EXPERIMENTAL MECHANICS IE..TILE IS A PARABOLA. EXPERIMENTAL MECHANICS A COURSE OF LECTURES DELIVERED AT THE ROYAL COLLEGE OF SCIENCE FOR IRELAND SIR ROBERT STAWELL BALL, LLD., ( F.R.S. ASTRONOMER ROYAL OF IRELAND FORMERLY PROFESSOR OF APPLIED MATHEMATICS AND MECHANISM IN TH! ROYAL COLLEGE OF SCIENCE FOR IRELAND (SCIENCE AND ART DEPARTMENT) WITH ILLUSTRATIONS SECOND EDITION Xondon 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 2066709 viii PREFACE. 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 require- ments 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. Culvenvell. ROBERT S. BALL. OBSERVATORY, Co. DUBLIN, yd 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 i LECTURE II. THE RESOLUTION OF FORCES. Introduction. One Force resolved into Two Forces. Experi- mental 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 Com- position of Parallel Forces. Parallel Forces acting in opposite directions. The Couple. The Weighing Scales .... 34 LECTURE IV. THE FORCE OF GRA VITY. Introduction. Specific Gravity. The Plummet and Spirit Level. The Centre of Gravity. Stable and Unstable Equili- brium. Property of the Centre of Gravity in a Revolving Wheel 50 x TABLE OF CONTENTS. LECTURE V. THE FORCE OF FRICTION. PAGE The Nature of Friction. The Mode of Experimenting. Fric- tion 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. Con- cluding Remarks 65 LECTURE VI. THE PU LLE Y. 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-BLOCA: Introduction. The Single Movable Pulley. The Three-sheave Pulley-block. The Differential Pulley-block. The Epicy- cloidal 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 TABLE OF CONTENTS. si LECTURE X. THE WHEEL AND AXLE. PAGE 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 Sus- pension Bridge . , 218 xii TABLE OF CONTENTS. LECTURE XV. THE MOTION OF A FALLING BODY. PAGE 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 deter- mined 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 Com- position of Vibrations 299 TABLE OF CONTENTS. xiii LECTURE XX. THE MECHANICAL PRINCIPLES OF A CLOCK. PAGE Introduction. The Compensating Pendulum. The Escapement. The Train of Wheels. The Hands. The Striking Tarts .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 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. i. I SHALL endeavour in this course of lectures to illus- trate the elementary laws of mechanics by means of experi- ments. 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 2 EXPERIMENTAL MECHANICS. [LECT. 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 frame-work. 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 work- shop. A vast deal ef 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 I.] THE DEFINITION OF FORCE. 3 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 under- stand 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 pro- duce 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 re- lease 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 B 2 4 EXPERIMENTAL MECHANICS. [LECT. 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 measure- ments. 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. I.] EQUILIBRIUM OF TWO FORCES. 5 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 Ibs. is applied at the point A, Fig. i, tending to make A move in the direction AB. A standard line c of certain A c length is to be taken. It is supposed that a line of this length represents a force of i Ib. 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 hav- ing 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 pass- ing 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 6 EXPERIMENTAL MECHANICS. [LECT. 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 i.] EQUILIBRIUM OF THREE FORCES. 7 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' FIG. 3. 1 We shall often, in these lectures, represent feet or inches in the manner usual among practical men i' is one foot, i" 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, o"'5 is the mode of expressing a length of half an inch; 3' i"'9 is to be read "three feet one inch and nine-tenths of an inch." EXPERIMENTAL MECHANICS. [LECT. 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. } i lb., 2 Ibs., &c., with rings at the top, are used ; one or more of these can easily be suspended from the hooks as occasion may require. 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 postions, we find R 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 I.] EQUILIBRIUM OF THREE FORCES. 9 the others, experiment shows that the forces o P and o Q 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 i 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 o R is capable of producing by itself the same effect as the forces o P and OQ taken together. Therefore o R is equiva- lent to o P and o Q. 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 ? 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 parallelo- EXPERIMENTAL MECHANICS. [LECT. gram. Here we first perceive the great law that the resultant of two forces acting at a point is the diagonal of a parallelo- gram, of which they are the two sides. 14. This remarkable geometrical figure is called the parallelogram of forces. Stated in its general form, the pro- perty 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 01-5 lb., and place i Ib. 