Part 1

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Underscores “_” before and after a word or phrase indicate _italics_ in the original text. Small capitals have been converted to SOLID capitals. Illustrations have been moved so they do not break up paragraphs. Typographical errors have been silently corrected.

EXPERIMENTAL MECHANICS

EXPERIMENTAL MECHANICS

A COURSE OF LECTURES

_DELIVERED AT THE ROYAL COLLEGE OF SCIENCE FOR IRELAND_

BY SIR ROBERT STAWELL BALL, LL.D., F.R.S. ASTRONOMER ROYAL OF IRELAND

FORMERLY PROFESSOR OF APPLIED MATHEMATICS AND MECHANISM IN THE ROYAL COLLEGE OF SCIENCE FOR IRELAND (SCIENCE AND ART DEPARTMENT)

_WITH ILLUSTRATIONS_

SECOND EDITION

London MACMILLAN AND CO. AND NEW YORK 1888.

_The Right of Translation and Reproduction is reserved_

RICHARD CLAY AND SONS, LIMITED, LONDON AND BUNGAY.

_The First Edition was printed in 1871._

PREFACE.

I here present the revised edition of a course of lectures on Experimental Mechanics which I delivered in the Royal College of Science at Dublin eighteen years ago. The audience was a large evening class consisting chiefly of artisans.

The teacher of Elementary Mechanics, whether he be in a Board School, a Technical School, a Public School, a Science College, or a University, frequently desires to enforce his lessons by exhibiting working apparatus to his pupils, and by making careful measurements in their presence.

He wants for this purpose apparatus of substantial proportions visible from every part of his lecture room. He wants to have it of such a universal character that he can produce from it day after day combinations of an ever-varying type. He wishes it to be composed of well-designed and well-made parts that shall be strong and durable, and that will not easily get out of order. He wishes those parts to be such that even persons not specially trained in manual skill shall presently learn how to combine them with good effect. Lastly, he desires to economize his money in the matters of varnish, mahogany, and glass cases.

I found that I was able to satisfy all these requirements by a suitable adaptation of the very ingenious system of mechanical apparatus devised by the late Professor Willis of Cambridge. The elements of the system I have briefly described in an Appendix, and what adaptations I have made of it are shown in almost every page and every figure of the book.

In revising the present edition I have been aided by my friends Mr. G. L. Cathcart, the Rev. M. H. Close, and Mr. E. P. Culverwell.

ROBERT S. BALL.

OBSERVATORY, CO. DUBLIN, _3rd August, 1888_.

TABLE OF CONTENTS.

LECTURE I. _THE COMPOSITION OF FORCES._ PAGE Introduction.—The Definition of Force.—The Measurement of Force.—Equilibrium of Two Forces.—Equilibrium of Three Forces.—A Small Force can sometimes balance Two Larger Forces 1

LECTURE II. _THE RESOLUTION OF FORCES._

Introduction.—One Force resolved into Two Forces.—Experimental Illustrations.—Sailing.—One Force resolved into Three Forces not in the same Plane.—The Jib and Tie-rod 16

LECTURE III. _PARALLEL FORCES._

Introduction.—Pressure of a Loaded Beam on its Supports.—Equilibrium of a Bar supported on a Knife-edge.—The Composition of Parallel Forces.—Parallel Forces acting in opposite directions.—The Couple.—The Weighing Scales 34

LECTURE IV. _THE FORCE OF GRAVITY._

Introduction.—Specific Gravity.—The Plummet and Spirit-Level.—The Centre of Gravity.—Stable and Unstable Equilibrium.—Property of the Centre of Gravity in a Revolving Wheel 50

LECTURE V. _THE FORCE OF FRICTION._

The Nature of Friction.—The Mode of Experimenting.—Friction is proportional to the pressure.—A more accurate form of the Law.—The Coefficient varies with the weights used.—The Angle of Friction.—Another Law of Friction.—Concluding Remarks 65

LECTURE VI. _THE PULLEY._

Introduction.—Friction between a Rope and an Iron Bar.—The Use of the Pulley.—Large and Small Pulleys.—The Law of Friction in the Pulley.—Wheels.—Energy 85

LECTURE VII. _THE PULLEY-BLOCK._

Introduction.—The Single Movable Pulley.—The Three-sheave Pulley-block.—The Differential Pulley-block.—The Epicycloidal Pulley-block 99

LECTURE VIII. _THE LEVER._

The Lever of the First Order.—The Lever of the Second Order.—The Shears.—The Lever of the Third Order 119

LECTURE IX. _THE INCLINED PLANE AND THE SCREW._

The Inclined Plane without Friction.—The Inclined Plane with Friction.—The Screw.—The Screw-jack.—The Bolt and Nut 131

