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BY THE SAME AUTHOR

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THE STRAIGHT LINE AND CIRCLE.

Chapters I-IX^ of^ Part^ I^ of^ The^ Elements^ of^ Coordinate^ Geometry.

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AN ARITHMETIC FOR (^) SCHOOLS. Seventeenth (^) Impression. Globe 8vo. With or without (^) Answers, 5*. Or in^ two^ Parts, with (^) Answers, 3s. each. The (^) Examples alone, 3s. Qd. The Answers (^) alone, Qd.

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LONDON: MACMILLAN (^) AND (^) CO., Limited

Pitt Press Mathematical Series

THE (^) ELEMENTS

OF

STATICS AND DYNAMICS.

PART I. ELEMENTS OF STATICa

Cambridge University^ Press

Fetter Lane, London

New York

Bombay, Calcutta,^ Madras Toronto Macmillan Tokyo

Maruzen Company, Ltd

All (^) rights reserved

First (^) Edition, Dec. 1890. Second (^) Edition, Sept. 1892, Third (^) Edition, June, 1893. Fourth (^) Edition, enlarged, Jan. 1895. Reprinted 1897,^ 1899,^ 1900, 1902,^ 1904. Fifth Edition^ (revised and^ enlarged), (^) July, 1901). Reprinted 1907, 1908, 1911, 1914, (^) 1918, (^1920) (twice) 1923, 1925, 1927, 1932

PRINTED IN^ GREAT^ BRITAIN

PREFACE TO PART I.

TN the (^) following work I have aimed at (^) writing a

-*- working text-book on Statics for the use of Junior

Students. Throughout the^ book^ will^ be^ found^ a^ large number

of examples ; most of them, with the exception of

many of^ those^ at^ the^ end^ of^ the^ Chapter on^ Friction and the Miscellaneous (^) Examples at the (^) end of the

volume, are^ of^ an^ easy type.

I (^) have tried to (^) make the book (^) complete as far as

it goes; it is suggested, however, that the student

should, on^ the^ first^ reading of^ the^ subject, omit^ every-

thing marked^ with^ an^ asterisk. I must (^) express my obligations to (^) my friend Mr (^) H. C. (^) Eobson, (^) M.A., Fellow and (^) Lecturer of

Sidney Sussex^ College, Cambridge, for^ his^ kindness

in (^) reading through the (^) proof-sheets, and (^) for (^) many

suggestions that^ he^ has^ made^ to^ me. Any corrections^ of^ errors, or^ hints^ for^ improvement

will be thankfully received.

S. (^) L. LONEY. Barnes, S.W. December, 1890.

CONTENTS.

STATICS. CHAP. PAGK I. Introduction 1

II. Composition and^ Resolution^ of^ Forces^.^.^8

III. Composition^ and^ Resolution^ of^ Forces^ (con-

tinued) 29

IV. Parallel Forces 47

V. Moments 58

VI. Couples 75

VII. Equilibrium of^ a^ rigid^ body^ acted^ upon^ by

THREE FORCES IN^ A^ PLANE^ ....^84

VIII. General conditions of equilibrium of^ a^ rigid

BODY ACTED ON BY A SYSTEM OF^ FORCES^ IN

ONE PLANE 97

IX. Centre of Gravity 119

Centre of (^) gravity of a (^) Triangle, Tetrahedron, etc 123 General formulae for the determination of^ the centre of (^) gravity 129

X. Centre of Gravity (continued).

Properties of^ the^ centre^ of^ gravity.^.^. Stable and unstable (^) equilibrium.^.^.^153 XI. Wore. .... (^) v .... 163

CONTENTS. CHAP.

XII. Machines 171

I. The^ Lever^174

II. (^) Pulleys and (^) Systems of^ Pulleys.^.^180 III. The^ Inclined^ Plane^.^.^.^. IV. The Wheel and^ Axle^.^.^.^. Weston's Differential (^) Pulley.^.^207 The Common Balance^ ....^209

The Steelyards 218

The Screw^224

. 241 . 246 . (^).