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. Let o P, o Q (Fig. 5) be the directions of the cords ; o P and o Q being each of the length which corresponds to i lb., while o s corresponds to i'5lb. Here, as before, o P and o Q together may be considered to counteract o s. But o s could have been counteracted by an equal and opposite force o R. Hence OR may be regarded as the single force equivalent to o P and o Q, that is, as their resultant ; and thus it is proved experimentally that these forces have a resultant. We can further verify that the resultant is the diagonal of the FIG. 5. I.] EQUILIBRIUM OF THREE FORCES. n parallelogram of which the equal forces are the sides. Construct a parallelogram on a piece of cardboard having its four sides equal, and one of the diagonals half as long again as one of the sides. This may be done very easily by first drawing one of the two triangles into which the diagonal divides the parallelogram. The diagonal is to be produced beyond the parallelogram in the direction o s. When the cardboard is placed close against the cords, the two cords will lie in the directions o P, o Q, while the produced diagonal will be in the vertical o s. Thus the application of the parallelogram of force is verified. 1 6. The same experiment shows that two unequal forces may be compounded into one resultant. For in Fig. 5 the two forces o P and o s may be considered to be counter- balanced by the force o Q ; in other words, o Q must be equal and opposite to a force which is the resultant of o P and o S. ^ 17. Let us place on the central hook G a weight of 5 Ibs., and weights of 3 Ibs. on the hook E and 4 Ibs. on F. This is actually the case shown in Fig. 3. The weights being unequal, we cannot immediately infer anything with reference to the position of the cords, but still we find, as be- fore, that the cords assume a definite position, to which they return when temporarily displaced. Let Fig. 6 represent the positions of the cords. No two of the angles are in this case equal. Still each of the forces is counterbalanced by the other two. Each is therefore equal and opposite to the resultant of the other two. Construct s FIG. 6. 12 EXPERIMENTAL MECHANICS. [LECT. the parallelogram on cardboard, as can be easily done by form- ing the triangle o P R, whose sides are 3, 4, and 5, and then drawing o Q and R Q parallel to R P and o p. Produce the diagonal o R to s. This parallelogram being placed behind the cords, you see that the directions of the cords coincide with its sides and diagonal, thus verifying the parallelogram of forces in a case where all the forces are of different magnitudes. 1 8. It is easy, by the application of a set square, to prove that in this case the cords attached to the 3 Ib. and 4lb. weights are at right angles to each other. We could have inferred, from the parallelogram of force, that this must be the case, for the sides of the triangle o P R are 3, 4, and 5 respectively, and since the square of 5 is 25, and the squares of 3 and of 4 are 9 and 16 respectively, it follows that the square of one side of this triangle is equal to the sum of the squares of the two opposite sides, and therefore this is a right-angled triangle (Euclid, i. 48). Hence, since P R is parallel to o Q, the angle P o Q must also be a right angle. A SMALL FORCE SOMETIMES BALANCES TWO LARGER FORCES. 19. Cases might be multiplied indefinitely by placing various amounts of weight on the hooks, constructing the parallelogram on cardboard, and comparing it with the cords as before. We shall, however, confine ourselves to one more illustration, which is capable of very remarkable appli- cations. Attach i Ib. to each of the hooks E and F ; the cord joining them remains straight until drawn down by placing a weight on the centre hook. A very small weight will suffice to do this. Let us put on half-a-pound ; the position the cords i.] A SMALL FORCE BALANCING TWO LARGER. 13 then assume is indicated in Fig. 7. As before, each force is equal and opposite to the resultant of the other two. Hence a force of half-a-pound is the K resultant of two forces each of i Ib. The apparent paradox is explained by noticing that the forces of i Ib. are very nearly opposite, and therefore to a large extent counteract each other. Constructing the cardboard parallelogram we may easily verify that the principle of the parallelogram of forces holds in this case also. 20. No matter how small be the weight we suspend from the middle of a horizontal cord, you see that the cord is deflected : and no matter how great a tension were applied, it would be impossible to straighten the cord. The cord could break, but it could not again become horizontal. Look at a telegraph wire ; it is never in a straight line between two consecutive poles, and its curved form is more evident the greater be the distance between the poles. But in putting up a telegraph wire great straining force is used, by means of special machines for the purpose ; yet the wires cannot be straightened: because the weight of the heavy wire itself acts as a force pulling it downwards. Just as the cord in our experiments cannot be straight when any force,' however small, is pulling it downwards at the centre, so it is impos- sible by any exertion of force to straighten the long wire. Some further illustrations of this principle will be given in our next lecture, and with one application of it the present will be concluded. 