New Tertiary Mathematics

Further Applied Mathematics

1st Edition - January 1, 1981
Authors: C. Plumpton, P. S. W. Macliwaine
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 4 7 7 3 - 4

New Tertiary Mathematics, Volume 2, Part 2: Further Applied Mathematics deals with various topics of theoretical mechanics and probability, from statics and the dynamics of a rigid… Read more

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New Tertiary Mathematics, Volume 2, Part 2: Further Applied Mathematics deals with various topics of theoretical mechanics and probability, from statics and the dynamics of a rigid body to the dynamics of a particle with one and two degrees of freedom. Many examples of varying difficulty are worked in the text and exercises are added after each major topic is covered. This book is comprised of five chapters and opens with a discussion on statics, with particular reference to the analysis of systems of forces in three dimensions, along with virtual work, stability, and the catenary. Complicated equilibrium problems are considered. The reader is then introduced to the dynamics of a particle in one and two dimensions, as well as the implications of the Galilean transformation and the general theorems of motion for a system of particles. These theorems are applied to simple cases of the motion of a rigid body. The final chapter on probability examines normal and Poisson distributions, Markov chains, and miscellaneous problems. This monograph will be a useful resource for mathematical pupils and students engaged in private study.

Glossary of Symbols and Abbreviations

Chapter 7 Further Statics

7:1 Forces in Three Dimensions

Exercise 7.1

7:2 Conditions of Equilibrium for a Rigid Body Acted Upon by a System of Coplanar Forces

Exercise 7.2

7:3 Hinged Bodies—Equilibrium Problems Involving More than One Body

1. Smooth Hinges

2. Rough Hinges

3. The Action of One Part of a Body on Another

4. Problems of Equilibrium Involving More than One Body

5. The Equilibrium of the Hinge

Exercise 7.3

7:4 The Principle of Virtual Work

The Equilibrium of a Particle

Applications of the Principle of Virtual Work

Exercise 7.4

7:5 Determination of Equilibrium Positions

7:6 Types of Equilibrium—Stability

Exercise 7.6

7:7 A Uniform Flexible Inelastic String Hanging under Gravity

The Tightly Stretched Wire

Exercise 7.7

7:8 A Rope in Contact with a Rough Surface

Exercise 7.8

Miscellaneous Exercise 7

Chapter 8 Further Dynamics of a Particle with One Degree of Freedom

8:1 Damped Harmonic Oscillations

Forced Oscillations

Exercise

8:2 Motion in a Straight Line under Variable Forces

Exercise 8.2(a)

Miscellaneous Problems

Motion under Gravity in a Resisting Medium

Exercise 8.2(b)

8:3 The Rectilinear Motion of Bodies with Variable Mass

Exercise 8.3

Miscellaneous Exercise 8

Chapter 9 Dynamics of a Particle with Two Degrees of Freedom

9:1 Further Differentiation of Vectors—Applications to Kinematics

1. Polar Coordinates

2. Intrinsic Coordinates

Exercise 9.1

9:2 Motion Referred to Cartesian Axes

Exercise 9.2

9:3 Motion of a Particle on a Smooth Curve

Exercise 9.3

9:4 Motion under a Central Force—Polar Coordinates

1. A Central Orbit is a Plane Curve

2. The Angular Momentum Integral

3. The Theorem of Areas

Exercise 9.4

9:5 The Motion of Projectiles

1. The Range on an Inclined Plane

Exercise 9.5(a)

2. The Bounding Parabola

Exercise 9.5(b)

9:6 Oblique Impact of Elastic Bodies

Exercise 9.6

9:7 The Galilean Transformation

1. Frames with Uniform Relative Velocity

2. Frames with Uniform Relative Acceleration

9:8 The Motion of a System of Particles—General Theorems

1. Notation

2. Linear Momentum

3. Kinetic Energy

4. Moment of Momentum

5. The Motion of the Center of Mass

6. Motion About the Center of Mass

7. Motions Generated By Simultaneously Applied Impulses

9:9 The Motion of Connected Particles

Exercise 9.9

9:10 The Two-Dimensional Motion of a Projectile in a Resisting Medium

Exercise 9.10

Miscellaneous Exercise 9

Chapter 10 An Introduction to the Dynamics of a Rigid Body

10:1 Rotation of a Lamina about a Fixed Axis

10:2 Momentum and Energy Equations for Angular Motion of a Lamina

Exercise 10.2

10:3 The Compound Pendulum

Exercise 10.3

10:4 The Force Exerted on the Axis of Rotation

Exercise 10.4

10:5 Impulse and Angular Momentum

Exercise 10.5

10:6 Note on the Relationship Between the Equations of Angular Motion of a Rigid Body and the Equations of Motion of a Particle Moving in a Straight Line

10:7 The Motion of a Lamina in Its Own Plane—Instantaneous Center of Rotation

Exercise 10.7

10:8 The General Motion of a Rigid Lamina in Its Own Plane

10:9 Application to Miscellaneous Problems

Exercise 10.9

Miscellaneous Exercise 10

Chapter 11 Further Probability

11:1 The Binomial and Geometric Probability Distributions

Exercise 11.1

11:2 Continuous Probability Distributions

Exercise 11.2

11:3 The Poisson Distribution

Exercise 11.3

11:4 The Normal Distribution

Exercise 11.4

11:5 Transition Matrices for Markov Chains

11:6 Steady-State Vector and Matrix

11:7 An Equivalence Relation on Transition States

Exercise 11.7

11:8 Further Probability Problems

Exercise 11.8

Miscellaneous Exercise 11

Answers to the Exercises

Index