURVILINEAR MOTION OF
PARTICLES
Acceleration Vectors
11.4B Derivatives of Vector Functions
11.4C Rectangular Components of
Velocity and Acceleration
11.4D Motion Relative to a Frame in
Translation
11.5 NON-RECTANGULAR
COMPONENTS
11.5A Tangential and Normal
Components
11.5B Radial and Transverse
Components
Objectives
• Describe the basic kinematic relationships between
position, velocity, acceleration, and time.
• Solve problems using these basic kinematic
relationships and calculus or graphical methods.
• Define position, velocity, and acceleration in terms of
Cartesian, tangential and normal, and radial and
transverse coordinates.
• Analyze the relative motion of multiple particles by
using a translating coordinate system.
• Determine the motion of a particle that depends on
the motion of another particle.
• Determine which coordinate system is most appropri-
ate for solving a curvilinear kinematics problem.
• Calculate the position, velocity, and acceleration of a
particle undergoing curvilinear motion using Cartesian,
tangential and normal, and radial and transverse
coordinates.
Introduction
Chapters 1 to 10 were devoted to statics, i.e., to the analysis of bodies at
rest. We now begin the study of dynamics, which is the part of mechanics
that deals with the analysis of bodies in motion.
Although the study of statics goes back to the time of the Greek
philosophers, the first significant contribution to dynamics was made by
Galileo (1564–1642). Galileo’s experiments on uniformly accelerated bod-
ies led Newton (1642–1727) to formulate his fundamental laws of motion.
Dynamics includes two broad areas of study:
1. Kinematics, which is the study of the geometry of motion. The principles
of kinematics relate the displacement, velocity, acceleration, and time
of a body’s motion, without reference to the cause of the motion.
2. Kinetics, which is the study of the relation between the forces acting
on a body, the mass of the body, and the motion of the body. We use
kinetics to predict the motion caused by given forces or to determine
the forces required to produce a given motion.
Chapters 11 through 14 describe the dynamics of particles; in
Chap. 11, we consider the kinematics of particles. The use of the word
particles does not mean that our study is restricted to small objects; rather,
it indicates that in these first chapters we study the motion of bodies—
possibly as large as cars, rockets, or airplanes—without regard to their size
or shape. By saying that we analyze the bodies as particles, we mean that
we consider only their motion as an entire unit; we neglect any rotation
about their own centers of mass. In some cases, however, such a rotation is
not negligible, and we cannot treat the bodies as particles. Such motions are
analyzed in later chapters dealing with the dynamics of rigid bodies.

In the first part of Chap. 11, we describe the rectilinear motion of
a particle; that is, we determine the position, velocity, and acceleration of
a particle at every instant as it moves along a straight line. We first use
general methods of analysis to study the motion of a particle; we then
consider two important particular cases, namely, the uniform motion and
the uniformly accelerated motion of a particle (Sec. 11.2). We then discuss
the simultaneous motion of several particles and introduce the concept
of the relative motion of one particle with respect to another. The first part
of this chapter concludes with a study of graphical methods of analysis
and their application to the solution of problems involving the rectilinear
motion of particles.
In the second part of this chapter, we analyze the motion of a par-
ticle as it moves along a curved path. We define the position, velocity, and
acceleration of a particle as vector quantities and introduce the derivative
of a vector function to add to our mathematical tools. We consider applica-
tions in which we define the motion of a particle by the rectangular com-
ponents of its velocity and acceleration; at this point, we analyze the
motion of a projectile (Sec. 11.4C). Then we examine the motion of a
particle relative to a reference frame in translation. Finally, we analyze
the curvilinear motion of a particle in terms of components other than
rectangular. In Sec. 11.5, we introduce the tangential and normal compo-
nents of an object’s velocity and acceleration and then examine the radial
and transverse components.
11.1 RECTILINEAR MOTION
OF PARTICLES
A particle moving along a straight line is said to be in rectilinear motion.
The only variables we need to describe this motion are the time, t, and
the distance along the line, x, as a function of time. With these variables,
we can define the particle’s position, velocity, and acceleration, which
completely describe the particle’s motion. When we study the motion of
a particle moving in a plane (two dimensions) or in space (three dimensions),
we will use a more general position vector rather than simply the distance
along a line.
Acceleration
At any given instant t, a particle in rectilinear motion occupies some
position on the straight line. To define the particle’s position P, we choose
a fixed origin O on the straight line and a positive direction along the line.
We measure the distance x from O to P and record it with a plus or minus
sign, according to whether we reach P from O by moving along the line
in the positive or negative direction. The distance x, with the appropriate
sign, completely defines the position of the particle; it is called the position
coordinate of the particle. For example, the position coordinate
corresponding to P in Fig. 11.1a is x 5 15 m; the coordinate corresponding
to P9 in Fig. 11.1b is x9 5 22 m.
Fig. 11.1 Position is measured from a fixed
origin. (a) A positive position coordinate;
(b) a negative position coordinate.
O
O
P
x
x
(a)
(b)
1 m
P'
x'
x
1 m

When we know the position coordinate x of a particle for every value
of time t, we say that the motion of the particle is known. We can provide
a “timetable” of the motion in the form of an equation in x and t, such as
x 5 6t2
2 t3
, or in the form of a graph of x versus t, as shown in Fig. 11.6.
The units most often used to measure the position coordinate x are the
meter (m) in the SI system of units†
and the foot (ft) in the U.S. customary
system of units. Time t is usually measured in seconds (s).
Now consider the position P occupied by the particle at time t and
the corresponding coordinate x (Fig. 11.2). Consider also the position P9
occupied by the particle at a later time t 1 Dt. We can obtain the position
coordinate of P9 by adding the small displacement Dx to the coordinate x
of P. This displacement is positive or negative according to whether P9 is
to the right or to the left of P. We define the average velocity of the
particle over the time interval Dt as the quotient of the displacement Dx
and the time interval Dt as
Average velocity 5
Dx
Dt
If we use SI units, Dx is expressed in meters and Dt in seconds; the
average velocity is then expressed in meters per second (m/s). If we use
U.S. customary units, Dx is expressed in feet and Dt in seconds; the
average velocity is then expressed in feet per second (ft/s).
We can determine the instantaneous velocity v of a particle at the
instant t by allowing the time interval Dt to become infinitesimally small. Thus,
Instantaneous velocity 5 v 5 lim
Dt y0
Dx
Dt
The instantaneous velocity is also expressed in m/s or ft/s. Observing that
the limit of the quotient is equal, by definition, to the derivative of x with
respect to t, we have
Velocity of a particle
along a line
v 5
dx
dt
(11.1)
We represent the velocity v by an algebraic number that can be positive or
negative.‡
A positive value of v indicates that x increases, i.e., that the particle
moves in the positive direction (Fig. 11.3a). A negative value of v indicates
that x decreases, i.e., that the particle moves in the negative direction
(Fig. 11.3b). The magnitude of v is known as the speed of the particle.
Consider the velocity v of the particle at time t and also its velocity
v 1 Dv at a later time t 1 Dt (Fig. 11.4). We define the average acceleration
of the particle over the time interval Dt as the quotient of Dv and Dt as
Average acceleration 5
Dv
Dt
v 5
dx
dx
d
dt
†
See Sec. 1.3.
‡
As you will see in Sec. 11.4A, velocity is actually a vector quantity. However, since we are
considering here the rectilinear motion of a particle where the velocity has a known and fixed
direction, we need only specify its sense and magnitude. We can do this conveniently by using
a scalar quantity with a plus or minus sign. This is also true of the acceleration of a particle
in rectilinear motion.
Fig. 11.2 A small displacement Dx from
time t to time t 1 Dt.
O
P
x
x
(t) (t + ∆t)
P'
∆x
Photo 11.1 The motion of this solar car can
be described by its position, velocity, and
acceleration.
Fig. 11.3 In rectilinear motion, velocity can
be only (a) positive or (b) negative along the
line.
(a)
(b)
P
P
x
x
v > 0
v < 0
Fig. 11.4 A change in velocity from v to
v 1 Dv corresponding to a change in time
from t to t 1 Dt.
(t) (t + ∆t)
v + ∆v
P'
P
x
v