21. One of the most important practical problems in mechanics is to make a small force overcome a greater. There are a number of ways in which this may be 14 EXPERIMENTAL MECHANICS. [LECT. accomplished for different purposes, and to the consideration of them several lectures of this course will be devoted. Perhaps, however, there is no arrangement more simple than that which is furnished by the principles we have been considering. We shall employ it to raise a 28 Ib. weight by means of a 2 Ib. weight. I do not say that this particular application is of much practical use. I show it to you rather as a remarkable deduction from the parallelogram of forces than as a useful machine. A rope is attached at one end of an upright, A (Fig. 8), and passes over a pulley B at the same vertical height about 1 6' distant. A weight of 28 Ibs. is fastened to the free end of the rope, and the supports must be heavily weighted or otherwise secured from moving. The rope AB is apparently straight and horizontal, in consequence of its weight being inappreciable in comparison with the strain (28 Ibs.) to which it is subjected; this position is indicated in the figure by the dotted line AB. We now suspend from c at the middle of the rope a weight of 2 Ibs. Instantly the rope moves to the position represented in the figure. But this it cannot do without at the same moment raising slightly the 28 Ibs., for, since two sides of a triangle, CB, I.] A SMALL FORCE BALANCING TWO LARGER. 15 CA, are greater than the third side, AB, more of the rope must lie between the supports when it is bent down by the 2 Ib. weight than when it was straight. But this can only have taken place by shortening the rope between the pulley B and the 28 Ib. weight, for the rope is firmly secured at the other end. The effect on the heavy weight is so small that it is hardly visible to you from a distance. We can, however, easily show by an electrical arrangement that the big weight has been raised by the little one. 22. When an electric current passes through this alarum you hear the bell ring, and the moment I stop the current the bell stops. I have fastened one piece of brass to the 28 Ib. weight, and another to the support close above it, but unless the weight be raised a little the two will not be in contact ; the electricity is intended to pass from one of these pieces of brass to the other, but it cannot pass unless they are touching. When the rope is straight the two pieces of brass are separated, the current does not pass, and our alarum is dumb ; but the moment I hang on the 2 Ib. weight to the middle of the rope it raises the weight a little, brings the pieces of brass in contact, and now you all hear the alarum. On removing the 2 Ibs. the current is interrupted and the noise ceases. 23. I am sure you must all have noticed that the 2 Ib. weight descended through a distance of many inches, easily visible to all the room ; that is to say, the small weight moved through a very considerable distance, while in so doing it only raised the larger one a very small distance. This is a point of the very greatest importance ; I there- fore take the first opportunity of calling your attention to it. 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. INTRODUCTION. 24. As the last lecture was princi- pally concerned with discussing how one force could replace two forces, so in the present we shall examine the converse question, How may two forces replace one force ? Since the diagonal of a parallelogram represents a single force equivalent to those represented by the sides, it is obvious that one force may be resolved into two others, pro- vided it be the diagonal of the paral- lelogram formed by them. 25. We shall frequently employ in the present lecture, and in some of those that follow, the spring balance, which is represented in Fig. 9 : the weight is attached to the hook, and when the balance is suspended by the L. li.] ONE FORCE RESOLVED INTO TWO FORCES. 17 ring, a pointer indicates the number of pounds on a scale. This balance is very convenient for showing the strain along a cord ; for this purpose the balance is held by the ring while the cord is attached to the hook. It will be noticed that the balance has two rings and two corresponding hooks. The hook and ring at the top and bottom will weigh up to 300 Ibs., corresponding to the scale which is seen. TJie hook and ring at the side correspond to another scale on the other face of the plate : this second scale weighs up to about 50 Ibs., consequently for a weight under 50 Ibs. the side hook and ring are employed, as they give a more accurate result than would be obtained by the top and bot- tom hook and ring, which are intended for larger weights. These ingenious and useful balances are sufficiently accurate, and can easily be tested by raising known weights. Besides the instrument thus described, we shall sometimes use one of a smaller size, and we shall be able with this aid to trace the existence and magnitude of forces in a most convenient manner. ONE FORCE RESOLVED INTO TWO FORCES. 