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### Theoretical Mechanics, Kinematics, Dynamics and Static's

Beam Deflections and Structural Analysis

Engineering Statics

Machine Design and Mechanics

Theoretical Mechanics, Kinematics, Dynamics and Static's

375 Pages

Includes 720 Solved Problems with an introduction to Lagrange's Equations and Hamiltonian Theory

Murray R. Spiegel

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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 static's 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.

TOC

Chapter 1 VECTORS, VELOCITY AND ACCELERATION

Mechanics, kinematics, dynamics and static's 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. Static's 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. Static's in a uniform gravitational field.

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. Over-damped, 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.

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. Boundary-value 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.

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.

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