# Theoretical Mechanics for Sixth Forms

### in Two Volumes

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Theoretical Mechanics for Sixth Forms, Volume 2 deals with mathematics as a double subject for sixth-form pupils. This book is a continuation of the revised edition of the course in Theoretical Mechanics. This volume discusses the principle of virtual work, stability of equilibrium, vector algebra, and the motion of bodies with variable mass—subjects that can help the student prepare for the freshman year in a university or in one of the polytechnic schools. Compared to the earlier edition, this volume discusses in more detail topics such as the motion of a particle in two dimensions and systems of coplanar forces. This book also emphasizes vector algebra and its many applications, explaining the concept of a vector, the Cartesian coordinates and components, the definitions of vectors and scalars, and vector applications to kinetics. This text also contains exercise problems with answers. This volume can be used for sixth-form students in the U.K. and is also suitable for third year students of a sixth form course as preparation for further studies.

## Table of Contents

Preface

Chapter XV. The Motion of a Rigid Body about a Fixed Axis

15.1. The Definition of a Rigid Body

15.2. The Kinetic Energy of a Rigid Body Rotating about a Fixed Axis

15.3. Calculation of Moments of Inertia in Particular Cases

15.4. Conservation of Energy

15.5. The Compound Pendulum

15.6. The Equation of Motion for a Rigid Body Rotating about a Fixed Axis

15.7. The Force Exerted on the Axis of Rotation

15.8. Impulse and Momentum

15.9. Note on the Relationship between the Equations of Motion of Rotation of a Rigid Body and the Equations of Motion for a Particle Moving in a Straight Line

Chapter XVI. Free Motion of a Rum Body in Two Dimensions

16.1. Angular Velocity and Angular Acceleration

16.2. Instantaneous Center of Rotation

16.3. The Motion of a Rigid Lamina Referred to Its Center of Mass

16.4. The Kinetic Energy of a Lamina Moving in Its Plane

16.5. The Angular Momentum about an Arbitrary Axis Perpendicular to Its Plane of a Lamina Moving in Its Own Plane

16.6. Three-dimensional Motion

16.7. Impulsive Motion

16.8. Application to Problems

Chapter XVII. Motion with Variable Acceleration

17.1. Kinematics

17.2. Dynamical Problems

17.3. Rate of Working by a Variable Force

17.4. Equations of Motion Deduced from the Energy Equation

17.5. Damped Harmonic Oscillations

Chapter XVIII. Motion of a Particle in Two Dimensions

18.1. A Note on Vectors

18.2. Motion of a Particle in Two Dimensions—special Representations

18.3. Motion of a Particle on a Smooth Curve

18.4. The Angular Momentum of a Particle about an Axis

18.5. The Motion of a Projectile

Chapter XIX. Systems of Coplanar Forces

19.1. Forces Acting on a Particle

19.2. Coplanar Forces Acting on a Rigid Body

19.3. The Equivalence of Sets of Coplanar Forces

19.4. The Equilibrium of a Rigid Body under the Action of a System of Coplanar Forces

19.5. A Law of Vector Addition

19.6. The Moment of a Force as a Vector

Chapter XX. The Equilibrium of Rigid Bodies

20.1. Conditions of Equilibrium

20.2. Hinged Bodies

20.3. Problems of Equilibrium Involving More than One Body

20.4. The Equilibrium of the Hinge

Chapter XXI. The Principle of Virtual Work

21.1. The Work Done by a Force Acting on a Particle

21.2. The Potential Energy of a Particle

21.3. Extension to Rigid Bodies

21.4. The Principle of Virtual Work

21.5. Applications of the Principle of Virtual Work

Chapter XXII. Stability of Equilibrium

22.1. The Concept of Stability

22.2. Stability of Equilibrium

22.3. Rolling Stability - The Rocking Stone

Chapter XXIII. Light Frameworks

23.1. Graphical Method of Obtaining the Resultant of a System of Coplanar Forces

23.2. Light Frameworks

23.3. Calculation of the Stresses in the Bars of a Light Framework

23.4. Graphical Investigation of the Stresses in a Light Framework

23.5. The Method of Sections

Chapter XXIV. Continuously Distributed Forces

24.1. Introduction - Shearing Force and Bending Moment

24.2. Relations between Bending Moment, Shearing Force and Loading

24.3. A Uniform Flexible Inelastic String Hanging under Gravity

24.4. The Tightly Stretched Wire

24.5. A Uniformly Loaded Inextensible String

Chapter XXV. The Motion of Bodies with Variable Mass

25.1. Introduction

25.2. Rectilinear Motion of a Particle of Varying Mass—rockets

25.3. The Rectilinear Motion of Chains

Chapter XXVI. Vector Algebra

26.1. The Concept of a Vector

26.2. Cartesian Coordinates and Components

26.3. The Formula of Three-dimensional Coordinate Geometry

26.4. The Definitions of Vectors and Scalars

26.5. The Addition and Subtraction of Vectors

26.6. Some Applications of Vectors to Geometry

26.7. The Scalar Product

26.8. The Vector Product

26.9. Triple Products

26.10. Applications to Kinematics

Answers to the Exercises

Index

## Product details

- No. of pages: 432
- Language: English
- Copyright: © Pergamon 1972
- Published: January 1, 1972
- Imprint: Pergamon
- eBook ISBN: 9781483137803

## About the Authors

### C. Plumpton

### W. A. Tomkys

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