If we use SI units, Dv is expressed in m/s and Dt in seconds; the average
acceleration is then expressed in m/s2
. If we use U.S. customary units, Dv
is expressed in ft/s and Dt in seconds; the average acceleration is then
expressed in ft/s2
.
We obtain the instantaneous acceleration a of the particle at the
instant t by again allowing the time interval Dt to approach zero. Thus,
Instantaneous acceleration 5 a 5 lim
Dt y0
Dv
Dt
The instantaneous acceleration is also expressed in m/s2
or ft/s2
. The limit
of the quotient, which is by definition the derivative of v with respect to t,
measures the rate of change of the velocity. We have
Acceleration of a
particle along a line
a 5
dv
dt
(11.2)
or substituting for v from Eq. (11.1),
a 5
d2
x
dt2
(11.3)
We represent the acceleration a by an algebraic number that can be posi-
tive or negative (see the footnote on the preceding page). A positive value
of a indicates that the velocity (i.e., the algebraic number v) increases.
This may mean that the particle is moving faster in the positive direction
(Fig. 11.5a) or that it is moving more slowly in the negative direction
(Fig. 11.5b); in both cases, Dv is positive. A negative value of a indicates
that the velocity decreases; either the particle is moving more slowly in
the positive direction (Fig. 11.5c), or it is moving faster in the negative
direction (Fig. 11.5d).
Sometimes we use the term deceleration to refer to a when the speed
of the particle (i.e., the magnitude of v) decreases; the particle is then moving
more slowly. For example, the particle of Fig. 11.5 is decelerating in parts
b and c; it is truly accelerating (i.e., moving faster) in parts a and d.
a 5
dv
v
dt
a 5
d2
x
dt2
Fig. 11.5 Velocity and acceleration can be in the same or different directions.
(a, d) When a and v are in the same direction, the particle speeds up;
(b, c) when a and v are in opposite directions, the particle slows down.
v
P
x
P'
v'
a > 0
(a)
x
v
P
P'
v'
a > 0
(b)
x
v
P P'
v'
a < 0
(c)
x
v
P
P'
v'
a < 0
(d)

We can obtain another expression for the acceleration by eliminating
the differential dt in Eqs. (11.1) and (11.2). Solving Eq. (11.1) for dt, we
have dt 5 dx/v; substituting into Eq. (11.2) gives us
a 5 v
dv
dx
(11.4)
a 5 v
dv
v
dx
dx
d
Concept Application 11.1
Consider a particle moving in a straight line, and assume that its position
is defined by
x 5 6t2
2 t3
where t is in seconds and x in meters. We can obtain the velocity v at any
time t by differentiating x with respect to t as
v 5
dx
dt
5 12t 2 3t2
We can obtain the acceleration a by differentiating again with respect to t.
Hence,
a 5
dv
dt
5 12 2 6t
In Fig. 11.6, we have plotted the position coordinate, the velocity, and the
acceleration. These curves are known as motion curves. Keep in mind,
however, that the particle does not move along any of these curves; the
particle moves in a straight line.
Since the derivative of a function measures the slope of the corre-
sponding curve, the slope of the x–t curve at any given time is equal to
the value of v at that time. Similarly, the slope of the v–t curve is equal
to the value of a. Since a 5 0 at t 5 2 s, the slope of the v–t curve must
be zero at t 5 2 s; the velocity reaches a maximum at this instant. Also,
since v 5 0 at t 5 0 and at t 5 4 s, the tangent to the x–t curve must be
horizontal for both of these values of t.
A study of the three motion curves of Fig. 11.6 shows that the motion
of the particle from t 5 0 to t 5 ∞ can be divided into four phases:
1. The particle starts from the origin, x 5 0, with no velocity but with
a positive acceleration. Under this acceleration, the particle gains a
positive velocity and moves in the positive direction. From t 5 0 to
t 5 2 s, x, v, and a are all positive.
2. At t 5 2 s, the acceleration is zero; the velocity has reached its
maximum value. From t 5 2 s to t 5 4 s, v is positive, but a is
negative. The particle still moves in the positive direction but more
slowly; the particle is decelerating.
3. At t 5 4 s, the velocity is zero; the position coordinate x has reached
its maximum value (32 m). From then on, both v and a are negative;
the particle is accelerating and moves in the negative direction with
increasing speed.
4. At t 5 6 s, the particle passes through the origin; its coordinate x is
then zero, while the total distance traveled since the beginning of the
motion is 64 m (i.e., twice its maximum value). For values of t larger
than 6 s, x, v, and a are all negative. The particle keeps moving in
the negative direction—away from O—faster and faster. �
Fig. 11.6 Graphs of position,
velocity, and acceleration as
functions of time for Concept
Application 11.1.
x(m)
v(m/s)
t(s)
t(s)
t(s)
32
24
16
8
0
12
2
2
4
4
6
6
0
–12
a(m/s2)
12
0
–24
–12
–24
–36
2 4 6

11.1B Determining the Motion of a
Particle
We have just seen that the motion of a particle is said to be known if we
know its position for every value of the time t. In practice, however, a
motion is seldom defined by a relation between x and t. More often, the
conditions of the motion are specified by the type of acceleration that the
particle possesses. For example, a freely falling body has a constant
acceleration that is directed downward and equal to 9.81 m/s2
or 32.2 ft/
s2
, a mass attached to a stretched spring has an acceleration proportional
to the instantaneous elongation of the spring measured from its equilibrium
position, etc. In general, we can express the acceleration of the particle
as a function of one or more of the variables x, v, and t. Thus, in order
to determine the position coordinate x in terms of t, we need to perform
two successive integrations.
Let us consider three common classes of motion.
1. a 5 f(t). The Acceleration Is a Given Function of t. Solving Eq. (11.2)
for dv and substituting f(t) for a, we have
dv 5 adt
dv 5 f(t)dt
Integrating both sides of the equation, we obtain
edv 5 ef(t)dt
This equation defines v in terms of t. Note, however, that an arbitrary
constant is introduced after the integration is performed. This is due to
the fact that many motions correspond to the given acceleration a 5 f(t).
In order to define the motion of the particle uniquely, it is necessary to
specify the initial conditions of the motion, i.e., the value v0 of the
velocity and the value x0 of the position coordinate at t 5 0. Rather
than use an arbitrary constant that is determined by the initial conditions,
it is often more convenient to replace the indefinite integrals with
definite integrals. Definite integrals have lower limits corresponding to
the initial conditions t 5 0 and v 5 v0 and upper limits corresponding
to t 5 t and v 5 v. This gives us
#
v
v0
dv 5#
t
0
f(t)dt
v 2 v0 5#
t
0
f(t)dt
which yields v in terms of t.
We can now solve Eq. (11.1) for dx as
dx 5 vdt
and substitute for v the expression obtained from the first integration.
Then we integrate both sides of this equation via the left-hand side with
respect to x from x 5 x0 to x 5 x and the right-hand side with respect
to t from t 5 0 to t 5 t. In this way, we obtain the position coordinate
x in terms of t; the motion is completely determined.

