Tips and tricks
The tutorial statics
THE TUTORIAL STATICS
BY WILLIAM BRIGGS,
GENERAL EDITOR OF THE TUTORIAL SERIES, PRINCIPAL OF UNIVERSITY CORRESPONDENCE COLLEGE,
AND
G. H. BRYAN,
SMITH'S PRIZEMAN, LATE FELLOW OF ST. PETER'S COLLEGE, CAMBRIDGE,
LONDON: W. B. CLIVE, UNIVERSITY CORRESPONDENCE COLLEGE PRESS.
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PREFACE
The present book may be regarded as in many respects an enlarged edition of the authors' Text Book of Statics. It now includes most of those portions of Statics which can be satisfactorily treated without the use of higher analytical methods. The earlier chapters can still be read by students having little or no knowledge of Trigonometry, as the few instances in which trigonometrical expressions occur involve practically nothing more than the mere definitions of the sine, cosine, and tangent. We are indebted to Mr. A. Cracknell, M.A., who is responsible for the later chapters. These deal with the general conditions of equilibrium of forces in one plane; the principles of virtual work and virtual velocities, the laws of friction, and the centres of gravity of such bodies as circular arcs, spherical zones, and the like.
Although "Virtual" Work thus forms a separate chapter, the Principle of Work is freely employed throughout the book as furnishing verifications or alternative proofs of results which have been established independently. Those who wish to omit all considerations relating to Work till the end will have no difficulty in doing so, as the sections in question have been kept distinct and can readily be picked out by their headings. We may, perhaps, call attention to the modification of Newton's proof of the Parallelogram of Forces, which will, we think, be found more instructive than most "dynamical" proofs, which generally practically consist in the mere statement that the proposition is included in Newton's Second Law.
Two sizes of type are used, all the important bookwork being printed in the larger type, while hints, explanations, examples, alternative proofs, and a few of the less important theorems are printed in smaller type. The more fundamental propositions - such as the Parallelogram of Forces - have their numbers as well as their headings printed in dark type (thus - 136). These the student should be be able to reproduce from memory.
The "Summaries of Results" at the end of each chapter are intended to facilitate a final revision of the whole subject, but they must only be used with great caution when the text has been thoroughly mastered.
Where results are given as corollaries, it is not in most cases sufficient in proving them to quote the result of the proposition on which they depend; the student will do well to write out complete proofs, employing, as far as desirable, the same methods of proof as are used for the propositions themselves.
It is hoped that the earlier chapters, which have been taken almost intact from the Text Book of Statics (now out of print) will appeal to the same class of readers who formerly used the latter book, and that the additional matter will render the book serviceable to those students who wish to pursue the subject further.
October, 1896.
CONTENTS.
Chapter I.
Chapter I.
THE PARALLELOGRAM OF FORCES.
Representation of forces by straight lines. Proof and experimental verification of Parallelogram of Forces. Triangle and Polygon of Forces; their converses and applications. Resultant of two perpendicular forces
Chapter II.
RECTANGULAR RESOLUTION OF FORCES.
Resolved part of a force. Magnitude of the resultant of two forces including any angle. Lami's Theorem. Principle of Work for a particle
Chapter III.
THE INCLINED PLANE.
Equilibrium under a horizontal force, or force up the plane. Particular Cases of Equilibrium (problems)
Chapter IV.
THE TRANSMISSION OF FORCE.
Rigid Bodies. Equilibrium of three forces. The "Wedge. Heavy bodies
Chapter V.
MOMENTS OF FORCES IN ONE PLANE
Definition, algebraic sign, and geometrical representation of Moments. Varignon's Theorem. Work of a Moment
Chapter VI.
PARALLEL FORCES
Their resultant when like and when unlike. Conditions of equilibrium of three parallel forces. Experimental verifications
Chapter VII.
