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Transcript
Copyright © 1967 by McGraw-Hill,
Inc. All rights reserved. Printed in
the United States of America. No
part of this publication may be
reproduced, stored in a retrieval
system, or transmitted, in any form
or by any means, electronic,
mechanical, photocopying,
recording, or otherwise, without the
prior written permission of the
publisher.
60232
8 9 10 11 12 13 14 15 SH SH 7 5
Preface
In the 17th century, Sir Isaac
Newton formulated his now famous
laws of mechanics. These
remarkably simple laws served to
describe and predict the motions of
observable objects in the universe,
including those of the planets of
our solar system.
Early in the 20th century it was
discovered that various theoretical
conclusions derived from Newton's
laws were not in accord with certain
conclusions deduced from theories
of electromagnetism and atomic
phenomena which were equally
well founded experimentally. These
discrepancies led to Einstein's
relativistic mechanics which
revolutionized the concepts of space
and time, and to quantum
mechanics. For objects which move
with speeds much less than that of
light and which have dimensions
large compared with those of atoms
and molecules Newtonian
mechanics, also called classical
mechanics, is nevertheless quite
satisfactory. For this reason it has
maintained its fundamental
importance in science and
engineering.
It is the purpose of this book to
present an account of Newtonian
mechanics and its applications. The
book is designed for use either as a
supplement to all current standard
textbooks or as a textbook for a
formal course in mechanics. It
should also prove useful to students
taking courses in physics,
engineering, mathematics,
astronomy, celestial mechanics,
aerodynamics and in general any
field which needs in its formulation
the basic principles of mechanics.
Each chapter begins with a clear
statement of pertinent definitions,
principles and theorems together
with illustrative and other
descriptive material. This is
followed by graded sets of solved
and supplementary problems. The
solved problems serve to illustrate
and amplify the theory, bring into
sharp focus those fine points
without which the student
continually feels himself on unsafe
ground, and provide the repetition
of basic principles so vital to
effective learning. Numerous proofs
of theorems and derivations of
basic results are included in the
solved problems. The large number
of supplementary problems with
answers serve as a complete review
of the material of each chapter.
Topics covered include the
dynamics and statics of a particle,
systems of particles and rigid
bodies. Vector methods, which lend
themselves so readily to concise
notation and to geometric and
physical interpretations, are
introduced early and used
throughout the book. An account of
vectors is provided in the first
chapter and may either be studied
at the beginning or referred to as
the need arises. Added features are
the chapters on Lagrange's
equations and Hamiltonian theory
which provide other equivalent
formulations of Newtonian
mechanics and which are of great
practical and theoretical value.
Considerably more material has
been included here than can be
covered in most courses. This has
been done to make the book more
flexible, to provide a more useful
book of reference and to stimulate
further interest in the topics.
I wish to take this opportunity to
thank the staff of the Schaum
Publishing Company for their
splendid cooperation.
M. R. Spiegel
Rensselaer Polytechnic Institute
February, 1967
CONTENTS
Page
Chapter 1 VECTORS, VELOCITY
AND ACCELERATION
Mechanics, kinematics, dynamics
and statics. Axiomatic foundations
of mechanics. Mathematical
models. Space, time and matter.
Scalars and vectors. Vector algebra.
Laws of vector algebra. Unit vectors.
Rectangular unit vectors.
Components of a vector. Dot or
scalar product. Cross or vector
product. Triple products.
Derivatives of vectors. Integrals of
vectors. Velocity. Acceleration.
Relative velocity and acceleration.
Tangential and normal acceleration.
Circular motion. Notation for time
derivatives. Gradient, divergence
and curl. Line integrals.
Independence of the path. Free,
sliding and bound vectors.
Chapter 2 NEWTON'S LAWS OF
MOTION. WORK, ENERGY
AND MOMENTUM 33
Newton's laws. Definitions of force
and mass. Units of force and mass.
Inertial frames of reference.
Absolute motion. Work. Power.
Kinetic energy. Conservative force
fields. Potential energy or potential.
Conservation of energy. Impulse.
Torque and angular momentum.
Conservation of momentum. •
Conservation of angular
momentum. Non-conservative
forces. Statics or equilibrium of a
particle. Stability of equilibrium.
Chapter 6 MOTION IN A
UNIFORM FIELD. FALLING
BODIES
AND PROJECTILES
Uniform force fields. Uniformly
accelerated motion. Weight and
acceleration due to gravity.
Gravitational system of units.
Assumption of a flat earth. Freely
falling bodies. Projectiles. Potential
and potential energy in a uniform
force field. Motion in a resisting
medium. Isolating the system.
Constrained motion. Friction.
Statics in a uniform gravitational
field.
62
Chapter 4 THE SIMPLE
HARMONIC OSCILLATOR AND
THE SIMPLE PENDULUM 86
The simple harmonic oscillator.
Amplitude, period and frequency of
simple harmonic motion. Energy of
a simple harmonic oscillator. The
damped harmonic oscillator. Overdamped, critically damped and
under-damped motion. Forced
vibrations. Resonance. The simple
pendulum. The two and three
dimensional harmonic oscillator.
Chapter 5 CENTRAL FORCES AND
PLANETARY MOTION 116
Central forces. Some important
properties of central force fields.
Equations of motion for a particle
in a central field. Important
equations deduced from the
equations of motion. Potential
energy of a particle in a central
field. Conservation of energy.
Determination of the orbit from the
central force. Determination of the
central force from the orbit. Conic
sections, ellipse, parabola and
hyperbola. Some definitions in
astronomy. Kepler's laws of
planetary motion. Newton's
universal law of gravitation.