Other documents randomly have
different content

memory and become voluntarily ignorant. No; let, rather, the
opposite course be taken! Let us popularize psychology as well!
11
These Advanced People
A. Free Love is all right in theory, but all wrong in practice. B. On the
contrary! I think it is all right in practice, but all wrong in theory.
12
Sex in Literature
In English literature, until very modern times, sex was treated only
within the limits of a very well-understood convention. From this
convention the physiological was strictly excluded. Yet, of our
classical writers, even in the most artificial periods, it cannot be said
that they did not understand sex. No matter how "unreal" they
might be in writing about Love, the physiological contingencies of
Love were unmistakably implied in their works, but only, it is true,
implied. The moderns, however, saw in this treatment of Love
nothing but a convention, a "lie"; and they became impatient of the
artificiality, as if art could be anything but artificial! To what was the
change of attitude due? Not to a failure in the artistic convention:
that was perfectly sound. No, it was the reader who had failed: a
generation of readers had arisen who had not learnt the art of
reading, who did not understand reading as a cultured amateur of
the eighteenth century, for instance, understood it. Literature was to
this reader a document, not an art. He had no eye for what is
written between the lines—for symbolism, idealization, "literature."
And it was to satisfy him that the realistic school arose: it arose,
indeed, out of himself. In the realist the modern reader has become
writer: the man who could not learn the art of reading has here
essayed the more difficult art of writing—documentary art!

13
History of a Realist
Who will write a series of biographies of modern writers, illustrating
this thesis: that they are nothing more than modern readers wielding
a hasty pen? Such a set of memoirs would almost compensate us for
having read the works of these writers. How interesting, for
instance, it would be to know how many years—surely it would be
years?—they spent in trying to understand literature before they
dedicated themselves to its service. How interesting, again, to
discover how many hours each day X, the celebrated novelist,
devotes to contemplation, how many to writing for the newspapers,
and how many to his present masterpiece. What! one hour's thought
has actually preceded five hours' dictation! This revelation is, after
all, not so startling. On second thought, these memoirs seem
superfluous; we can read everything we wish to know of the
moderns in their works.
Yet, for our better amusement, will not some one write his one and
only novel, giving the true history of the novelist? A novel against
novels! But for that we need a second Cervantes, yet how unlike the
first! For on this occasion it is not Don Quixote that must be
satirized, but Sancho Panza.
14
Novelists by Habit
All of us who read are novelists more or less nowadays: that is to
say, we collect "impressions," "analyse" ourselves, make a pother
about sex, and think that people, once they are divorced, live
happily ever after. The habit of reading novels has turned us into
this! When one of us becomes articulate, however—in the form of a
novel—he only makes explicit his kinship with the rest; he proclaims
to all the world that he is a mediocrity.

15
The Only Course
All the figures in this novel are paltry; we despise them, and, if we
were in danger of meeting them in real life, would take steps to
avoid them; yet such is the author's adroitness that we are led on
helplessly through the narrative, through unspeakable sordidness of
circumstance and soul, hating ourselves and him, and feeling
nothing better than slaves. To rouse our anxiety lest Herbert lose
five pounds, or Mabel find it impossible to get a new dress, this is
art, this is modern art! But to feel anxiety about such things is
ignoble; and to live in a sordid atmosphere, even if it be of a book, is
the part of a slave. And yet we cannot but admire. For in this novel
what subtlety in the treatment there must be overlying the
fundamental vulgarity of the theme! How is Art, which should make
Man free, here transformed into a potent means for enslaving him!
It is impossible to yield oneself to the sway of a modern realist
without a loss in one's self-respect. To what is due this conspicuous
absence of nobility in modern writers? But is the question, indeed,
worth the asking? For to the artist and to him who would retain
freedom of soul, there is only one course with the paltry in literature
—to avoid it.
16
The Average Man
It is surely one of G. K. Chesterton's paradoxes that he praises the
average man. For he is not himself an average man, but a man of
genius; he does not write of the average man, but of grotesques; he
is not read by the average man, but by intellectuals and the
nonconformist middle-class. The true prophets of the average man
are the popular realistic novelists. For they write of him and for him
—yes, even when they write "for themselves," when they are
"serious artists." Who, then, but them should extol him? It is their
métier.

17
The "New" Writers
The fault of the most modern writers—and especially of the novelists
—is not that they are too modern, but that they are too traditional.
It is true, they are not traditional in the historical manner of G. K.
Chesterton, who wishes to destroy one tradition—the modern
tradition—in order to get back to another—the mediæval. To Mr.
Chesterton tradition is a matter of selection; the dead tradition
seems to him nobler than the living; and, deliberately, therefore, he
would return to it. The new writers, however, follow a tradition also,
though a much narrower one; they, too, believe in the past, but only,
alas, in the immediate past; they are slaves to the generation which
preceded theirs. In short, that which is disgusting in them is their
inability to rise high enough to see their little decade or two, and to
challenge it, if they cannot from the standpoint of a nobler future,
then, at least, from that of the noblest past. But how weak must a
generation be which is not strong enough to challenge and
supersede Arnold Bennett, for instance.
18
The Modern Reader
What is it that the modern reader demands from those who write for
him? To be challenged, and again to be challenged, and evermore to
be challenged—but on no account to be asked to accept a challenge,
on no account to be expected to take sides! A seat at the
tournament is all that he asks, where he may watch the most
sincere and intrepid spirits of his time waging their desperate battle
and spilling their life blood upon the sand. How he loves them when,
with high gesture, they fling down their gauntlets and utter their
blasphemies! His heart then exults within him; but, why? Simply
because he is a connoisseur; simply because he collects gauntlets!

19
The Public
Of the modern writers who are in earnest, Mr. Chesterton has had
the most ironical fate: he has been read by the people who will
never agree with him. To the average man for whom he writes he is
an intellectual made doubly inaccessible by his orthodoxy and his
paradoxy. It is the advanced, his bête noire, who read him, admire
him, and—disagree with him.
20
Reader and Writer
The modern reader loves to be challenged. The modern writer, if he
is in earnest, however, is bound to challenge him. This is his greatest
burden; that he must fall a victim of the advanced idlers. But one
day he thinks he see a way of escape. He has noticed that the
reader desires not only to be challenged, but to be able to
understand the challenge at a glance. And here he sees his
advantage. I shall write, he says, to himself, in a manner beautiful,
exact, and yet not easily understood; so I shall throw off the
intellectual coquettes and secure my audience of artists, for my style
is beautiful; an audience of critics, for my style is exact; an audience
of patient, resolute, conscientious intellects, for my style is difficult.
This, perhaps, was the conscious practice of Nietzsche. But he did
not foresee that, for the benefit of the intellectual coquettes, who
must have hold of new thoughts by one end or another, a host of
popularizers would be born; he did not reckon with the
Nietzscheans!
21
Popularity
How amazingly popular he is. Even the man in the street reads him.
Yes; but it is because he has first read the man in the street.

