;'1 !':" ! SCHAUM'S OUTLINE OF THEORY AND PROBLEMS of THEORETICAL MECHANICS SI (METRIC) EDITION with an introduction to Lagrange's Equations and Hamiltonian Theory • by MURRAY R. SPIEGEL, Ph.D. Professor of Mathematics Rensselaer Polytechnic Institute Adapted for SI Units by Y. PROYKOVA, Ph.D. McGRAW-HILL BOOK COMPANY .. ___ ,... "1'-..... l o· It! f D • A H r l T!). 't I O!AIMUIG CENr):,t ; t MALT,u. .. .. . New York . St Louis' San Francisco . Auckland' Bogota Guatemala . Hambur[ . Lisbon . London Madrid· Mexico' Montreal' New Delhi' Panama' Paris San Juan . Sao Paulo . Singapore . Sydney . Tokyo . Toronto
12
Embed
THEORY AND PROBLEMS of THEORETICAL MECHANICSllrc.mcast.edu.mt/digitalversion/Table_of_Contents_134236.pdf;'1 !':" ! SCHAUM'S OUTLINE OF THEORY AND PROBLEMS of THEORETICAL MECHANICS
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
;'1
!':" !
SCHAUM'S OUTLINE OF
THEORY AND PROBLEMS
of
THEORETICAL MECHANICS
SI (METRIC) EDITION
with an introduction to Lagrange's Equations
and Hamiltonian Theory
• by
MURRAY R. SPIEGEL, Ph.D. Professor of Mathematics
Rensselaer Polytechnic Institute
Adapted for SI Units by
Y. PROYKOV A, Ph.D.
McGRAW-HILL BOOK COMPANY
.. ~_. ___ ~--..:.". ,... "1'-.....
l ~! o· It! f D • A H r l T!). 't
I O!AIMUIG CENr):,t ;
t MALT,u. ~ ~--~--~--- .. -~ .. -~ . -~;;.""':"
New York . St Louis' San Francisco . Auckland' Bogota Guatemala . Hambur[ . Lisbon . London
Madrid· Mexico' Montreal' New Delhi' Panama' Paris San Juan . Sao Paulo . Singapore . Sydney . Tokyo . Toronto
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
-
CONTENTS
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 acc~leration .. Circular motion. Notation for time derivatives. Gradient, divergence and curl. Line integrals. Independence of the path. Free, sliding and bound vectors.
NEWTON'S LAWS OF MOTION. WORK, ENERGY
Page
1
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.
MOTION IN A UNIFORM FIELD. FALLING BODIES AND PROJECTILES .......................................... 62 Uniform force fields. Uniformly accelerated motion. Weight and acceleration due to gravity. 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.
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. Over-damped, critically damped and under-damped motion. Forced vibrations. Resonance. The simple pendulum. The two and three dimensional harmonic oscillator.
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.
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
CONTENTS
Page
MOVING COORDINATE SYSTEMS ........................... 144 Non-inertial coordinate systems. Rotating coordinate systems. l)erivative 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.
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.
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. Boundary-value problems. Fourier series. Odd and even functions. Convergence of Fourier series.
PLANE MOTION OF RIGID BODIES Rigid bodies. Translations and rotations. Euler's theorem. Instantaneous axis of rotation. Degrees of freedom. General motion of a rigid body. Chasle'stheorem. Plane motion of a rigid body. Moment of inertia. Radius of gyration. Theorems on moments of inertia. Parallel axigo theorem. Perpendicular axes theorem. Special moments of inertia. Couples. Kinetic energy and ang.ular 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'Aiembert's principle. Principle of minimum potential energy. Stability.
224
SP ACE 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 equatiens of motion. Force free motion. The invariable line and plane. Po in sot'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.
f
- !
? •
Chapter 11
Chapter 12
CONTENTS
Page
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.
HAMILTONIAN THEORY 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-J acobi equation. Solution of the Hamilton-Jacobi equation. Case where Hamiltonian is independent of time. Phase integrals. Action and angle variables.