SYSTEMS OF PARALLEL FORCES
Equilibrium of a loaded beam or heavy lever. Resultant and centre of parallel forces at given points in a straight line. Couples; their moments and properties. Resultant of two or more Couples
Chapter VIII
MACHINES
Their mechanical advantage. The Lever; the three classes of Lever. The Wheel and Axle; the Windlass and Capstan. Principle of Work for Machines
Chapter IX.
THE PULLEY
Fixed and moveable. The three Systems of Pulleys. Work as applied to Pulleys. The Screw
Chaptee X.
CENTRES OF PARALLEL FORCES
Definition of the Centre of Gravity. Every heavy body has one and only one C.G. Mass-Centre Defined. C.G. of a Straight Line, Circle, or Sphere.' C.G. of a Parallelogram
Chapter XI.
PROPERTIES OF THE CENTRE OF GRAVITY
Equilibrium of a Heavy Body about a Fixed Point. To find the C.G. of a Lamina. Equilibrium of a body resting on its Base. Stable, Unstable, and Neutral Equilibrium
Chapter XII.
DETERMINATION OF THE CENTRE OF GRAVITY
Triangular Lamina, Triangular Wire, Polygon. C.G. of a portion' of a body, and of a system of weights in one plane. Work done in raising C.G. of weights. C.G. of a Triangular Pyramid or Cone
Chapter XIII.
BALANCES
Requisites of a good Balance. False Balances. The Common or Roman Steelyard.
The Danish Steelyard
Chapter XIV.
CONDITIONS OF EQUILIBRIUM OF FORCES IN ONE PLANE
Properties of Couples ; Graphical Methods
Chapter XV.
THE PRINCIPLES OF VIRTUAL WORK AND VIRTUAL VELOCITIES
Chapter XVI.
THE LAWS OF FRICTION
Equilibrium on Rough Inclined Plane, and in Rough Screw-Press
Chapter XVII.
FURTHER DETERMINATIONS OF CENTRE OF GRAVITY
Frustum of Cone, Circular Arc and Sector, Spherical Zone and Sector
Representation of forces by straight lines. Proof and experimental verification of Parallelogram of Forces. Triangle and Polygon of Forces; their converses and applications. Resultant of two perpendicular forces
Chapter II.
RECTANGULAR RESOLUTION OF FORCES.
Resolved part of a force. Magnitude of the resultant of two forces including any angle. Lami's Theorem. Principle of Work for a particle
Chapter III.
THE INCLINED PLANE.
Equilibrium under a horizontal force, or force up the plane. Particular Cases of Equilibrium (problems)
Chapter IV.
THE TRANSMISSION OF FORCE.
Rigid Bodies. Equilibrium of three forces. The "Wedge. Heavy bodies
Chapter V.
MOMENTS OF FORCES IN ONE PLANE
Definition, algebraic sign, and geometrical representation of Moments. Varignon's Theorem. Work of a Moment
Chapter VI.
PARALLEL FORCES
Their resultant when like and when unlike. Conditions of equilibrium of three parallel forces. Experimental verifications
Chapter VII.
SYSTEMS OF PARALLEL FORCES
Equilibrium of a loaded beam or heavy lever. Resultant and centre of parallel forces at given points in a straight line. Couples; their moments and properties. Resultant of two or more Couples
Chapter VIII
MACHINES
Their mechanical advantage. The Lever; the three classes of Lever. The Wheel and Axle; the Windlass and Capstan. Principle of Work for Machines
Chapter IX.
THE PULLEY
Fixed and moveable. The three Systems of Pulleys. Work as applied to Pulleys. The Screw
Chaptee X.
CENTRES OF PARALLEL FORCES
Definition of the Centre of Gravity. Every heavy body has one and only one C.G. Mass-Centre Defined. C.G. of a Straight Line, Circle, or Sphere.' C.G. of a Parallelogram
Chapter XI.
PROPERTIES OF THE CENTRE OF GRAVITY
Equilibrium of a Heavy Body about a Fixed Point. To find the C.G. of a Lamina. Equilibrium of a body resting on its Base. Stable, Unstable, and Neutral Equilibrium
Chapter XII.
