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a

hSCHAUM'S OUTLINE OF 1(i

THEORY AND PROBLEMSOF

THEORETICAL MECHANICSwith an introduction to

Lagrange's Equations

and Hamiltonian Theory

BY

MURRAY

R.

SPIEGEL, Ph.D.

Professor of Mathematics

Rensselaer Polytechnic Institute

SCHAUM'S OUTLINE SERIESMcGRAW-HILL BOOK COMPANYNewYork, St. Louis, San Francisco, Toronto, Sydney

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.

602328 9 10 11 12 13 14 15

SH SH

7 5

PrefaceIn 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 defrom 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 andrived

engineering.

purpose of this book to present an account of Newtonian mechanics and its 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.It is the

applications.

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 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.rigid bodies.

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 for their splendid cooperation.

Schaum Publishing Company

M. R. SpiegelRensselaer Polytechnic Institute February, 1967

CONTENTSPageChapter

1

VECTORS, VELOCITY AND ACCELERATIONMechanics, kinematics, dynamics and statics. Axiomatic foundations of meMathematical 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 boundchanics.vectors.

Chapter

2

NEWTON'S LAWS OF MOTION. WORK, ENERGY AND MOMENTUMNewton'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.

33

Chapter

6

MOTION IN A UNIFORM AND PROJECTILESUniform forcefields.

FIELD. FALLING BODIES62

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.

Chapter

4

THE SIMPLE HARMONIC OSCILLATOR AND THE SIMPLE PENDULUMThe simple harmonic oscillator. Amplitude, period and frequency of simple harmonic motion. Energy of a simple harmonic oscillator. The damped harmonic oscillator. Over-damped, critically damped and under-damped motion. Forced vibrations. Resonance. The simple pendulum. The two and threedimensional harmonic oscillator.

86

Chapter

5

CENTRAL FORCES AND PLANETARY MOTIONCentral 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.

116

CONTENTSPageChapter

6

MOVING COORDINATE SYSTEMSNon-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

144

Foucault pendulum.

Chapter

7

SYSTEMS OF PARTICLESDiscrete and continuous systems. Density. Rigid and elastic bodies. Degrees of freedom. Center of mass. Center of gravity. Momentum of a system of Angular particles. Motion of the center of mass. Conservation of momentum. system of particles. Total external torque acting on a system. momentum of a Relation between angular momentum and total external torque. Conservation particles. Work. Poof angular momentum. Kinetic energy of a system of the center of mass. tential energy. Conservation of energy. Motion relative to Constraints. Holonomic and non-holonomic constraints. Virtual disImpulse. work. Equiplacements. Statics of a system of particles. Principle of virtual

165

librium in conservative

fields.

Stability of equilibrium. D'Alembert's principle.

Chapter

8

APPLICATIONS TO VIBRATING SYSTEMS, ROCKETS AND COLLISIONSmass. Rockets. Vibrating systems of particles. Problems involving changing string. particles. Continuous systems of particles. The vibrating Collisions of problems. Fourier series. Odd and even functions. Con-

194

Boundary-value vergence of Fourier

series.

Chapter

9

PLANE MOTION OF RIGID BODIES

224

Instantaneous Rigid bodies. Translations and rotations. Euler's theorem. Degrees of freedom. General motion of a rigid body. Chasle's axis of rotation. inertia. Radius of gyratheorem. Plane motion of a rigid body. Moment of moments of inertia. Parallel axis theorem. Perpendicular tion. Theorems on Kinetic energy and axes theorem. Special moments of inertia. Couples. fixed about a fixed axis. Motion of a rigid body about a angular momentum angular momentum. Principle of conservation of energy.axis.

Principle of

Work and

power.

Impulse.

pound pendulum. General plane motion of a rigid body. Principle of virtual work Space and body centrodes. Statics of a rigid body. minimum potential energy. Stability. and D'Alembert's principle. Principle of

Conservation of angular momentum. The comInstantaneous center.

Chapter

10

SPACE MOTION OF RIGID BODIES

253

Pure rotation General motion of rigid bodies in space. Degrees of freedom. of a rigid body with one point of rigid bodies. Velocity and angular velocity inertia. Products of inertia. Moment fixed Angular momentum. Moments of of rotation. Principal axes of of inertia matrix or tensor. Kinetic energy momentum and kinetic energy about the principal axes. The inertia. Angular motion. Force free motion. The inellipsoid of inertia. Euler's equations of Herpolhode. Spa