311
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
INDEX ........................................................ 361
-< I
INDEX
Absolute motion, 34, 60 Acceleration, 1, 7, 17-20
along a space curve, 8, 20 angular, 8, 145,. 148 apparent, 149 centrifugal,145 centripetal, 8, 20, 21,_ 150 Coriolis, 145, 150 due to gravity, 62 in cylindrical coordinates, 32 in moving coordinate systems,
about principal axes, 255 conservation of, 37, 45-47, 168, 228, 237 of a rigid body, 227, 236, 254, 259, 260 of a system of particles, 168, 169, 176, 179 of the earth about its axis, 150 principle of, 227, 229, 236, 238 relationship to torque, 37, 46,
168, 169, 176 Angular speed, 8 (see also Angular velocity) Angular velocity, 144, 148
227, 229, 236, 240 kinetic, (see Kinetic energy) of simple harmonic oscillator, 87, 99 potential (see Potential energy) total, 36
English systems, 63, 339 Epoch, 87, 88, 93 Equality of vectors, 2 Equilibrant,47 Equilibrium, 37, 171
in a uniform gravitational field, 65, 66, 74, 75 of a particle, 37, 38, 47, 48 of a rigid body, 229, 241, 242 of a system of particles, 170, 180, 181 position, 86 stable, 38, 48, 49, 60, 141,171,230
about principal axes, 255 in terms of Euler angles, 258, 268 in terms of generalized velocities, 283, 287, 288 of a rigid body, 227, 236, .259, 26'0 of a system of particles, 168, 169, 179, 182 of rotation, 229 of translation, 229 relationship to work, 35, 41, 169 relativistic, 55
Magnetic field. 83 Main diagonal, of moment of inertia
matrix,254 Major axis, of ellipse, 118
of hyperbola, 139
J
Mass, 2, 33 axiomatic definition of, 49 center of (see Center of mass) changing, 194 of the earth, 129 reduced, 182, 231 rest, 54, 61 units'of,33
Mathefhatical models, 1 Matrix, moment of inertia, 254 Matter, 1, 2 Mechanics, 1
relativistic, 34 Membrane, vibrating, 195 Meteorite, 121 Metric system, 339 Minor axis, of ellipse, 119
of hyperbola, 139 Mks system, 33, 62, 339, 340 Mode of vibration, normal, 194 Models, mathematical, 1 Modulation, amplitude, 102 Modulus of elasticity, 86 Moment, of couple, 226
of force, 36 of momentum, 37 (see also Angular momentum)
Momental ellipsoid, 256 Moments of inertia, 225, 231-233, 254, 259,
260, 263, 264 matrix, 254 principal (see Principal momert'ts of inertia) special, 226 theorems on, 225, 233-235
Momentum, 33, 167 angular (see Angular momentum) conjugate, 284, 288 conservation of, 37, 1.67, 173 generalized, 284, 288 moment of, 37 (see also Angular momentum) of a system of particles, 167, 169, 172, 173 principle of, 238
natural, 89 of damped motion, 89 of harmonic oscillator, 87 of motion in a magnetic field, 83 of simple harmonic motion, 86, 87 of simple pendulum, 91, 105, 106 orbital, 135, 136 sidereal, 120
integrals, 316, 328, 329 out of, 93 space, 312, 318-320
365
366
Piano string, vibrations of, 195 (see also Vibrating string)
Piecewise continuous functions, 197 Planck's constant, 338 Planets, 119, 343 Poinsot's construction, 257 Point, 1, 2 Po~on bracket, 331, 332 Polhr coordinates, 25, 26 g~adient in, 54 velocity and acceleration in, 26
Polhode, 257, 266 Position, 2
coordinates, 312 vector, 4
Potential, 35 (see also Potential energy) relation to stability, 38 scalar, 35, 309 vector, 309
Potential energy, 35, 36, 43-45 (see also Potential)
in a central force field, 117, 123-125 in a uniform force field, 64, 69 of a system of particles, 169, 176-178 principle of minimum, 230 relation of to work, 35, 44
Pound, 33 weight, 63
Poundal, 33 Power, 34, 41-43, 227, 237
relation to work, 42 Precession, 156, 256, 270, 272
frequency of, 257, 265, 270, 273, 274 Principal axes of inertia, 255, 260-263 Principal diagonal, 254 Principal moments of inertia, 255, 260-263
method of Lagrange multipliers for, 280 Principal normal, unit, 7, 8, 20 Products of inertia, 254, 259, 260 Products of vectors, by a scalar, 3
cross (see Cross products) dot (see Dot products)
Projectiles, 62, 63 maximum height of, 68 motion of, 68, 69, 71, 72 on an inclined plane, 75, 76, 81 range of (see Range of projectile) time of flight, 68
Sines, law of, 27 . Sliding. vector, 9, 10 Slug, 63 Solar system, 119 S"lution of differential equation, 344 Space, 1, 2 Space axes, 257 Space centrode or locus, 229, 240, 241 Space cone, 257, 266 Special relativity (see Relativity) Speed, 7 (see also Velocity)
angular, 8 escape, 134 of light, 34 orbital, 135, 136
in a uniform gravitational field, 65, 66, 74, 75 of a particle, 37, 38, 47, 48 of a rigid body, 229, 241, 242 of a system of particles, 170, 180, 181
Statistical mechanics, 311 Steady-state solution, 89 Stiffness factor, 86 Sum of vectors, 2
obtained graphically. and analytically, 12, 48 Sun, 119. 342 Superposition principle, 199 Surface, normal to, 24 Symmetric matrix or tensor, 254 Systems of particles, 165-193
Tangent vector, unit, 7, 8, 19 Tautochrone problem, 113 Tension, 74, 76 Tensor, moment of inertia, 254 Terminal point, of a vector, 2 Theorems, 1 Time. 1, 2