26. We shall first illustrate how a single force may be resolved into a pair of forces ; for this purpose we shall use the arrangement shown in Fig. 10 (see next page). The ends of a cord are fastened to two small spring balances ; to the centre E of this cord a weight of 4 Ibs. is attached. At A and B are pegs from which the balances can be suspended. Let the distances AE, BE be each 12", and the distance AB 16". When the cord is thus placed, and the weight allowed to hang freely, each of the cords EA, EB is strained by an amount of force that is shown to be very nearly 3 Ibs. by the balances. But the weight of 4 Ibs. is the i a EXPERIMENTAL MECHANICS. [LECT. only weight acting ; hence it must be equivalent to two forces of very nearly 3 Ibs. each along the directions AE and BE. Here the two forces to which 4 Ibs. is equivalent are each of them less than 4 Ibs., though taken together they exceed it. 27. But remove the cords from AB and hang them on CD, the length CD being i' 10", then the forces shown along FC and FD are each 5 Ibs. ; here, therefore, one force of 4 Ibs. is equivalent to two forces each of 5 Ibs. In the last lecture (Art. 19) we saw that one force could balance two greater forces ; here we see the analogous case of one force being changed into two greater forces. Further, we learn that the number of pairs of forces into which one force may be decomposed is unlimited, for with every different distance between the pegs different forces will be indicated by the balances. Whenever the weight is suspended from a point half- way between the balances, the forces along the cords are I!.] EXPERIMENTAL ILLUSTRATIONS. 19 equal ; but by placing the weight nearer one balance than the other, a greater force will be indicated on that balance to which the weight is nearest. EXPERIMENTAL ILLUSTRATIONS. 28. The resolution or decomposition of one force into two forces each greater than itself is capable of being illustrated in a variety of ways, two of which will be here explained. In Fig. n an arrangement for this purpose is shown. A piece of stout twine AB, able to support from 20 Ibs. to 30 Ibs., is fastened at one end A to a fixed support, and at the other end B to the eye of a wire-strainer. A 2 EXPERIMENTAL MECHANICS. [LECT. wire-strainer consists of an iron rod, with an eye at one end and a screw and a nut at the other ; it is used for tightening wires in wire fencing, and is employed in this case for the purpose of stretching the cord. This being done, I take a piece of ordinary sewing-thread, which is of course weaker than the stout twine. I tie the thread to the middle of the cord at c, catch the other end in my fingers, and pull ; something must break something has broken : but what has broken ? Not the slight thread, it is still whole ; it is the cord which has snapped. Now this illustrates the point on which we have been dwelling. The force which I transmitted along the thread was insufficient to break it ; ii.] SAILING. 21 the thread transferred the force to the cord, but under such circumstances that the force was greatly magnified, and the consequence was that this magnified force was able to break the cord before the original force could break the thread. We can also see why it was necessary to stretch the cord. In Fig. 10 the strains along the cords are greater when the cords are attached at c and D than when they are attached at A and B ; that is to say, the more the cord is stretched towards a straight line, the greater are the forces into which the applied force is resolved. 29. We give a second example, in illustration of the same principle. In Fig. 12 is shown a chain 8' long, one end of which B is attached to a wire-strainer, while the other end is fastened to a small piece of pine A, which is o"'5 square in section, and 5" long between the two upright irons by which it is supported. By means of the nut of the wire-strainer I straighten the chain as I did the string of Fig. n, and for the same reason. I then put a piece of twine round the chain and pull it gently. The strain brought to bear on the wood is so great that it breaks across. Here, the small force of a few pounds, transmitted to the chain by pulling the string, is magnified to upwards of a hundred- weight, for less than this would not break the wood. The explanation is precisely the same as when the string was broken by the thread. SAILING. 30. The action of the wind upon the sails of a vessel affords a very instructive and useful example of the de- composition of forces. By the parallelogram of forces we are able to explain how it is that a vessel is able even to sail EXPERIMENTAL MECHANICS. [LECT. against the wind. A force is that which tends to produce motion, and motion generally takes place in the line of the force. In the case of the action of wind on a vessel through the medium of the sails, we have motion produced which is not necessarily in the direction of the wind, and which may be to a certain extent opposed to it. This apparent paradox requires some elucidation. FIG. 13. 31. Let us first suppose the wind to be blowing in a direction shown by the arrows of Fig. 13, perpendicular to the line AB in which the ship's course lies. In what direction must the sail be set ? It is clear that the sail must not be placed along the line AB, for then the only effect of the wind would be to blow the vessel sideways ; nor could the sail be placed with its edge to the wind, that II.] SAILING. 