LECTURE X. _THE WHEEL AND AXLE._

Introduction.—Experiments upon the Wheel and Axle.—Friction upon the Axle.—The Wheel and Barrel.—The Wheel and Pinion.—The Crane.—Conclusion 149

LECTURE XI. _THE MECHANICAL PROPERTIES OF TIMBER._

Introduction.—The General Properties of Timber.—Resistance to Extension.—Resistance to Compression.—Condition of a Beam strained by a Transverse Force 169

LECTURE XII. _THE STRENGTH OF A BEAM._

A Beam free at the Ends and loaded in the Middle.—A Beam uniformly loaded.—A Beam loaded in the Middle, whose Ends are secured.—A Beam supported at one end and loaded at the other 188

LECTURE XIII. _THE PRINCIPLES OF FRAMEWORK._

Introduction.—Weight sustained by Tie and Strut.—Bridge with Two Struts.—Bridge with Four Struts.—Bridge with Two Ties.—Simple Form of Trussed Bridge 203

LECTURE XIV. _THE MECHANICS OF A BRIDGE._

Introduction.—The Girder.—The Tubular Bridge.—The Suspension Bridge 218

LECTURE XV. _THE MOTION OF A FALLING BODY._

Introduction.—The First Law of Motion.—The Experiment of Galileo from the Tower of Pisa.—The Space is proportional to the Square of the Time.—A Body falls 16' in the First Second.—The Action of Gravity is independent of the Motion of the Body.—How the Force of Gravity is defined.—The Path of a Projectile is a Parabola 230

LECTURE XVI. _INERTIA._

Inertia.—The Hammer.—The Storing of Energy.—The Fly-wheel.—The Punching Machine 250

LECTURE XVII. _CIRCULAR MOTION._

The Nature of Circular Motion.—Circular motion in Liquids.—The Applications of Circular Motion.—The Permanent Axes 267

LECTURE XVIII. _THE SIMPLE PENDULUM._

Introduction.—The Circular Pendulum.—Law connecting the Time of Vibration with the Length.—The Force of Gravity determined by the Pendulum.—The Cycloid 284

LECTURE XIX. _THE COMPOUND PENDULUM AND THE COMPOSITION OF VIBRATIONS._

The Compound Pendulum.—The Centre of Oscillation.—The Centre of Percussion.—The Conical Pendulum.—The Composition of Vibrations 299

LECTURE XX. _THE MECHANICAL PRINCIPLES OF A CLOCK._

Introduction.—The Compensating Pendulum.—The Escapement.—The Train of Wheels.—The Hands.—The Striking Parts 318

APPENDIX I.

The Method of Graphical Construction 339 The Method of Least Squares 342

APPENDIX II.

Details of the Willis Apparatus used in illustrating the foregoing lectures 345

INDEX 355

EXPERIMENTAL MECHANICS.

LECTURE I. _THE COMPOSITION OF FORCES._

Introduction.—The Definition of Force.—The Measurement of Force.—Equilibrium of Two Forces.—Equilibrium of Three Forces.—A Small Force can sometimes balance Two Larger Forces.

INTRODUCTION.

1. I shall endeavour in this course of lectures to illustrate the elementary laws of mechanics by means of experiments. In order to understand the subject treated in this manner, you need not possess any mathematical knowledge beyond an acquaintance with the rudiments of algebra and with a few geometrical terms and principles. But even to those who, having an acquaintance with mathematics, have by its means acquired a knowledge of mechanics, experimental illustrations may still be useful. By actually seeing the truth of results with which you are theoretically familiar, clearer conceptions may be produced, and perhaps new lines of thought opened up. Besides, many of the mechanical principles which lie rather beyond the scope of elementary works on the subject are very susceptible of being treated experimentally; and to the consideration of these some of the lectures of this course will be devoted.

Many of our illustrations will be designedly drawn from very commonplace sources: by this means I would try to impress upon you that mechanics is not a science that exists in books merely, but that it is a study of those principles which are constantly in action about us. Our own bodies, our houses, our vehicles, all the implements and tools which are in daily use—in fact all objects, natural and artificial, contain illustrations of mechanical principles. You should acquire the habit of carefully studying the various mechanical contrivances which may chance to come before your notice. Examine the action of a crane raising weights, of a canal boat descending through a lock. Notice the way a roof is made, or how it is that a bridge can sustain its load. Even a well-constructed farm-gate, with its posts and hinges, will give you admirable illustrations of the mechanical principles of framework. Take some opportunity of examining the parts of a clock, of a sewing-machine, and of a lock and key; visit a saw-mill, and ascertain the action of all the machines you see there; try to familiarize yourself with the principles of the tools which are to be found in any workshop. A vast deal of interesting and useful knowledge is to be acquired in this way.