DETERMINATION OF THE CENTRE OF GRAVITY
Triangular Lamina, Triangular Wire, Polygon. C.G. of a portion' of a body, and of a system of weights in one plane. Work done in raising C.G. of weights. C.G. of a Triangular Pyramid or Cone
Chapter XIII.
BALANCES
Requisites of a good Balance. False Balances. The Common or Roman Steelyard.
The Danish Steelyard
Chapter XIV.
CONDITIONS OF EQUILIBRIUM OF FORCES IN ONE PLANE
Properties of Couples ; Graphical Methods
Chapter XV.
THE PRINCIPLES OF VIRTUAL WORK AND VIRTUAL VELOCITIES
Chapter XVI.
THE LAWS OF FRICTION
Equilibrium on Rough Inclined Plane, and in Rough Screw-Press
Chapter XVII.
FURTHER DETERMINATIONS OF CENTRE OF GRAVITY
Frustum of Cone, Circular Arc and Sector, Spherical Zone and Sector
CHAPTER XIII - BALANCES
In this chapter we shall describe the various contrivances by which bodies are usually weighed. Remembering that weight is proportional to mass, we observe that the operation of weighing by balancing a body with known weights affords in every case a correct measure of the mass or quantity of matter in the body in pounds or grammes or other chosen units, and that the observed weight is independent of any local variations in the intensity of gravity.
The common balance (see Fig. 147, p. 190) consists essentially of a beam or lever AB fixed so that it can turn about a fulcrum placed a little above its middle point. From its ends are suspended two scale pans; the goods to be weighed are placed in one of these, and are balanced by placing suitable weights in the other, till the beam assumes a horizontal position.
In delicately constructed balances, the fulcrum and points of suspension consist of wedge-shaped pieces of hard steel (called " knife blades "), whose edges rest on, hard plates of steel.
The requisites of a good balance are that it be (i.) true, (ii.) stable, (iii.) sensitive, (iv.) rigid.
182. Conditions that the balance may be true -A balance is said to be true if the beam assumes a Horizontal position when equal weights are placed in the two scale pans. This requires that -
(i.) The two arms of the beam must he of equal length, that is, A0 = BO, or the fulcrum must be in a line HO, bisecting at right angles the line AB, which joins the points of suspension of the two scale pans.
(ii.) The scale pans must be of equal weight.
(iii.) The C.G. of the beam must be vertically under the fulcrum when the beam is horizontal, and therefore also in HO.
When these conditions are satisfied, equal weights placed anywhere whatever in the scale pans will balance each other with the beam horizontal.
183. Conditions that the balance may be stable. - A balance is said to be stable if the beam tends of its own accord to fall into its equilibrium position. A balance would evidently be useless for weighing if its equilibrium position were unstable or even neutral.
A balance is said to be more or less stable according to the comparative readiness or reluctance of the beam to assume its equilibrium position.
Stability is secured by placing 0, the fulcrum of the beam, a little above the points G, H, at which the resultant weights of the beam and the two pans act respectively.
Conditions that the balance may be sensitive. - It is not sufficient that the beam should be horizontal when the weights in the scale pans are equal. It must also indicate when they are unequal, by the beam assuming a non-horizontal position. This is expressed by saying that the balance must be sensitive (or, as some writers call it, "sensible"). In a sensitive balance, even a small additional weight placed in one scale pan should turn
the beam through a perceptible angle, and the smallness of the weight which suffices to do so affords a measure of the sensitiveness of the balance and of the degree of accuracy attainable in weighing with it.
185. Rigidity. - The balance must have a beam sufficiently strong not to bend under the weights which it has to carry. For this purpose a short thick beam would be preferable to a long thin one, but it would of course be less sensitive. To secure the greatest strength consistent with lightness, the beam is usually made in the form shown in Fig. 147.
186. False balances. - Double weighing. - A balance will evidently be false if -
(i.) The arms are of unequal length,
(ii.) The scale pans are of unequal weight,
(iii.) The beam is improperly balanced (i.e., G not on OH).