Attraction of spheres and other
objects. Motion in an inverse
square field.
CONTENTS
Page
Chapter 6 MOVING COORDINATE
SYSTEMS 144
Non-inertial coordinate systems.
Rotating coordinate systems.
Derivative operators. Velocity in a
moving system. Acceleration in a
moving system. Coriolis and
centripetal acceleration. Motion of a
particle relative to the earth.
Coriolis and centripetal force.
Moving coordinate systems in
general. The Foucault pendulum.
Chapter 7 SYSTEMS OF
PARTICLES 165
Discrete and continuous systems.
Density. Rigid and elastic bodies.
Degrees of freedom. Center of
mass. Center of gravity. Momentum
of a system of particles. Motion of
the center of mass. Conservation of
momentum. Angular momentum of
a system of particles. Total external
torque acting on a system. Relation
between angular momentum and
total external torque. Conservation
of angular momentum. Kinetic
energy of a system of particles.
Work. Potential energy.
Conservation of energy. Motion
relative to the center of mass.
Impulse. Constraints. Holonomic
and non-holonomic constraints.
Virtual displacements. Statics of a
system of particles. Principle of
virtual work. Equilibrium in
conservative fields. Stability of
equilibrium. D'Alembert's principle.
Chapter 8 APPLICATIONS TO
VIBRATING SYSTEMS,
ROCKETS AND COLLISIONS 194
Vibrating systems of particles.
Problems involving changing mass.
Rockets. Collisions of particles.
Continuous systems of particles.
The vibrating string. Boundaryvalue problems. Fourier series. Odd
and even functions. Convergence of
Fourier series.
Chapter 9 PLANE MOTION OF
RIGID BODIES 224
Rigid bodies. Translations and
rotations. Euler's theorem.
Instantaneous axis of rotation.
Degrees of freedom. General
motion of a rigid body. Chasle's
theorem. Plane motion of a rigid
body. Moment of inertia. Radius of
gyration. Theorems on moments of
inertia. Parallel axis theorem.
Perpendicular axes theorem. Special
moments of inertia. Couples.
Kinetic energy and angular
momentum about a fixed axis.
Motion of a rigid body about a fixed
axis. Principle of angular
momentum. Principle of
conservation of energy. Work and
power. Impulse. Conservation of
angular momentum. The compound
pendulum. General plane motion of
a rigid body. Instantaneous center.
Space and body centrodes. Statics of
a rigid body. Principle of virtual
work and D'Alembert's principle.
Principle of minimum potential
energy. Stability.
Chapter 10 SPACE MOTION OF
RIGID BODIES 253
General motion of rigid bodies in
space. Degrees of freedom. Pure
rotation of rigid bodies. Velocity
and angular velocity of a rigid body
with one point fixed Angular
momentum. Moments of inertia.
Products of inertia. Moment of
inertia matrix or tensor. Kinetic
energy of rotation. Principal axes of
inertia. Angular momentum and
kinetic energy about the principal
axes. The ellipsoid of inertia. Euler's
equations of motion. Force free
motion. The invariable line and
plane. Poinsot's construction.
Polhode. Herpolhode. Space and
body cones. Symmetric rigid bodies.
Rotation of the earth. The Euler
angles. Angular velocity and kinetic
energy in terms of Euler angles.
Motion of a spinning top.
Gyroscopes.
CONTENTS
Page
Chapter // LAGRANGE'S
EQUATIONS 282
General methods of mechanics.
Generalized coordinates. Notation.
Transformation equations.
Classification of mechanical
systems. Scleronomic and
rheonomic systems. Holonomic and
non-holonomic systems.
Conservative and non-conservative
systems. Kinetic energy.
Generalized velocities. Generalized
forces. Lagrange's equations.
Generalized momenta. Lagrange's
equations for non-holonomic
systems. Lagrange's equations with
impulsive forces.
Chapter 12 HAMILTONIAN
THEORY 311
Hamiltonian methods. The
Hamiltonian. Hamilton's equations.
The Hamiltonian for conservative
systems. Ignorable or cyclic
coordinates. Phase space. Liouville's
theorem. The calculus of variations.
Hamilton's principle. Canonical or
contact transformations. Condition
that a transformation be canonical.
Generating functions. The
Hamilton-Jacobi equation. Solution
of the Hamilton-Jacobi equation.
Case where Hamiltonian is
independent of time. Phase
integrals. Action and angle
variables.
APPENDIX A
UNITS AND DIMENSIONS 339
APPENDIX B
ASTRONOMICAL DATA 342
appendix c SOLUTIONS OF
SPECIAL DIFFERENTIAL
EQUATIONS .... 344
APPENDIX D INDEX OF SPECIAL
SYMBOLS AND NOTATIONS 356
and ACCELERATION
MECHANICS, KINEMATICS,
DYNAMICS AND STATICS
Mechanics is a branch of physics
concerned with motion or change in
position of physical objects. It is
sometimes further subdivided into:
1. Kinematics, which is concerned
with the geometry of the motion,
2. Dynamics, which is concerned
with the physical causes of the
motion,
3. Statics, which is concerned with
conditions under which no motion
is apparent.
AXIOMATIC FOUNDATIONS OF
MECHANICS
An axiomatic development of
mechanics, as for any science,
should contain the following basic
ingredients:
1. Undefined terms or concepts.
This is clearly necessary since
ultimately any definition must be
based on something which remains
undefined.
2. Unproved assertions. These are
fundamental statements, u