22
Middle Age's Betrayals
It is not easy to tell by a glance what is the character of a young
man; his soul has not yet etched itself clearly enough upon his body.
But one may read a middle-aged man's soul with perfect ease; and
not only his soul but his history. For when a man has passed five-
and-forty, he looks—not what he is, perhaps—but certainly what he
has been. If he has been invariably respectable, he is now the very
picture of respectability. If he has been a man about town or a
secret toper, the fact is blazoned so clearly on his face that even a
child can read it. If he has studied, his very walk, to use a phrase of
Nietzsche's, is learned. As for the poet, we know how terribly
poetical he looks in middle age—poor devil! Well, to every one of
you, I say, Beware!
23
The Novelists and the Artist
Is it the modern novelists who are to be blamed for the degraded
image of the artist which lives in the minds of the cultured populace?
Turgenieff in "On the Eve," and Henry James in "Roderick Hudson"
display the artist simply as a picturesque waster, an oh so charming,
impulsive, childlike, naïve waster. But, in doing so, they surely
confused the artist with the man of artistic temperament. Of the
artistic temperament, however, the great artists had very often little
or nothing—far less, certainly, than either Shubin or Roderick. The
great examples of last century, the Goethes, Ibsens, and Nietzsches,
knew that there were qualities more essential to them than
temperament; discipline, for instance, perseverance, truth to
themselves, self-control. How is it possible, indeed, without these
virtues—virtues of the most difficult and heroic kind—for the artist to
bring his gifts to maturity, to become great? His discipline to beauty
must be as severe as the discipline of the saint to holiness. And,
then, how has his sensuousness been misconstrued and vulgarized;

and treated precisely, indeed, as if it were the licentiousness of a
present-day Tom Jones! That artists can be thought about in such a
way proves only one thing, namely, in what poor esteem they are
now held. We need a new ideal of the artist; or, failing that, an old
one, that of Plato, perhaps, or of Leonardo, or of Nietzsche.
24
Decadence and Health
It is in the decadent periods that the most triumphantly healthy men
—one or two—appear. The corrupt Italy of the Renaissance gave
birth to Leonardo; the Europe of Gautier, Baudelaire and Wilde
produced Nietzsche. In decadent eras both disease and health
become more self-conscious; they are cultivated, enhanced and
refined. It has been said that the best way to remain healthy is not
to think of health. But lack of self-consciousness speaks here.
Perhaps the Middle Ages were as diseased as our own—only they did
not know it! Is decadence nothing more than the symptom of a self-
conscious age? And is "objectivity" the antidote? Well, we might
believe this if we could renounce our faith that mankind will yet
become healthy—if we could become optimists in the present-day
sense!
25
Art in Modern Society
An object of beauty has in modern surroundings a dangerous
seduction which it did not possess in less hideous eras. In this is
there to be found a contributory explanation of Decadence—the
decadent being one who feels the power of beauty intensely, and
the repulsion from his environment as intensely, and who plunges
into the enjoyment of beauty madly, with abandonment? In a
society, however, which was not hideous as ours is, and in which
beauty was distributed widely over all the aspects and forms of

existence, the intoxication of beauty would not be felt with the same
terrible intensity; a beautiful object would be enjoyed simply as one
among many lovely things. In short, it would be enjoyed in the
manner of health, not in that of sickness. It is the contrast that is
dangerous; the aridity of modern life arouses a terrible thirst, which
is suddenly presented with the spectacle of a beauty unaccountable
and awful; and this produces a dislocation and convulsion of the
very soul. So that the present-day artist, if he would retain his health
—if he would remain an artist—must curb his very love of the
beautiful, and treat beauty, when he meets it, as he always does, in
the gutter, a little cynically. Otherwise he will lose his wits, and Art
will become his Circe. Therefore, mockery and hard laughter—alas,
that it must be so!
26
Art in Industry
In those wildernesses of dirt, ugliness and obscenity, our industrial
towns, there are usually art galleries, where the daintiest and most
beautiful things, the flowers of Greek statuary, for instance, bloom
among the grime like a band of gods imprisoned in a slum. The
spectacle of art in such surroundings sometimes strikes us as being
at once ludicrous and pathetic, like something delicate and lovely
sprawling in the gutter, or an angel with a dirty face.
27
Conventions
The revolt against conventions in art, thought, life and manners may
be due to at least more than one cause. It is usually ascribed to
"vitality" which "breaks through" forms, because it desires to be
"free." But common sense tells us that more than two or three of
our friends abjure convention for an altogether different reason—to
be candid, on account of a lack of vitality resulting in laziness and

the inability to endure restraint of any kind. And, for the others, we
shall judge their "vitality" to be justified when they build new
conventions worthy of observance, instead of running their heads
finally into illimitable space. Or does their strength not go just so
far? There is something suspicious about this vitality which cannot
create: it resembles impotence so much! Heaven preserve the
moderns from their "vitality"!
28
"Vitality"
When moderns talk of the "vitality" of their most lauded writer, what
they mean is finally the size of his muscles, physical energy, or, at
the most, strong emotions; not vigour of mind. Well, let us on no
account make the opposite mistake and revile the large muscle and
energetic feelings: they are admirable things. Let us point out,
however, that vitality of emotion undisciplined by vitality of thought
leads nowhere, is often disruptive and cannot build. But to build is
our highest duty and our peculiar form of freedom—we who have
realized that there is no freedom without power. As for the old
freedom—it is only the slaves who are not already tired of it.
29
Decadence
The decisive thing, determining whether an artist shall be major or
minor, is very often not artistic at all, but moral. Yes, though it shock
our modern ears, let this be proclaimed! The more "temperament"
an artist has, the more character he requires to govern it, to make it
fruitful for him, if he would not have it get beyond control, and
wreck both him and itself. And, consequently, the great artists show,
as a rule, less "temperament" than the minor; they appear more
self-contained and less "artistic." Indeed, they smile with the hint of
irony at the merely "artistic."