In all such cases the true weight of a body may be found by either of the following methods of double weighing.
The first method is to place the body in one scale pan and balance it with suitable counterpoises (e.g., small shot or fine sand) placed in the opposite pan. Now remove the body and replace it by weights sufficient to balance the counterpoises, and to bring the beam to the same position as before. These weights are evidently equal to the required weight of the body, however false the balance used, for they act under exactly the same circumstances and produce exactly the same effect.
The second method is less simple, but it enables us to test the trueness of the balance. The body is weighed first in one scale pan and then in the other. If the two observed weights are equal, the balance is true, and each is equal to the true weight of the body. If not, the balance is false, and we have the following cases to consider.
The common balance (see Fig. 147, p. 190) consists essentially of a beam or lever AB fixed so that it can turn about a fulcrum placed a little above its middle point. From its ends are suspended two scale pans; the goods to be weighed are placed in one of these, and are balanced by placing suitable weights in the other, till the beam assumes a horizontal position.
In delicately constructed balances, the fulcrum and points of suspension consist of wedge-shaped pieces of hard steel (called " knife blades "), whose edges rest on, hard plates of steel.
The requisites of a good balance are that it be (i.) true, (ii.) stable, (iii.) sensitive, (iv.) rigid.
182. Conditions that the balance may be true -A balance is said to be true if the beam assumes a Horizontal position when equal weights are placed in the two scale pans. This requires that -
(i.) The two arms of the beam must he of equal length, that is, A0 = BO, or the fulcrum must be in a line HO, bisecting at right angles the line AB, which joins the points of suspension of the two scale pans.
(ii.) The scale pans must be of equal weight.
(iii.) The C.G. of the beam must be vertically under the fulcrum when the beam is horizontal, and therefore also in HO.
When these conditions are satisfied, equal weights placed anywhere whatever in the scale pans will balance each other with the beam horizontal.
183. Conditions that the balance may be stable. - A balance is said to be stable if the beam tends of its own accord to fall into its equilibrium position. A balance would evidently be useless for weighing if its equilibrium position were unstable or even neutral.
A balance is said to be more or less stable according to the comparative readiness or reluctance of the beam to assume its equilibrium position.
Stability is secured by placing 0, the fulcrum of the beam, a little above the points G, H, at which the resultant weights of the beam and the two pans act respectively.
Conditions that the balance may be sensitive. - It is not sufficient that the beam should be horizontal when the weights in the scale pans are equal. It must also indicate when they are unequal, by the beam assuming a non-horizontal position. This is expressed by saying that the balance must be sensitive (or, as some writers call it, "sensible"). In a sensitive balance, even a small additional weight placed in one scale pan should turn
the beam through a perceptible angle, and the smallness of the weight which suffices to do so affords a measure of the sensitiveness of the balance and of the degree of accuracy attainable in weighing with it.
185. Rigidity. - The balance must have a beam sufficiently strong not to bend under the weights which it has to carry. For this purpose a short thick beam would be preferable to a long thin one, but it would of course be less sensitive. To secure the greatest strength consistent with lightness, the beam is usually made in the form shown in Fig. 147.
186. False balances. - Double weighing. - A balance will evidently be false if -
(i.) The arms are of unequal length,
(ii.) The scale pans are of unequal weight,
(iii.) The beam is improperly balanced (i.e., G not on OH).
In all such cases the true weight of a body may be found by either of the following methods of double weighing.
The first method is to place the body in one scale pan and balance it with suitable counterpoises (e.g., small shot or fine sand) placed in the opposite pan. Now remove the body and replace it by weights sufficient to balance the counterpoises, and to bring the beam to the same position as before. These weights are evidently equal to the required weight of the body, however false the balance used, for they act under exactly the same circumstances and produce exactly the same effect.
The second method is less simple, but it enables us to test the trueness of the balance. The body is weighed first in one scale pan and then in the other. If the two observed weights are equal, the balance is true, and each is equal to the true weight of the body. If not, the balance is false, and we have the following cases to consider.
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