23 is, along the line o w, for then the wind would merely glide along the sail without producing a propelling force. Let, then, the sail be placed between the two positions, as in the direction p Q. The line o w represents the magnitude of the force of the wind pressing on the saiL We shall suppose for simplicity that the sail extends on both sides of o. Through o draw o R perpendicular to p Q, and from w let fall the perpendicular w x on p Q, and w R on o R. By the principle of the parallelogram of forces, the force o w may be decomposed into the two forces o x and o R, since these are the sides of the parallelogram of which o w, the force of the wind, is the diagonal. We may then leave o w out of consideration, and imagine the force of the wind to be replaced by the pair of forces o x and o R ; but the force ox cannot produce an effect, it merely represents a" force which glides along the surface of the sail, not one which pushes against it ; so far as this component goes, the sail has its edge towards it, and therefore the force produces no effect. On the other hand, the sail is perpendicular to the force o R, and this is therefore the efficient component. The force of the wind is thus measured by o R, both in magnitude and direction : this force represents the actual pressure on the mast produced by the sail, and from the mast communicated to the ship. Still o R is not in the direction in which the ship is sailing : we must again de- compose the force in order to find its useful effect. This is done by drawing through R the lines R L and R M parallel to OA and ow, thus forming the parallelogram OMRL. Hence, by the parallelogram of forces, the force. OR is equi- valent to the two forces o L and o M. The effect of o L upon the vessel is to propel it in a direction perpendicular to that in which it is sailing. We must, therefore, endeavour to counteract this force as far as 24 EXPERIMENTAL MECHANICS. [LECT. possible. This is accomplished by the keel, and the form of the ship is so designed as to present the greatest possible resistance to being pushed sideways through the water : the deeper the keel the more completely is the effect of o L annulled. Still o L would in all cases produce some leeway were it not for the rudder, which, by turning the head of the vessel a little towards the wind, makes her sail in a direction sufficiently to windward to counteract the small effect of o L in driving her to leeward. Thus o L is disposed of, and the only force remaining is o M, which acts directly to push the vessel in the required direction. Here, then, we see how the wind, aided by the resistance of the water, is able to make the vessel move in a direction perpendicular to that in which the wind blows. We have seen that the sail must be set somewhere between the direction of the wind and that of the ship's motion. It can be proved that when the direction of the sail supposed to be flat and vertical, is such as to bisect the angle w o B, the magnitude of the force o M is greater than when the sail has any other position. 32. The same principles show how a vessel is able to sail against the wind : she cannot, of course, sail straight against it, but she can sail within half a right angle of it, or perhaps even less. This can be seen from Fig. 14. The small arrows represent the wind, as before. Let o w be the line parallel to them, which measures the force of the wind, and let the sail be placed along the line P Q ; o w is decomposed into o x and o Y, o x merely glides along the sail, and o Y is the effective force. This is decomposed into o L and o M ; o L is counteracted, as already explained, and o M is the farce that propels the vessel onwards. Hence we see that there is a force acting to push the vessel onwards, even though the movement be partly against the wind. SAILING. -5 It will be noticed in this case that the force o L acting to leewards exceeds OM pushing onwards. Hence it is that vessels with a very deep keel, and therefore opposing very great resistance to moving leewards, can sail more closely to the wind than others not so constructed ; a vessel should be formed so that she shall move as freely as possible in the direction of her length, for which reason she is sharpened at the bow, and otherwise shaped for gliding through the water easily ; this is in order that o M may have to overcome as little resistance as possible. If the sail were flat and vertical it should bisect the angle A o w for the wind to act in the most efficient manner. Since, then, a vessel can sail to- wards the wind, it follows that, by taking a zigzag course, she can proceed from one port to another, even though the wind be blowing from the place to which she would go towards the place from which she comes. This well- known manoeuvre is called "tacking." You will understand that in a sailing-vessel the rudder has a more important part EXPERIMENTAL MECHANICS. [LECT. to play than in a steamer : in the latter it is only useful for changing the direction of the vessel's motion, while in the former it is not only necessary for changing the direction, but must also be used to keep the vessel to her course by counteracting the effect of leeway. ONE FORCE RESOLVED INTO THREE FORCES NOT IN THE SAME PLANE. 