V. VI. VII.

XIII. Friction

Laws of Friction

Equilibrium on^ a^ rough^ Inclined^ Plane^. Efficiency of^ machines^ ..... Machines with friction

XIV. Problems with Friction 260

XV. Miscellaneous.

Smooth Hinges 274

Funicular (^) Polygon 279 Tensions of Elastic^ Strings ....^282 Graphic Constructions.^ Link^ and^ Force^ Poly- gons XVI. Some^ Additional^ Propositions^ .... Formal (^) proof of^ the^ Parallelogram of^ Forces^. Centre of gravity of a^ Circular^ Arc, and^ of^ a

Sector and^ Segment of^ a^ Circle

Centre of (^) gravity of a Zone of a^ Sphere Centres of^ gravity of^ a^ Hollow^ and^ a^ Solid Hemisphere 308 Virtual Work^310 Roberval's Balance 314

Easy Miscellaneous^ Examples^318

Harder Miscellaneous Examples^320

Answers ... i^ xx

2 STATICS

A (^) body may be (^) regarded as an (^) indefinitely large number of (^) indefinitely small (^) portions, or as a (^) conglomeration of

particles.

6. A (^) Rigid Body is a (^) body whose (^) parts always

preserve an^ invariable^ position with^ respect to^ one^ another.

This (^) conception, like that of a (^) particle, is idealistic.

In nature no body is perfectly rigid. Every body yields,

perhaps only very slightly, if^ force^ be^ applied to^ it.^ If^ a rod, made^ of^ wood, have^ one^ end^ firmly fixed^ and^ the^ other

end be pulled, the wood stretches slightly if the rod be

made of iron the deformation is (^) very much less. To (^) simplify our (^) enquiry we shall assume that all the bodies with^ which^ we have to^ deal^ are^ perfectly rigid.

7. (^) Equal Forces.^ Two^ forces^ are^ said^ to^ be^ equal

when, if^ they act^ on^ a^ particle in^ opposite directions,^ the

particle remains^ at^ rest.

8. Mass. The mass of a (^) body is the (^) quantity of

matter in the body. The unit of mass^ used^ in^ England is

a pound and is^ defined^ to^ be^ the^ mass^ of^ a^ certain^ piece of

platinum kept in^ the^ Exchequer^ Office.

Hence the mass of a (^) body is (^) two, three, four... (^) lbs., when it^ contains^ two, three, four...^ times^ as^ much^ matter

as the standard lump of platinum.

In France, and other foreign countries, the^ theoretical

unit of mass used is^ a^ gramme, which^ is^ equal to^ about

15*432 grains. The practical unit is a kilogramme (

grammes), which^ is^ equal^ to^ about^ 2*2046^ lbs.

9. (^) Weight. The idea of^ weight is^ one^ with^ which everyone is^ familiar.^ We^ all^ know^ that^ a^ certain^ amount

of exertion is required to prevent any body from falling to

the ground. The earth attracts every body to itself with

INTRODUCTION 3

a force (^) which, as^ we^ shall^ see^ in^ Dynamics, is^ proportional

to the mass^ of^ the^ body.

The force with which the earth attracts any body to

itself is called the weight of the body.

10. Measurement of Force. "We shall choose, as^ our

unit of force in Statics, the^ weight of^ one^ pound. The^ unit

of force is therefore equal to^ the^ force^ which^ would^ just

support a^ mass^ of^ one^ pound when^ hanging^ freely. We shall^ find^ in^ Dynamics that^ the^ weight of^ one

pound is^ not^ quite^ the^ same^ at^ different^ points^ of^ the

earth's surface. In (^) Statics, however, we shall not have to (^) compare forces at different (^) points of the earth's (^) surface, so that^ this variation in the (^) weight of a (^) pound is of no (^) practical importance ; we

shall therefore neglect this variation and assume the weight

of a pound to be constant.

11. In practice the expression "^ weight of one pound "

is, in^ Statics, often^ shortened^ into^ "one^ pound." The

student will therefore understand that "a force of 10 lbs." means "a force (^) equal to the (^) weight of 10 lbs."

