Vector Mechanics for Engineers Dynamics

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Vector Mechanics for Engineers Dynamics

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The main objective of a first course in mechanics should be to develop in the engineering student the ability to analyze any problem in a simple and logical manner and to apply to its solution a few, well-understood, basic principles. It is hoped that this text, as well as the preceding volume, Vector Mechanics for Engineers: Statics, will help the instructor achieve this goal.

Vector algebra was introduced at the beginning of the first volume and used in the presentation of the basic principles of statics, as well as in the solution of many problems, particularly three-dimensional problems. Similarly, 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 results in a more concise derivation of the fundamental principles. It also makes it possible to analyze many problems in kinematics and kinetics which could not be solved by the standard scalar methods. The emphasis in this text, however, remains on the correct understanding 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.

KINEMATICS OF PARTICLES 435
11.1 Introduction to Dynamics 435
RECTILINEAR MOTION OF PARTICLES 436
11.2 Position. Velocity, and Acceleration 436
11.3 Determination of the Motion of a Particle 440
11.4 Uniform Rectilinear Motion 448
11.5 Uniformly Accelerated Rectilinear Motion 448
1 1 .6 Motion of Several Particles 450
11.7 Graphical Solution of Rectilinear-Motion Problems 456
*11.8 Other Graphical Methods 458

CURVILINEAR MOTION OF PARTICLES 464
11.9 Position Vector, Velocity, and Acceleration 464
11.10 Derivatives of Vector Functions 467
11.11 Rectangular .Components, of VelpciJy. and.Acceleration 470
11.12 Motion Relative to a Frame in Translaiion.. . 471
11.13 Tangential and Normal Components, 480
11.14 Radial and transverse Components' 483

KINETICS OF PARTICLES; NEWTON'S SECOND LAW 494
12.1 Newton's Second Law of Motion 494
12.2 Linear Momentum of a' Particle. ' Rate-or-Change -of Linear
Momentum 496
12.3 Systems of Units 497
12.4 Equations of Motion 500
12.5 Dynamic Equilibrium
1 2.6 Angular Momentum of a Particle. Rate of Change of
Angular Momentum
12.7 Equations of Motion in Terms of Radial and Transverse
Components 518
1 2.8 Motion under a Central Force. Conservation of
Angular Momentum
1 2.9 Newton's Law of Gravitation
*12.10 Trajectory of a Particle under a Central Force
*12.11 Application to Space Mechanics
*12.12 Kepler's Laws of Planetary Motion

KINETICS OF PARTICLES: ENERGY
AND MOMENTUM METHODS 541
13.1 Introduction
1 3.2 Work of a Force
1 3.3 Kinetic Energy of a Particle. Principle of Work and Energy 546
1 3.4 Applications of the Principle of Work and Energy
1 3.5 Power and Efficiency
1 3.6 Potential Energy
*13.7 Conservative Forces
1 3.8 Conservation of Energy
13.9 Motion under a Conservative Central Force. Application to
Space Mechanics
13.10 Principle of Impulse and Momentum 584
13.11 Impulsive Motion
13.12 Impact
13.13 Direct Central Impact
13.14 Oblique Central Impact 597
13.15 Problems Involving Energy and Momentum

SYSTEMS OF PARTICLES 611
14.1 Application of Newton's Laws to the Motion of a System of
Particles. Effective Forces
14.2 Linear and Angular Momentum of a System of Particles
14.3 Motion of the Mass Center of a System of Particles
14.4 Angular Momentum of a System of Particles about Its Mass Center
14.5 Conservation of Momentum for a System of Particles
14.6 Kinetic Energy of a System of Particles 626
14.7 Work-Energy Principle. Conservation of Energy for a
System of Particles 627
1 4.8 Principle of Impulse and Momentum for a System of Particles g28
14.9 Variable Systems of Particles 635
14.10 Steady Stream of Particles 635
14.11 Systems Gaining or Losing Mass 639

KINEMATICS OF RIGID BODIES 655
15.1 Introduction 655
15.2 translation 657
15.3 Rotation about a Fixed Axis 658
15.4 Equations Defining the Rotation of a Rigid Body about a
Fixed Axis 660
15.5 General Plane Motion 666
1 5.6 Absolute and Relative Velocity in Plane Motion
15.7 Instantaneous Center of Rotation in Plane Motion
1 5.8 Absolute and Relative Acceleration in Plane Motion
V 15.9 Analysis of Plane Motion in Terms of a Parameter
15.10 Rate of Change of a Vector with Respect to a Rotating Frame 694
15.1 1 Plane Motion of a Particle Relative to a Rotating Frame.
Coriolis Acceleration 696
* 15.1 2 Motion about a Fixed Point 705
15.13 General Motion 707
"15.14 Three-dimensional Motion of a Particle Relative to a Rotating Frame. Coriolis Acceleration 716
"15.15 Frame of Reference in General Motion

PLANE MOTION OF RIGID BODIES:
FORCES AND ACCELERATIONS 729
16.1 Introduction 72g
1 6.2 Equations of Motion for a Rigid Body 730
16.3 Angular Momentum of a Rigid Body in Plane Motion
1 6.4 Plane Motion of a Rigid Body. D'Alembert's Principle
*16.5 A Remark on the Axioms of the Mechanics of Rigid Bodies 735
1 6.6 Solution of Problems Involving the Motion of a Rigid Body
16.7 Systems of Rigid Bodies
16.8 Constrained Plane Motion 756

PLANE MOTION OF RIGID BODIES:
ENERGY AND MOMENTUM METHODS 779
17.1 Principle of Work and Energy for a Rigid Body
1 7.2 Work of Forces Acting on a Rigid Body 780
1 7.3 Kinetic Energy of a Rigid Body in Plane Motion 782
1 7.4 Systems of Rigid Bodies 783
17.5 Conservation of Energy 784
1 7.6 Power 785
1 7.7 Principle of Impulse and Momentum for the Plane Motion
of a Rigid Body 798
1 7.8 Systems of Rigid Bodies 801
1 7.9 Conservation of Angular Momentum 802
17.10 Impulsive Motion
17.11 Eccentric Impact 813

KINETICS OF RIGID BODIES IN
THREE DIMENSIONS 828
Introduction 828
Angular Momentum of a Rigid Body in Three Dimensions 829
Application of the Principle of Impulse and Momentum
to the Three-dimensional Motion of a Rigid Body 832
Kinetic Energy of a Rigid Body in Three Dimensions 833
Motion of a Rigid Body in Three Dimensions 843
Euler's Equations of Motion. Extension of D'Alembert's
Principle to the Motion of a Rigid Body in Three Dimensions 845
Motion of a Rigid Body about a Fixed Point 847
Rotation of a Rigid Body about a Fixed Axis 848
Motion of a Gyroscope. Eulerian Angles 858
Steady Precession of a Gyroscope 860
Motion of an Axisymmetncal Body under No Force 862

MECHANICAL VIBRATIONS 875
19.1 Introduction 875

VIBRATIONS WITHOUT DAMPING 876
19.2 Free Vibrations of Particles. Simple Harmonic Motion 876
19.3 Simple Pendulum (Approximate Solution)
*19.4 Simple Pendulum (Exact Solution)
19.5 Free Vibrations of Rigid Bodies
19.6 Application of the Principle of Conservation of Energy
19.7 Forced Vibrations
DAMPED VIBRATIONS
*19.8 Damped Free Vibrations
"19.9 Damped Forced Vibrations
*19.10 Electrical Analogues
Appendix A Some Useful Definitions and
Properties of Vector Algebra
Appendix B Moments of Inertia of Masses
Index
Answers to Even-numbered Problems