33. Up to the present we have only been considering forces which lie in the same plane, but in nature we meet Fia. 15. with forces acting in all directions, and therefore we must not be satisfied with confining our inquiries to the simpler case. We proceed to show, in two different ways, how a force can be decomposed into three forces not in the same plane, though passing through the same point. The first mode of doing so is as follows. To three points A, B, c ONE FORCE RESOLVED INTO THREE. (Fig. 15) three spring balances are attached; A, B, c are not in the same straight line, though they are at the same ver- tical height : to the spring balances cords are attached, which unite in a point o, from which a weight w is suspended. This weight is supported by the three cords, and the strains along these cords are indicated by the spring balances. The greatest strain is on the shortest cord and the least strain on the longest Here the force w Ibs. produces three forces which, taken together, exceed its own amount. If I add an equal weight w, I find, as we might have anticipated, that the strains indicated by the scales are precisely double what they were before. Thus we see that the proportion of the force to each of the components into which it is decomposed does not depend on the actual magnitude of the force, but on the relative direction of the force and its components. 34. Another mode of show- ing the decomposition of one force into three forces not in the same plane is represented in Fig. 1 6. The tripod is formed of three strips of pine, 4' x o" - 5 x o //- 5, secured by a piece of wire running through each at the top ; one end of this wire hangs down, and carries a hook to which is attached a weight of 28 Ibs. This weight is supported by the wire, but the strain on the wire must be borne by the three wooden rods : hence there is a force acting downwards through the wooden rods. We cannot render this manifest by a contrivance like 28 EXPERIMENTAL MECHANICS. [LECT. the spring scales > because it is a push instead of a pull. However, by raising one of the legs I at once become aware that there is a force acting downwards through it. The weight is, then, decomposed into three forces, which act downwards through the legs ; these three forces are riot in a plane, and the three forces taken together are larger than the weight. 35. The tripod is often used for supporting weights ; it is convenient on account of its portability, and it is very steady. You may judge of its strength by the model represented in the figure, for though the legs are very slight, yet they sup- port very securely a considerable weight. The pulleys by means of which gigantic weights are raised are often sup- ported by colossal tripods. They possess stability and steadiness in addition to great strength. 36. An important point may be brought out by contrast- ing the arrangements of Figs. 15 and 16. In the one case three cords are used, and in the other three rods. Three rods would have answered for both, but three cords would not have done for the tripod. In one the cords are strained, and the tendency of the strain is to break the cords, but in the other the nature of the force down the rods is entirely different ; it does not tend to pull the rod asunder, it is trying to crush the rod, and had the weight been large enough the rods would bend and break. I hold one end of a pencil in each hand and then try to pull the pencil asunder ; the pencil is in the condition of the cords of Fig. 15; but if instead of pulling I push my hands together, the pencil is like the rods in Fig. 16. 37. This distinction is of great importance in mechanics. A rod or cord is in a state of tension is called a " tie " ; while a rod in a state of compression is called a " strut." Since a rod can resist both tension and compression it can serve ii.] THE JIB AND TIE ROD. 29 either as a tie or as a strut, but a cord or chain can only act as a tie. A piilar is always a strut, as the superincumbent load makes it to be in a state of compression. These distinctions will be very frequently used during this course of lectures, and it is necessary that they be thoroughly understood. THE JIB AND TIE ROD. 38. As an illustration of the nature of the "tie" and " strut," and also for the purpose of giving a useful example of the decomposition of forces, I use the apparatus of Fig. 17 (see next page). It represents the principle of the framework in the common lifting crane, and has numerous applications in practical mechanics. A rod cf wood BC 3' 6" long and i" X i' section is capable of turning round its support at the bottom B by means of a joint or hinge : this rod is called the "jib " ; it is held at its upper end by a tie AC 3' long, which is attached to the support above the joint. A B is one foot long. From the point c a wire descends, having a hook at the end on which a weight can .be hung. The tie is attached to the spring balance, the index of which shows the strain. The Spring balance is secured by a wire-strainer, by turn- ing the nut of which the length of the wire can be shortened or lengthened as occasion requires. This is necessary, because when different weights are suspended from the hook the spring is stretched more or less, and the screw is then employed to keep the entire length of the tie at 3'. The remainder of the tie consists of copper wire. 39. Suppose a weight of 20 Ibs. be suspended from the hook w, it endeavours to pull the top of the jib downwards : but the tie holds it back, consequently the tie is put into a EXPERIMENTAL MECHANICS. [LECT. state of tension, as indeed its name signifies, and the magni- tude of that tension is shown to be 60 Ibs. by the spring- balance. Here we find again what we have already so often referred to , namely, one force developing another force that is greater than itself, for the strain along the tie is three times Flu. 17. as great as the strain in the vertical wire by which it was produced. 