It is, perhaps, when the traditions of artistic morality and discipline
have broken down, when the "temperament" has, therefore, become
unfettered and lawless, that decadence in art is born. The sincerity
of the artist, his chief virtue, is gone—the sincerity which commands
him to create only under the pressure of an artistic necessity, which
tells him, in other words, to produce nothing which is not genuine.
Without sincerity, severity and patience, nothing great in art can be
created. And it is precisely in these virtues that the decadent is
lacking. A love of beauty is his only credential as an artist, but,
undisciplined, it degenerates very soon into a love of mere effect. An
effect of beauty at all costs, whether it be the true beauty or not!
That becomes his object. Without a root in any soil, he aspires to the
condition of the water lily, and, in due time, becomes a full-blown
æsthete. Is it because he is incapable of becoming anything else?
Has he in despair grown "artistic" simply because he is not an artist?
Is Decadence the most subtle disguise of impotence? And are
decadents those who, if they had submitted to an artistic discipline
of sincerity, would never have written at all? Of some of them this is
true, but of others it is not; and in that lies the tragedy of
Decadence. Wilde himself was, perhaps, a decadent by
misadventure; for on occasion he could rise above decadence into
sincerity. "The Ballad of Reading Gaol" proves that. He was the
victim of a bad æsthetic morality, to which, it is true, he had a
predisposition. And if this is true of him, it is true, also, of his
followers. A baleful artistic ethic still rules, demoralizing the young
artist at the moment when he should be disciplining himself; and
turning, perhaps, some one with the potentiality of greatness into a
minor artist. By neglecting the harder virtues, the decadents have
made minor art inevitable and great art almost impossible.
The old tradition of artistic discipline must be regained, then, or a
new and even more severe tradition inaugurated. A text-book of
morality for artists is now overdue. When it has been written, and
the new discipline has been hailed and submitted to by the artists,
who can say if greatness may not again be possible?

30
Decadence Again
How is the dissolution of the tradition of artistic discipline to be
explained? To what cause is it to be traced? Perhaps to the more
general dissolution of tradition which has taken place in modern
times. When theological dogmas and moral values are thrown into
the melting-pot, and the discipline of centuries is dissolved into
anarchy, it is natural that artistic traditions should perish along with
them. Decadence follows free-thought: it appears at the time when
the old values lie deliquescent and the new values have not yet
risen, the dry land has not yet appeared. But this does not happen
always: the old traditions of morality, theology, politics and industry
are overthrown, the beginnings of a new tradition appear tentatively,
everything fixed has vanished, the wildest hopes and the most
chilling despair are the common possession of one and the same
generation—but, throughout, the artistic tradition is held securely
and confidently, it remains the one thing fixed in a world of
dissolution. Then an art arises greater even than that of the eras of
tradition. The pathos of the dying and the inexpressible hope of the
newly born find expression side by side; all chains are broken, and
the world appears suddenly to be immeasurable. Is this what
happened at the Renaissance?
31
Wilde
The refined degeneracy of Oscar Wilde might be explained on the
assumption that he was at once over—and under—civilized: he had
acquired all the exquisite and superfluous without the necessary
virtues. These "exquisite" virtues are unfortunately dangerous to all
but those who have become masters of the essential ones; they are
qualities of the body more than of the mind; they are developments
and embellishments of the shell of man. In acquiring them, Wilde
ministered to his body merely, and, as a consequence, it became

more and more powerful and subtle—far more powerful and subtle
than his mind. Eventually this body—senses, passions and appetite—
actually became the intellectual principle in him, of which his mind
was merely a drugged and stupefied slave!
32
Wilde and the Sensualists
The so-called Paganism of our time, the movement towards
sensualism of the followers of Wilde, is not an attempt, however
absurd, to supersede Christianity; nor is it even in essence anti-
Christian. At the most it is a reaction—not a step beyond current
religion into a new world of the spirit, but a changing from one foot
to the other, a reliance on the senses for a little, so that the over-
laboured soul may rest. And there is still much of Christianity in this
modern Paganism. Its devotees are too deeply corrupted to be
capable either of pure sensuousness or of pure spirituality. They
speak of Christ like voluptuaries, and of Eros like penitents. But it is
impossible now to become a Pagan: one must remember Ibsen's
Julian and take warning. Two thousand years of "bad conscience," of
Christian self-probing, with its deepening of the soul, cannot be
disavowed, forgotten, unlived. For Paganism a simpler spirit, mind
and sensuousness are required than we can reproduce. We cannot
feel, we cannot think, above all, we cannot feel without thinking of
our feelings, as the Pagans did. Our modern desire to take out our
soul and look at it separates us from the naïve classic sensuousness.
What, then, does modern sensualism mean? What satisfaction does
it bring to those, by no means few in number, its "followers"? A
respite, an escapade, a holiday from Christianity, from the inevitable.
For Christianity is assumed by them to be the inevitable, and it fills
them with the loathing which is evoked by the enforced
contemplation of things tyrannical and permanent. To escape from it
they plunge madly into sensuality as into a sea of redemption. But
the disgust which drives them there will eventually drive them forth
again—into asceticism and the denial of the senses. Christianity will

then appear stronger than ever, having been purged of its
"uncleanness." Yes, the sensualists of our time are the best
unconscious friends of Christianity, its "saviours," who have taken its
sins upon their shoulders.
There still remain the few who do not assume Christianity to be
inevitable, who desire, no matter how hopeless the fight may seem,
to surmount it, and who see that men have played too long the
game of reaction. "To cure the senses by the soul and the soul by
the senses" seems to them a creed for invalids. And, therefore, that
against which, above all, they guard, is a mere relapse into
sensualism. Not by fleeing from Christianity do they hope to reach
their goal; but by understanding it, perhaps by "seeing through" it,
certainly by benefiting in so far as they can by it, and, finally,
emancipating themselves from it. They know that the soil no longer
exists out of which grew the flower of Paganism, and that they must
pass through Christianity if they would reach a new sensuality and a
new spirituality. But their motto is, Spirituality first, and, after that,
only as much sensuality as our spirituality can govern! They hold
that as men become more spiritual they may safely become more
sensual; but that, to the man without spirit, sensuality and
asceticism are alike an indulgence and a curse. That the spirit should
rule—such is their desire; but it must rule as a constitutional
governor, not as an arbitrary tyrant. For the senses, too, as Heine
said, have their rights.
33
Arnold Going Down the Hill
One section of the realist school—that represented by Bennett and
John Galsworthy—may be described as a reaction from asceticism.
Men had become tired of experiencing Life only in its selected and
costly "sensations," and sought an escape from "sensations," sought
the ordinary. But another section of the school—George Moore, for
example—was merely a bad translation of æstheticism. Equally tired
of the exquisite, already having sampled all that luxury in

"sensation" could provide, the artists now sought new "sensations"—
and nothing else—in the squalid. It was the rôle of the æsthetes to
go downhill gracefully, but when they turned realists they ceased
even to do that. They went downhill sans art. Yet, in doing so, did
they not rob æstheticism of its seductiveness? And should we not,
therefore, feel grateful to them? Alas, no; for to the taste of this
age, grace and art have little fascination: it is the heavy, unlovely
and sordid that seduces. To disfigure æstheticism was to popularize
it. And now the very man in the street is—artistically speaking—
corrupted: a calamity second in importance only to the corruption of
the artists and thinkers.
34
Pater and the Æsthetes
How much of Walter Pater's exclusiveness and reclusiveness was a
revulsion from the ugliness of his time—an ugliness which he was
not strong enough to contemplate, far less to fight—it is hard to say.
Perhaps his phase of the Decadence may be defined as largely a
reaction against industrialism, just as that of Wilde may be defined
as largely a reaction against Christianity: but, in the former case as
in the latter, that against which the reaction was made was assumed
to be permanent. Indeed, by escaping from industrialism instead of
fighting it, Pater and his followers made its persistence only a little
more secure. It is true, there are excuses enough to palliate their
weakness: the delicateness of their own nerves and senses, making
them peculiarly liable to suffering, the ugliness and apparent
invulnerability of industrialism, the beauty and repose of the world of
art wherein they might take refuge and be happy. Art as
forgetfulness, art as Lethe, the seduction of that cry was strong! But
to yield to it was none the less unforgivable: it was an act traitorous
not only to society but to art itself. For what was the confession
underlying it? That the society of today and of tomorrow is, and
must be, barren; that no great art can hereafter be produced; that
there is nothing left but to enjoy what has been accomplished!

