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VOURSELF

BOOK'S

MECHANICSABBOTT

MECHANICS" T h e author rightly considers t h a t the approach t o mechanics should be largely experimental, and t h e r e f o r e in the early chapters he describes experiments w i t h simple apparatus w h i c h can be easily constructed o r obtained by the student. This is a splendid book, and anyone w o r k i n g his way t h r o u g h it conscientiously should acquire a sound knowledge of those mechanical principles which are applied in many branches of industry and in the technical branches of the forces." Scottish Educational Journal " T h e book is well illustrated w i t h singularly clear line diagrams and full use is made of varied types w h i c h make the t e x t very readable." The Journal of Education " A n excellent book . . . it is w e l l adapted t o any student of elementary mechanics, and t o teaching y o u r s e l f . " The Faraday House Journal

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THE TEACH YOURSELF BOOKS EDITED BY LEONARD CUTTS

MECHANICS

Uniform with this volume and in the same series

Teach Yourself ALGEBRA Teach Yourself ARITHMETIC Teach Yourself CALCULUS Teach Yourself GEOMETRY Teach Yourself MATHEMATICS Teach Yourself READY R E C K O N E R Teach Yourself: THE SLIDE RULE Teach Yourself STATISTICS Teach Yourself TRIGONOMETRY

TEACH

YOURSELF

MECHANICSBy P. A B B O T T , B.A.

iA

THE

ENGLISH

UNIVERSITIESSTREET

PRESS LTD

102 N E W G A T E

LONDON,

E.C.I

First printed This impression

ig4i 1959

ALL RIGHTS RESERVED

Printed in Great Britain for the English Universities Press, Limited, by Richard Clay and Company, Ltd., Bungay, Suffolk

INTRODUCTIONTHIS

book has been written designedly for the private student and especially for those who need to acquire a knowledge of the mechanical principles which underlie much of the work in many branches of industry and in the technical branches of the fighting forces. It is difficult in these troubled times to obtain the tuition provided by Scientific and Technical Institutions for such subjects as Mechanics. Nor is there any completely satisfactory substitute. Nevertheless the author hopes and believes that enough can be learnt from the present volume to be of very real practical use as an introduction to the subject. Like other volumes in the "Teach Yourself " series, it seeks to give such help as is possible to those- students who are anxious to acquire a knowledge of the subject but must rely, in the main, on their own studies. The difficulty is increased in such a subject when access is not possible to laboratories and apparatus. The view of the author of this book, founded on long experience, is that the approach to mechanics should be largely experimental. To a considerable extent the subject has been built up through centuries of human progress by experiment, often prompted by the needs of mankind. Experiment has preceded theory, as witness the discoveries of Archimedes, Torricelli and Galileo. What is to take the place of this practical basis for the subject? In the early chapters of this book directions and descriptions are given of experiments with simple apparatus such as most students with a little ingenuity can construct or obtain. By this means the student is led to formulate some of the simple fundamental principles upon which the subject is built. When the apparatus required is more complicated and cannot be obtained by the student it seems to the author that in many cases there is little value in theV

vi

INTRODUCTION

description of experiments which have been performed by others, with the numerical results which must be accepted without personal verification. It seems simpler and just as convincing to state the principles which are demonstrated by such experiments, leaving the verification for later study. Mathematical proofs which involve a knowledge and experience of the subject which may be beyond the average student are omitted, the truth of the principles which they demonstrate being assumed. Another important factor in a book of this kind is that of the amount of mathematical knowledge which may be assumed as being possessed by the average reader. The minimum amount required for the greater part of this volume is ordinary arithmetic, elementary algebra including the solution of quadratic equations with a knowledge of ratio, variation and a certain amount of fundamental geometry. In some sections of the book a knowledge of elementary trigonometry is essential. To assist the student cross references, when it seems desirable, are given to the appropriate sections in the companion book on Trigonometry in this series. In those cases in which practical drawing can be employed as an alternative to a trigonometrical solution, this is indicated. Inevitable limitations of space have made it necessary to exclude many practical and technical applications of the principles evolved. It has been considered more profitable for private students, who in many cases are familiar with practical applications, that as much space as possible should be given to explanations of the theoretical aspects of the subject. The examples to be worked by the student are designed to enable him to test his knowledge of the theorems on which they are based and to consolidate his knowledge of them. Academic exercises,, depending for their solution mainly on mathematical ingenuity have been excluded. The author desires to express his thanks to Mr. W. D. Hills, B.Sc. for permission to use diagrams 22 and 138 from his book on " Mechanics and Applied Mathematics," published by the University of London Press; also to

INTRODUCTION

vii

Messrs. Cussons for the use of blocks of some of the admirable apparatus which has been designed by them for use in teaching mechanics. The author is also indebted to Mr. C. E. Kerridge, B.Sc. for the use of the example on p. 207 from " National Certificate Mathematics," Vol. I, and to the University of London for their consent to the inclusion of a few of their Examination questions.