40. What is the condition of the jib ? It is evidently being pushed downwards on its joint at B ; it is therefore in a state of compression ; it is a strut. This will be evident if we think for a moment how absurd it would be to 11.] THE JIB AND TIE ROD. 31 deavour to replace the jib by a string or chain : the whole arrangement would collapse. The weight of 20 Ibs. is there- fore decomposed by this contrivance into two other forces, one of which is resisted by a tie and the other by a strut. 41. We have no means of showing the magnitude of the strain along the strut, but we shall prove that it can be computed by means of the parallelogram of force ; this will also explain how it is that the tie is strained by a force three times that of the weight which is used. Through c (Fig. 18) draw c P parallel to the tie A B, and P Q parallel to the strut C B then B p is the diagonal of the parallelogram whose sides are each equal to B c and B Q. If therefore we consider the force of 20 Ibs. to be represented by B p, the two forces into which it is decomposed will be shown by B Q and B c ; but A B is equal to B Q, since each of them is equal to c P ; also E F is equal to A c. Hence the weight of 20 Ibs. being re- presented by A c, the strain along the tie will be represented by the length A B, and that along the strut by the length B c. Remembering that AB is 3' long, c B 3' 6", and AC i', it follows that the strain along the tie is 60 Ibs., and along the 32 EXPERIMENTAL MECHANICS. [LECT. strut 70 Ibs., when the weight of 20 Ibs. is suspended from the hook. 42. In every other case the strains along the tie and strut can be determined, when the suspended weight is known, by their proportionality to the sides of the triangle formed by the tie, the jib, and the upright post, respectively. 43. In this contrivance you will recognize, no doubt, the framework of the common lifting crane, but that very essential portion of the crane which provides for the raising and lowering is not shown here. To this we shall return again in a subsequent lecture (Art. 332). You will of course understand that the tie rod we have been considering is entirely different from the chain for raising the load. 44. It is easy to see of what importance to the engineer the information acquired by means of the decomposition of forces may become. Thus in the simple case with which we are at present engaged, suppose an engineer were required to erect a frame which was to sustain a weight of 10 tons, let us see how he would be enabled to determine the strength of the tie and jib. It is of importance in designing any structure not to make any part unnecessarily strong, as doing so involves a waste of valuable material, but it is of still more vital importnnce to make every part strong enough to avoid the risk of accident, not only under ordinary circum- stances, but also under the exceptionally great shocks and strains to which every machine is liable. 45. According to the numerical proportions we have employed for illustration, the strain along the tie rod would be 30 tons when the load was 10 tons, and therefore the tie must at least be strong enough to bear a pull of 30 tons ; but it is customary, in good engineering practice, to make the machine of about ten times the strength that would just be sufficient to sustain the ordinary load. Hence the crank II.] THE JIB AND TIE ROD. 33 must be so strong that the tie would not break with a tension less than 300 tons, which would be produced when the crane was lifting 100 tons. So great a margin of safety is necessary on account of the jerks and other occasional great strains that arise in the raising and the lowering of heavy weights. For a crane intended to raise 10 tons, the engineer must therefore design a tie rod which not less than 300 tons would tear asunder. It has been proved by actual trial that a rod of wrought iron of average quality, one square inch in section, can just withstand a pull of twenty tons. Hence fifteen such rods, or one rod the section of which was equal to fifteen square inches, would be just able to resist 300 tons ; and this is therefore the proper area of section for the tie rod of the crane we have been considering. 46. In the same way we ascertain the actual thrust down the jib; it amounts to 35 tons, and the jib should be ten times as strong 'as a strut which would collapse under a strain of 35 tons. 47. It is easy to see from the figure that the tie rod is pulling the upright, and tending, in fact, to make it snap off near B. It is therefore necessary that the upright support A B (Fig. 17) be secured very firmly. LECTURE III. PARALLEL FORCES. Introduction. Pressure of a Loaded Beam on its Supports. Equi- librium of a Bar supported on a Knife-edge. The Composition or Parallel Forces. Parallel Forces acting in opposite directions. The Couple. The Weighing Scales. INTRODUCTION. 48. THE parallelogram of forces enables us to find the resultant of two forces which intersect: but since parallel forces do not intersect, the construction does not avail to determine the resultant of two parallel forces. We can, however, find this resultant very simply by other means. 