Against that presumption, not the Philistines but the great artists will
cry as the last word of Nihilism.
Pater's creed marks, therefore, a degradation of the conception of
art. Art as something exclusive, fragile and a little odd, the
occupation of a few æsthetic eccentrics—this is the most pitiable
caricature! To make themselves understood by one another, this little
clique invented a jargon of their own; in this jargon Pater's books
are written, and not only his, but those of his followers to this day. It
is a style lacking, above all, in good taste; it very easily drops into
absurdity; indeed, it is always on the verge of absurdity. It has no
masculinity, no hardness; and it is meant to be read by people a
little insincerely "æsthetic," who are conscious that they are open to
ridicule, and who are accordingly indulgent to the ridiculous; the
Fabians of art. To admire Pater's style, it is necessary first to put
oneself into the proper attitude.
35
Creator and Æsthete
The true creators and the mere æsthetes agree in this, that they are
not realists. Neither of them copies existence in its external details:
wherein do they differ? In that the creators write of certain realities
behind life, and the æsthetes—of the words standing for these
realities.
36
Hypocrisy of Words
The æsthetes, and Pater and Wilde in particular, made a cult of the
use of decorative words. They demanded, not that a word should be
true, nor even that it should be true and pretty at the same time,
but simply that it should be pretty. It cannot be denied that writers
here and there before them had been guilty of using a fine word
where a common one was most honest; but this had been generally

regarded as a forgiveable, "artistic" weakness. Wilde and his
followers, however, chose "exquisite" words systematically, in
conformity to an artistic dogma, and held that literature consisted in
doing nothing else. And that was dangerous; for truth was thereby
banished from the realm of diction and a hypocrisy of words arose.
In short, language no longer grasped at realities, and literature
ceased to express any thing at all, except a writer's taste in words.
37
The Average Man
In this welter of dissolving values, the intellectuals of our time find
themselves struggling, and liable at any moment to be engulfed. A
few of them, however, have snatched at something which, in the
prevailing deliquescence, appears to be solid—the average man.
Encamped upon him, they have won back sanity and happiness. But
their act is nevertheless simply a reaction; here the real problem has
not yet been faced! What is it that makes the average man more
sane and happy than the modern man? The possession of dogmas,
says G. K. Chesterton; let us therefore have dogmas! But, alas, for
them he goes back and not forward. And not only back, but back to
the very dogmas against which modern thought, and Decadence
with it, are a reaction, nay, the inevitable reaction. What! has Mr.
Chesterton, then, postponed the solution of the problem? And on
the heels of his remedy does there tread the old disease over again?
Perhaps it is so. The acceptance of the old dogmas will be followed
by a new reaction from them, a new disintegration of values
therefore, and a new Decadence. The hands of the clock can be put
back, it is true; but they will eventually reach the time when the
hour shall strike again for the solution of the modern problem.
And that is the criticism which modern men must pass upon Mr.
Chesterton; that he interposed in the course of their malady to bring
relief with a remedy which was not a remedy. The modern problem
should have been worked out to a new solution, to its own solution.
Instead of going back to the old dogmas, we should have strained

on towards the new. And if, in this generation, the new dogmas are
still out of sight, if we have meantime to live our lives without peace
or stability, does it matter so very much? To do so is, perhaps, our
allotted task. And as sacrifices to the future we justify our very
fruitlessness, our very modernity!
II
ORIGINAL SIN
38
Original Sin
Original Sin and the Future are essentially irreconcilable conceptions.
The believer in the future looks upon humanity as plastic: the good
and the bad in man are not fixed quantities, always, in every age,
past and future, to be found in the same proportions: an "elevation
of the type man" is, therefore, possible. But the believer in Original
Sin regards mankind as that in which—the less said about the good,
the better—there is, at any rate, a fixed substratum of the bad. And
that can never be lessened, never weakened, never conquered.
Therefore, man has to fight constantly to escape the menace of an
ever-present defeat. A battle in which victory is impossible; a contest
in which man has to climb continually in order not to fall lower;
existence as the tread mill: that is what is meant by Original Sin.
And as such it is the great enemy of the Future, the believers in
which hold that there is not this metaphysical drag. But it is more. At
all things aspiring it sets the tongue in the cheek, gladly provides a
caricature for them, and becomes their Sancho Panza. To the great
man it says, through the mouths of its chosen apostles, the average
men, "What matter how high you climb! This load which you carry
even as we will bring you back to us at last. And the higher you

climb the greater will be your fall. Humanity cannot rise above its
own level." And therefore, humility, equality, radicalism, comradeship
in sin—the ideas of Christianity!
39
Again
Distrust of the future springs from the same root as distrust of great
men. It derives from the belief in the average man, which derives
from the belief in Original Sin. The egalitarian sentiment strives
always to become unconditional. It claims not only that all men are
equal, but that the men who live now are no more than the equals
of those who lived one, or five, thousand years ago, and no less
than the equals of those who will live in another one, or five,
thousand years. And it desires that this should be so: its jealousy
embraces not only the living, but the dead and the unborn.
40
Again
Society is a conspiracy, said Emerson, against the great man. And to
blast him utterly in the centre of his being, it invented Original Sin.
Is Original Sin, then, a theological dogma or a political device?
41
Equality
Is equality, in truth, a generous dogma? Does it express, as every
one assumes, the solidarity of men in their higher attributes? It is
time to question this, and to ask if inequality be not the more noble
and generous belief. For, surely, it is in their nobler qualities that
men are most unequal. It was not in his genius that Shakespeare
was only the equal, for instance, of his commentators; it was in the
groundwork of his nature, in those feelings and desires without

which he would not have been a man at all, in the things which
made him human, but which did not make him Shakespeare: in a
word, in that which is for us of no significance. Equality in the
common part of man's nature, equality in sin, equality before God—
it is the same thing—that is the only equality which can be admitted.
And if its admission is insisted upon by apologists for Christianity,
that is because to the common part of man's nature they give so
much importance, because they are believers in Original Sin. In their
equality there is accordingly more malice than generosity. The belief
that no one is other than themselves, the will that no one shall be
other than themselves—there is nothing generous in that belief and
that will. For man, according to them, is guilty from the womb. And
what, then, is equality but the infinitely consoling consciousness of
tainted creatures that every one on this earth is tainted?
The believer in Original Sin will, of course, deny this, and say that in
his philosophy men are equals also in their higher rôle as "sons of
God." But is this so? Is salvation, like sin, common to all men? Is it
not, on the contrary, something conferred as the reward of a belief
and a choice—a belief and a choice which an Atheist, for instance,
simply cannot embrace? So that here, touching the highest part of
men, their soul, there is introduced, by Christianity itself, a
distinction, an inequality—the distinction, the inequality between the
"saved" and the "lost." Men are equal inasmuch as they are all
damned, but they are not equal inasmuch as they are not all
redeemed.
Gazing at man, however, no longer through the eyes of the serpent,
shall we not be bound to find, if we look high enough, distinction,
superiority, inferiority, valuation? The dogma of equality is itself a
device to evade valuation. For valuation is difficult, and demands
generosity for its exercise. To recognize that one is greater than you,
and cheerfully to acknowledge it; to see that another is less than
you, and to treat the inferiority as a trifling thing, that is difficult,
that requires generosity. But one who believes in inequality will
always be looking for greatness in others; his eye, habituated to the
contemplation of lofty things, will become subtle in the detection of