THE DEFINITION OF FORCE.

2. It is necessary to know the answer to this question, What is a force? People who have not studied mechanics occasionally reply, A push is a force, a steam-engine is a force, a horse pulling a cart is a force, gravitation is a force, a movement is a force, &c., &c. The true definition of force is _that which tends to produce or to destroy motion_. You may probably not fully understand this until some further explanations and illustrations shall have been given; but, at all events, put any other notion of force out of your mind. Whenever I use the word Force, do you think of the words “something which tends to produce or to destroy motion,” and I trust before the close of the lecture you will understand how admirably the definition conveys what force really is.

3. When a string is attached to this small weight, I can, by pulling the string, move the weight along the table. In this case, there is something transmitted from my hand along the string to the weight in consequence of which the weight moves: that something is a force. I can also move the weight by pushing it with a stick, because force is transmitted along the stick, and makes itself known by producing motion. The archer who has bent his bow and holds the arrow between his finger and thumb feels the string pulling until the impatient arrow darts off. Here motion has been produced by the force of elasticity in the bent bow. Before he released the arrow there was no motion, yet still the bow was exerting force and _tending_ to produce motion. Hence in defining force we must say “that which _tends_ to produce motion,” whether motion shall actually result or not.

4. But forces may also be recognized by their capability or tendency to prevent or to destroy motion. Before I release the arrow I am conscious of exerting a force upon it in order to counteract the pull of the string. Here my force is merely manifested by _destroying the motion_ that, if it were absent, the bow would produce. So when I hold a weight in my hand, the force exerted by my hand destroys the motion that the weight would acquire were I to let it fall; and if a weight greater than I could support were placed in my hand, my efforts to sustain it would still be properly called force, because they _tended_ to destroy motion, though unsuccessfully. We see by these simple cases that a force may be recognized either by producing motion or by trying to produce it, by destroying motion or by tending to destroy it; and hence the propriety of the definition of force must be admitted.

THE MEASUREMENT OF FORCE.

5. As forces differ in magnitude, it becomes necessary to establish some convenient means of expressing their measurements. The pressure exerted by one pound weight at London is the standard with which we shall compare other forces. The piece of iron or other substance which is attracted to the earth with this force in London, is attracted to the earth with a greater force at the pole and a less force at the equator; hence, in order to define the standard force, we have to mention the locality in which the pressure of the weight is exerted.

It is easy to conceive how the magnitude of a pushing or a pulling force may be described as equivalent to so many pounds. The force which the muscles of a man’s arm can exert is measured by the weight which he can lift. If a weight be suspended from an india-rubber spring, it is evident the spring will stretch so that the weight pulls the spring and the spring pulls the weight; hence the number of pounds in the weight is the measure of the force the spring is exerting. In every case the magnitude of a force can be described by the number of pounds expressing the weight to which it is equivalent. There is another but much more difficult mode of measuring force occasionally used in the higher branches of mechanics (Art. 497), but the simpler method is preferable for our present purpose.

6. The straight line in which a force tends to move the body to which it is applied is called the direction of the force. Let us suppose, for example, that a force of 3 lbs. is applied at the point A, Fig. 1, tending to make A move in the direction AB. A standard line C of certain length is to be taken. It is supposed that a line of this length represents a force of 1 lb. The line AB is to be measured, equal to three times C in length, and an arrow-head is to be placed upon it to show the direction in which the force acts. Hence, by means of a line of certain length and direction, and having an arrow-head attached, we are able completely to represent a force.

EQUILIBRIUM OF TWO FORCES.

7. In Fig. 2 we have represented two equal weights to which strings are attached; these strings, after passing over pulleys, are fastened by a knot C. The knot is pulled by equal and opposite forces. I mark off parts CD, CE, to indicate the forces; and since there is no reason why C should move to one side more than the other, it remains at rest. Hence, we learn that two equal and directly opposed forces counteract each other, and each may be regarded as destroying the motion which the other is striving to produce. If I make the weights unequal by adding to one of them, the knot is no longer at rest; it instantly begins to move in the direction of the larger force.

8. When two equal and opposite forces act at a point, they are said to be in _equilibrium_. More generally this word is used with reference to any set of forces which counteract each other. When a force acts upon a body, at least one more force must be present in order that the body should remain at rest. If two forces acting on a point be not opposite, they will not be in equilibrium; this is easily shown by pulling the knot C in Fig. 2 downwards. When released, it flies back again. This proves that if two forces be in equilibrium their directions must be opposite, for otherwise they will produce motion. We have already seen that the two forces must be equal.