CONTENTSPARA. PACE

CHAPTER IINTRODUCTORY

1. The Meaning ol Mechanics 3. Weight and Force . 4. Transmission of Force . 6. Equilibrium . . 6. Measurement of a Force

. . . . . II

. . . . .

. . . . .

. . . .

13 14 15 16 . 1 6

CHAPTERTHE LEVER

8. 9. 10. 12. 13. 14. 16. 17. 20. 22.

Machines . . . . . . . . The Principles of the Lever . . . . Turning Moments . . . . . . The Weight of the Lever Pressure on the Fulcrum. Resultant Force . Centre of Force . . . . . . Bars Resting on Two Supports . . . . Orders of Levers . . . . . . . Practical Examples of Levers . . . . A Simple Pulley CHAPTER III

18 . 1 9 20 23 . 27 . 2 9 29 35 36 39

CENTRE OF GRAVITY

24. 25. 26. 27. 28. 29. 30. 32. 33. 35. 37. 38.

Centre of Parallel Forces . . . . . Centre of Gravity of a Number of Particles . . Centre of Gravity of a Uniform Rod . Centre of Gravity of Regular Geometrical Figures . Experimental Determination of the Centre of Gravity Centre of Gravity of a Rectangular Lamina . . Centre of Gravity of a Triangular Lamina . . Centre of Gravity of Composite Bodies . Use of Moments in Determining Centre of Gravity . Centre of Gravity of Regular Solids . Equilibrium of Bodies . . . . . . Stable, Unstable, and Neutral Equilibrium . . CHAPTER IV

44 45 45 46 46 47 48 50 53 59 61 62

RESULTANT OF NON-PARALLEL FORCES PARALLELOGRAM OF FORCE

40. 41.

Geometric Representation of a Force Vector Quantity . . . . .ix

. .

. .

.

66 67

CONTENTS 42. 43. 45. 47. 49. Non-parallel Forces acting on a Body . Parallellogram of Forces . . . . . Calculation of the Resultant of Two Forces Acting at a Point . . . . Resultant of a Number of Forces Acting at a. Point . Moments of Intersecting Forces . . . . CHAPTER V67

68 7i 75 77

COMPONENTS OF A FORCE, RESOLVED PARTS OF A FORCE

52. 53. 56. 59. 60. 63.

Components of a Force . . . . . Resolving a Force . . . . . . Forces Acting on a Body on a Slope Resolved Paxts. Resultant of Forces Acting at a Point Resultant of any Number of Concurrent Forces Equilibrium of Forces Acting at a Point CHAPTER VI

82 83 84 89 90

95

TRIANGLE OF FORCES; POLYGON OF FORCES; LAMI'S THEOREM

64. The Triangle of Forces 67. The Inclined Plane 69. Lami's Theorem . 71. The Polygon of Forces

98

101 . . . VII "5116 117 119 120 125

. .

. .

. .

. .

105 109

CHAPTERFRICTION

73. Friction as a Force 74. Limiting Friction 75. Coefficient of Friction 76. The Angle of Friction 78. The Laws of Friction 81. The Inclined Plane and Friction CHAPTERBODIES IN

VIII130

MOTION-VELOCITY

83. 84. 85. 86. 87. 88. 89.

Speed and Velocity Measurement of Speed and Velocity Uniform Velocity Average Velocity Distance-time Graphs . Velocity-time Graphs . Area under a Velocity-time Graph

131 132 133 133 137141

CONTENTSPARA.

xiPAGE

CHAPTER 90. 92. 93. 94. 97. 98. 100.

IXI46

ACCELERATION

Changes in Velocity . . . . . . Formula for Uniform Acceleration . . . . Average Velocity and Distance . . . . Distance passed over by a Uniformly Accelerated Body Acceleration due to Gravity . . . . . Acceleration of a Falling Body . . . . Motion of a Body Projected Upwards . CHAPTER XMOTION

147I48

149 153 154 156

N E W T O N ' S LAWS OF

103. 104. 105. 106. 108. 109. 110. 113.

Mass and Weight Measurement of Mass . . Mass and Inertia . . . Newton's First Law of Motion Newton's Second Law of Motion Momentum Units of Force Newton's Third Law of Motion CHAPTER XI