concealed nobility; while to the ignoble he will give only a glance—
and is it not good, where one may not help, to pass on the other
side? The egalitarians will cry that it is ungenerous to believe that
some men are vile; but it is a strange generosity which would
persuade us with them that all men are vile. Let us be frank. To
those who believe in the future, inequality is a holy thing; their
pledge that greatness shall not disappear from the earth; the
rainbow assuring them that Man shall not go down beneath the vast
tide of mankind. All great men are to them at once forerunners and
sacrifices; the imperfect forms which the Future has shattered in
trying to incarnate itself; the sublime ruins of future greatness.
42
If Men Were Equal
If men had been equal at the beginning, they would never have
risen above the savage. For in absolute equality even the concept of
greatness could not have come into being. Inequality is the source
of all advancement.
43
The Fall of Man
In very early times men must have had a deep sense of the
tragicality of existence: life was then so full of pain; death, as a rule,
so sudden and unforeseen, and the world generally so beset with
terrors. The few who were fortunate enough to escape violent death
had yet to toil incessantly to retain a footing on this unkind star. Life
would, accordingly, appear to them in the most sombre tones and
colours. And it was to explain this human misfortune, and not sin at
all, that the whole fable of Adam and Eve and the Fall was invented.
The doctrine of Original Sin was simply an interpretation which was
afterwards read into the story, an interpretation, perhaps, as
arbitrary as the orthodox interpretation of the Song of Songs.

How would the fable arise? Well, a primitive poet one day in a fit of
melancholy made the whole thing up. Out of his misery his desires
created for him an imaginary state, its opposite, the Garden of Eden.
But this state being created, the problem arose, How did Man fall
from it? And the Tree was brought in. But to the naïve, untheological
poet, this tree had nothing to do with metaphysics or with sin, the
child of metaphysics. It was simply a magical tree, and if Man ate of
the fruit of it, something terrible would happen to him. The Fall of
Man was a mystery to the poet, which he did not rationalize or
theologize. Well, Man succumbed to curiosity, and pain and
misfortune befell the human race. But we must not assume in the
modern manner that with the eating of the fruit early man
associated any idea of guilt. Rather the contrary; he regarded the
act simply as unfortunate, just as at the present day we regard as
unfortunate the foolish princess in some fairy tale. So the Fall was
not to him a crime, branding all mankind with a metaphysical
stigma.
That conception came much later, when the conscience had become
deeper, more subtle and more neurotic; when individualism had
been introduced into morality. And at that time, too, the ideal of the
Redeemer became vitiated. Early man, if he did envisage a
Redeemer, envisaged him as one who would set him back in the
Garden of Eden again, in the literal, terrestrial Garden of Eden, be it
understood: theology had not yet been etherealized. And this
Redeemer would redeem all men: the distinction of the individual
came afterwards. It was not until later, too, that this ideal was
"interpreted," and, as a concession to the conscience, salvation was
made a conditional thing: the reward of those who were successful
in a competition in credulity, in which the first prize went to the most
simple, most stupid. The "guilt" now implicated in the Fall was not
purged away from all men by the Redeemer, but only from such as
would "accept" it. And, lastly, with the passing of Jesus, the
redemption was still further de-actualized. It was found that
acceptance of the Redeemer did not reinstate Man in an earthly
Garden: paradise was, therefore, drawn on the invisible wires of

theology into the inaccessible heavens. Salvation lay at the other
side of the grave, and there it was safe from assault.
Nevertheless, what our primitive poet meant by the Fall and the
Redemption was probably something entirely different. The Fall to
him was the fall into misfortune, not into sin: the Redemption to him
was the redemption from misfortune, not from sin. And his
Redeemer would be, therefore—whom? Perhaps it is impossible for
us to imagine the nature of such a being.
This is not an interpretation, but an attempted explanation of the
story of the Fall.
44
Interpretations
How inexhaustible is myth! In the story of the Fall is a meaning for
every age and every creed. The interpretation called Original Sin is
only one of a thousand, and not the greatest of them. Let us dip our
bucket into the well.
The tree of the knowledge of good and evil—that was the tree of
morality! And morality was then the original sin? And through it Man
lost his innocence? The antithesis of morality and innocence is as old
as the world. And if we are to capture innocence again, if the world
is to become æsthetically acceptable to us, we must dispense more
and more with morality and limit its domain. This, one desperate
glance into the depths of the myth tells us. Instinct is upheld in it
against isolated reason and exterior law. Detached, "abstract"
Reason brought sin into the world, but Instinct, which is
fundamentally Love, Creation, Will to Power, is forever innocent,
beyond good and evil. It was when Reason, no longer the sagacity
of Instinct, no longer the eyes of Love, became its opponent and
oppressor, that morality arose and Man fell.
Or to take another guess, granted we read Original Sin in the Fall,
must we not read there, also, the way to get rid of it? If by Original

Sin Man fell, then by renouncing it let him arise again. But how
renounce it? What! Cannot Man renounce a metaphor?
Yet how powerful is metaphor! Man is ruled by metaphor. The gods
were nothing but that, some sublime, some terrible, some lovely, all
metaphors, Jehovah, Moloch, Apollo, Eros. Life is now stained
through and through with metaphor. And there are further
transfigurations still possible! Yet we would not destroy the beauty
already starring Life's skies, the lovely hues lent by Aphrodite, and
Artemis, and Dionysos, or the sublime colours of Jehovah and Thor.
But the heavy disfiguring blot tarnishing all, Love, Innocence,
Ecstasy, Wrath, that we would rather altogether extirpate and annul.
Original Sin we would cut off as a disfigurement and disease of Life.
Or, again, may not the myth be an attempt to glorify Man and to
clothe him with a sad splendour. And not Original Sin, but Original
Innocence is the true reading of the fable? Its raison d'être is the
Garden of Eden, not the Fall? To glorify Humanity at its source it set
there a Superman. The fall from innocence—that was the fall from
the Superman into Man. And how, then, is Man to be redeemed? By
the return of the Superman! Let that be our reading of the myth!
45
The Use of Myth
In the early world myth was used to dignify Man by idealizing his
origin. Henceforward it must be used to dignify him by idealizing his
goal. That is the task of the poets and artists.
46
Before the Fall
Innocence is the morality of the instincts. Original Sin—that was war
upon the instincts, morality become abstract, separate, self-centred,
accusing and tyrannical. This self-consciousness of morality, this
disruption in the nature of Man, was the Fall.