A book lying on the table is at rest. This book is acted upon by two forces which, being equal and opposite, destroy each other. One of these forces is the gravitation of the earth, which tends to draw the book downwards, and which would, in fact, make the book fall if it were not sustained by an opposite force. The pressure of the book on the table is often called the _action_, while the resistance offered by the table is the force of _reaction_. We here see an illustration of an important principle in nature, which says that _action and reaction are equal and opposite_.

EQUILIBRIUM OF THREE FORCES.

9. We now come to the important case where three forces act on a point: this is to be studied by the apparatus represented in Fig. 3. It consists essentially of two pulleys H, H, each about 2" diameter,[1] which are capable of turning very freely on their axles; the distance between these pulleys is about 5', and they are supported at a height of 6' by a frame, which will easily be understood from the figure. Over these pulleys passes a fine cord, 9' or 10' long, having a light hook at each of the ends E, F. To the centre of this cord D a short piece is attached, which at its free end G is also furnished with a hook. A number of iron weights, 0·5 lb., 1 lb., 2 lbs., &c., with rings at the top, are used; one or more of these can easily be suspended from the hooks as occasion may require.

[1] We shall often, in these lectures, represent feet or inches in the manner usual among practical men—1' is one foot, 1" is one inch. Thus, for example, 3' 4" is to be read “three feet four inches.” When it is necessary to use fractions we shall always employ decimals. For example, 0"·5 is the mode of expressing a length of half an inch; 3' 1"·9 is to be read “three feet one inch and nine-tenths of an inch.”

10. We commence by placing one pound on each of the hooks. The cords are first seen to make a few oscillations and then to settle into a definite position. If we disturb the cords and try to move them into some new position they will not remain there; when released they will return to the places they originally occupied. We now concentrate our attention on the central point D, at which the three forces act. Let this be represented by O in Fig. 4, and the lines OP, OQ, and OS will be the directions of the three cords.

On examining these positions we find that the three angles P O S, Q O S, P O Q, are all equal. This may very easily be proved by holding behind the cords a piece of cardboard on which three lines meeting at a point and making equal angles have been drawn; it will then be seen that the cords coincide with the three lines on the cardboard.

11. A little reflection would have led us to anticipate this result. For the three cords being each stretched by a tension of a pound, it is obvious that the three forces pulling at O are all equal. As O is at rest, it seems obvious that the three forces must make the angles equal, for suppose that one of the angles, P O Q for instance, was less than either of the others, experiment shows that the forces O P and OQ would be too strong to be counteracted by O S. The three angles must therefore be equal, and then the forces are arranged symmetrically.

12. The forces being each 1 lb., mark off along the three lines in Fig. 4 (which represent their directions) three equal parts O P, O Q, O S, and place the arrowheads to show the direction in which each force is acting; the forces are then completely represented both in position and in magnitude.

Since these forces make equilibrium, each of them may be considered to be counteracted by the other two. For example, O S is annulled by O Q and O P. But O S could be balanced by a force O R equal and opposite to it. Hence OR is capable of producing by itself the same effect as the forces O P and OQ taken together. Therefore O R is equivalent to O P and OQ. Here we learn the important truth that two forces not in the same direction can be replaced by a single force. The process is called the _composition of forces_, and the single force is called the _resultant_ of the two forces. O R is only one pound, yet it is equivalent to the forces O P and O Q together, each of which is also one pound. This is because the forces O P and O Q partly counteract each other.

13. Draw the lines P R and Q R; then the angles P O R and Q O R are equal, because they are the supplements of the equal angles P O S and Q O S; and since the angles P O R and Q O R together make up one-third of four right angles, it follows that each of them is two-thirds of one right angle, and therefore equal to the angle of an equilateral triangle. Also O P being equal to O Q and O R common, the triangles O P R and O Q R must be equilateral. Therefore the angle P R O is equal to the angle R O Q; thus P R is parallel to O Q; similarly Q R is parallel to O P; that is, O P R Q is a parallelogram. Here we first perceive the great law that the resultant of two forces acting at a point is the diagonal of a parallelogram, of which they are the two sides.

14. This remarkable geometrical figure is called the _parallelogram of forces_. Stated in its general form, the property we have discovered asserts that two forces acting at a point have a resultant, and that this resultant is represented both in magnitude and in direction by the diagonal of the parallelogram, of which two adjacent sides are the lines which represent the forces.

15. The parallelogram of forces may be illustrated in various ways by means of the apparatus of Fig. 3. Attach, for example, to the middle hook G 1·5 lb., and place 1 lb. on each of the remaining hooks E, F. Here the three weights are not equal, and symmetry will not enable us, as it did in the previous case, to foresee the condition which the cords will assume; but they will be observed to settle in a definite position, to which they will invariably return if withdrawn from it.