12. Forces (^) represented by straight lines. A force will be (^) completely known when we know (^) (i) its (^) magnitude,

(ii) its^ direction,^ and^ (iii) its^ point of^ application, i.e.^ the

point of^ the^ body at^ which^ the^ force^ acts.

Hence we can (^) conveniently (^) represent a force (^) by a

straight line^ drawn^ through its^ point of^ application; for

a straight line has both magnitude and direction.

Thus (^) suppose a (^) straight line OA (^) represents a (^) force, equal to^10 lbs.^ weight, acting at^ a^ point 0.^ A^ force^ of

5 lbs. weight acting in the same direction would be repre-

sented (^) by 0, where B bisects the distance (^) OA, whilst a 12

INTRODUCTION 5

string whatever^ be^ the^ point, A,^ 2>,^ or^ C^ of^ the^ string^ at

which the force is^ applied.

Now the force at A (^) required to (^) support the^ weight

is the same in each case ; hence it^ is^ clear^ that^ the^ effect

at A is the same whatever be the^ point of^ the^ string to

which the tension is^ applied and^ that^ the^ tension^ of^ the

string is^ therefore^ the^ same^ throughout^ its^ length.

Again, if^ the^ weight W^ be^ sup-

ported by a^ light^ string^ passing^ round

a smooth peg A, it is found that the

same force must be exerted at^ the^ other

end of the string whatever be^ the

direction (^) (AB, AC, or^ AD) in^ which

the string is pulled and^ that^ this^ force

is equal to the weight W.

[These forces^ may^ be^ measured^ by^ attaching^ the^ free

end of the string to a spring-balance.]

Hence the tension (^) of a (^) light string passing round a

smooth peg is^ the^ same^ throughout its^ length.

If two or more strings be knotted together the tensions

are not necessarily the same in each string.

The student must (^) carefully notice that the tension of a (^) string is

not proportional to^ its^ length. It^ is^ a^ common^ error^ to^ suppose that

the longer a^ string the^ greater is^ its^ tension^ ; it^ is^ true^ that^ we^ can

often (^) apply our force more (^) advantageously if we use a (^) longer piece of string, and^ hence^ a^ beginner^ often^ assumes^ that,^ other^ things^ being equal, the^ longer string has^ the^ greater tension.

16. Reaction. If one body lean, or be pressed, against

another body, each body experiences a force at the point of

contact (^) ; such a^ force^ is^ called^ a^ reaction.

The force, or action, that^ one^ body exerts^ on a^ second

body is^ equal and^ opposite to^ the^ force,^ or^ reaction,^ that

the second body exerts on the first.

6 STATICS

This statement^ will^ be^ found^ to^ be^ included^ in^ Newton's

Third Law of Motion (^) [Part II., Art. (^) 73]. Examples. If^ a^ ladder^ lean^ against^ a^ wall^ the^ force

exerted by the end of the ladder upon the wall is equal and

opposite to^ that^ exerted^ by the^ wall^ upon the^ end^ of^ the

ladder.

If a cube of wood is placed upon a table the force which

it exerts upon the table is equal and opposite to the force

which the table exerts on it.

17. (^) Equilibrium. When two or more forces act

upon a^ body and^ are^ so^ arranged that^ the^ body remains^ at

rest, the^ forces^ are^ said^ to^ be^ in^ equilibrium.

18. Introduction, or removal, of equal and opposite

forces. We^ shall^ assume^ that^ if^ at^ any point of^ a^ rigid body we^ apply two^ equal and^ opposite^ forces,^ they will

have no effect on^ the^ equilibrium of^ the^ body; similarly,

that if at any point of a body two equal and opposite

forces are (^) acting they may be removed.

19. Principle of the Transmissibility of Force. If a

force act^ at^ any^ point^ of a^ rigid^ body,^ it^ may^ be^ considered

to act at any other point in^ its^ line^ of action^ provided that

this latter (^) point be (^) rigidly connected with the (^) body. Let a force F act at a (^) point A of a^ body in^ a^ direction AX. Take (^) any point B in AX and^ at^ B^ introduce^ two