47
Beyond Original Sin
How far is Man still from his goal? How sexual, foul in word and
thought, naively hedonistic! How little of spirit is in him! How
clumsily his mind struggles in the darkness! How far he is still from
his goal!—This is a cry which the believer in Original Sin cannot
understand, because he accepts all this imperfection as inevitable,
as the baleful heritage of Man, from which he cannot escape.
The feeling of pure joy in life, the feeling that Life is a sacrament—
that also is forever denied to the believer in Original Sin. For Life is
not a sacrament to him, but a sin of which joy itself is only an
aggravation.
48
The Eternal Bluestocking
The bluestocking is as old as mankind. Her original was Eve, the first
dabbler in moral philosophy.
49
The Sin of Intellectualism
The first sin, the original sin was that of the intellectuals. The
knowledge of Good and Evil was not an instantaneous "illumination";
it was the result of long experiment and analysis: the apple took
perhaps hundreds of years to eat! Before that, in the happy day of
innocence, Good and Evil were not, for instinct and morality were
one and not twain. As time passed, however, the physically lazy, who
had been from the beginning, became weaker and wiser. Enforced
contemplation, the contemplation of those who were not strong
enough to hunt or to labour, made them more subtle than their
simple brethren; they formed themselves into a priesthood, and
created a theology. In these priests instinct was not strong: they

were invalids with powerful reason. But they had the lust for power;
they wished to conquer by means of their reason; therefore, they
said to themselves, belittle instinct, tyrannize over instinct, discover
an absolute "good" and an absolute "evil," become moral. Morality,
which had in the days of innocence been unconscious, the harmony
of the instincts, was now given a separate existence. The cry was
morality against the instincts. Thus triumphed the priests, the
intellectuals, by means of their reason. Original Sin was their sin—
the result of the analysis by which they had separated morality and
the instincts. If we are to speak of Original Sin at all, let it be in this
manner.
50
Once More
The belief in Original Sin—that was itself Man's original sin.
51
Apropos Gautier
He had just read "Mlle. de Maupin," "What seduction there is still for
Man in the senses!" he exclaimed. "How much more of an animal
than a spirit he must be to be charmed and enslaved by this book!"
Yet, what ground had he to conclude that because the sensual
intoxicates Man, therefore Man is more sensual than spiritual? For
we are most fatally attracted by what is most alien to us.
52
Psychology of the Humble
There is something very naïve in those who speak of humility as a
certain good and of pride as a proven evil. In the first place these
are not opposites at all; there are a hundred kinds of both, and
humility is sometimes simply a refined form of pride. Humility may

be prudence, or good taste, or timidity, or a concealment, or a
sermon, or a snub. How much of it, for instance, is simple prudence?
Is not this, indeed, its chief utility, that it saves men from the
dangers which accompany pride? On the day on which some one
discovered that "Pride goeth before a fall," humility became no mean
virtue. For if one become the servant and proclaim himself the least
of all, how can he still fall? Yet if he does it is a fall into greater
humility, and his virtue only shows the brighter. This is the sagacity
of the humble, that they turn even ignominy to their glorification.
Humility is most commonly used with a different meaning, however.
There are people who wish to be anonymous and uniform, and
people who desire to be personal and distinct. Or, more exactly, it is
their instincts that seek these ends. The first are humble in the
fundamental sense that they are instinctively so; the latter are proud
in the same sense. Humility, then, is the desire to be as others are
and to escape notice; and this desire can only be realized in
conformity. It is true, people become conceited after a while about
their very conformity, and would be wounded in their vanity if they
failed to comply with fashion; but vanity and humility are not
incompatible.
Pride, however, is something much more subtle. The naïve,
unconditional contemners of pride, who plead with men to cast it
out, have certainly no idea what would happen if they were obeyed.
For pride is the condition of all fruitful action. This thought must be
consciously or subconsciously present in the doer, What I do is of
value! I am capable of doing a thing which is worth doing! The
Christian, it is true, still acts, though he is convinced that all action is
sinful and of little worth. But it is only his mind that is convinced: his
instincts are by no means persuaded of the truth of this! For though
in the conscious there may be self-doubt, in the unconscious there
must be pride, or actions would not be performed at all. Moreover, in
all those qualities which are personal and not common—in
personality—pride is an essential ingredient. The pronoun "I" is itself
an affirmation of pride. The feeling, This is myself, this quality is my
quality, by possessing it I am different from you, these things

constitute my personality and are me: what a naïve assumption of
the valuableness of these qualities do we have there, how much
pride is there in that unconscious confession! And without this
instinctive pride, these qualities, personality could never have been
possible. In the heart of all distinct, valuable and heroic things, pride
lies coiled. Yes, even in the heart of humility, of the most refined,
spiritual humility. For such humility is not a conformity; it separates
and individualizes its possessor as effectually as pride could; it takes
its own path and not that of the crowd; and so its source must be in
an inward sense of worth, of independence: it is a form of pride. But
pride is so closely woven into life that to wound it is to wound life; to
abolish it, if that were possible, would be to abolish life. Well do its
subtler defamers know that! And when they shoot their arrows at
pride, it is Life they hope to hit.
53
Les Humbles
Humility is the chief virtue, said a humble man. Then are you the
vainest man, said his friend, for you are renowned for your humility.
Good taste demands from writers who praise humility a little
aggressiveness and dogmatism, lest they be taken for humble, and,
therefore, proud. On the other hand, if humility is the chief virtue, it
is immoral not to practise it. And, therefore, one should praise
humility, and practise it? Or praise it and not practise it? Or not
praise it and practise it? There is contradiction in every course. That
is the worst of believing in paradoxical virtues!
54
Against the Ostentatiously Humble
He who is truly humble conceals even his humility.

55
The Pessimists
In pessimistic valuations of Life, the alternative contemplated is
generally not between Life and Death, but between different types
of Life. The real goal of Schopenhauerism is not the extinction of
life, for death is a perfectly normal aspect of existence, and Life
would not be denied even if death became universal. In order to
deny Life and to triumph over it, the pessimist must continue at least
to exist, in a sort of death in life: he must be dead, but he must also
know it. That is the goal of Schopenhauerism; perhaps not so
difficult, perhaps frequently attained! "They have not enough life
even to die," said Nietzsche.
56
Sickness and Health
Some men have such unconquerable faith in Life that they defy their
very maladies, creating out of them forms of ecstasy: that is their
way of triumphing over them. Perhaps some poetry, certainly not a
little religion has sprung from this. In religions defaming the senses
and enjoining asceticism, or, in other words, a lowering of vitality,
the chronic sufferers affirm Life in their own way; for sickness is
their life: their praise of sickness is their praise of Life. And if they
sometimes morbidly invite death, that is because death is nothing
but another form of experience, of Life. To the sick, if they are to
retain self-respect and pride, these doctrines are perhaps the best
possible; it is only to the healthy that they are noxious. For the
healthy who are converted by them, become sick through them, yet
not so sick as to find comfort in them. The aspiration after an ascetic
life contends in these men with their old health, their desire to live
fully, and causes untold perplexities and conflicts; leaving them at
last with nothing but a despairing desire for release. Thus, a religion
of consolation becomes for the strong a Will to Death—the very
opposite of that which it was to those who created it.

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