49. Fig. 1 9 represents a wooden rod 4' long, sustained by resting on two supports A and B, and having the length A B divided into 14 equal parts. Let a weight of 14 Ibs. be hung on the rod at its middle point c ; this weight must be borne by the supports, and it is evident that they will bear LECT. HI.] PRESSURE OF A LOADED BEAM. 35 the load in equal shares, for since the weight is at the middle of the rod there is no reason why one end should be differently circumstanced from the other. Hence the total pressure on each of the supports will be 7 Ibs., together with half the weight of the wooden bar. 50. If the weight of 14 Ibs. be placed, not at the centre of the bar, but at some other point such as D, it is not then so easy to see in what proportion the weight is distributed between the supports. We can easily understand that the support near the weight must bear more than the remote one, but how much more ? When we are able to answer this question, we shall see that it will lead us to a knowledge of the composition of parallel forces. PRESSURE OF A LOADED BEAM ON ITS SUPPORTS. 51. To study this question we shall employ the apparatus shown in Fig. 20. An iron bar 5' 6" long, weighing 10 Ibs., rests in the hooks of the spring balances A,C, in the manner shown in the figure. These hooks are exactly five feet apart, so that the bar projects 3" beyond each end. The space between the hooks is divided into twenty equal portions, each of course 3" long. The bar is sufficiently strong"to bear the weight B of 20 Ibs. suspended from it by an S hook, without appreciable deflection. Before the weight of 20 Ibs. is suspended, the spring balances each show a strain of 5 Ibs. We would expect this, for it is evident that the whole weight of the bar amounting to i o Ibs. should be borne equally by the two supports. 52. When I place the weight in the middle, 10 divisions from each end, I find the balances each indicate 15 Ibs. But 5 Ibs. is due to the weight of the bar. Hence the 20 Ibs. is divided equally, as we have already stated that it T) 2 EXPERIMENTAL MECHANICS. [LECT. should be. But let the 20 Ibs. be moved to any other position, suppose 4 divisions from the right, and 16 from the left; then the right-hand scale reads 21 Ibs., and the left-hand reads 9 Ibs. To get rid of the weight of the bar itself, we must subtract 5 Ibs. from each. We learn there- fore that the 20 Ib. weight pulls the right-hand spring balance with a strain of 16 Ibs., and the left with a strain of 4 Ibs. Observe this closely; you see I have made the number of divisions in the bar equal to the number of pounds weight suspended from it, and here we find that when the weight is 16 divisions from the left, the strain of 1 6 Ibs. is shown on the right. At the same time the weight is 4 divisions from the right, and 4 Ibs. is the strain shown to the left. in.] PRESSURE OF A LOADED BEAM. 37 53. I will state the law of the distribution of the load a little more generally, and we shall find that the bar will prove the law to be true in all cases. Divide the bar into as many equal parts as there are pounds in the load, then the pressure in pounds on one end is the number of divisions that the load is distant from the other. 54. For example, suppose I place the load 2 divisions from one end : I read by the scale at that end 23 Ibs. ; subtracting 5 Ibs. for the weight of the bar, the pressure due to the load is shown to be 18 Ibs., but the weight is then exactly 18 divisions distant from the other end. We can easily verify this rule whatever be the position which the load occupies. 55. If the load be placed between two marks, instead of being, as we have hitherto supposed, exactly at one, the partition of the load is also determined by the law. Were it, for example, 3^5 divisions from one end, the strain on the other would be 3*5 Ibs. ; and in like manner for other cases. 56. We have thus proved by actual experiment this useful and instructive law of nature ; the same result could have been inferred by reasoning from the parallelogram of force, but the purely experimental proof is more in accordance with our scheme. The doctrine of the composition of parallel forces is one of the most fundamental parts of mechanics, and we shall have many occasions to employ it in this as well as in subsequent lectures. 57. Returning now to Fig. 19, with which we com- menced, the law we have discovered will enable us to find how the weight is distributed. We divide the length of the bar between the supports into 14 equal parts because the weight is 14 Ibs. ; if, then, the weight be at D, 10 divisions from one end A, and 4 from the other B, the 38 EXPERIMENTAL MECHANICS. [LECT. pressure at the corresponding ends will be 4 and 10. If the weight were 2-5 divisions from one end, and therefore 1 1 -5 from the other, the shares in which this load would be supported at the ends are 11*5 Ibs. and 2-5 Ibs. The actual pressure sustained by each end is, however, about 6 ounces greater if the weight of the wooden bar itself be taken into account. 58. Let us suspend a second weight from another point of the bar. We must then calculate the pressures at the ends which each weight separately would produce, and those at the same end are to be added together, and to half the weight of the bar, to find the total pressure. Thus, if one weight of 20 Ibs. were in the middle, and another of 14 Ibs. at a distance of n divisions from one end, the middle weight would produce 10 Ibs. at each end and the 14 Ibs. would pro- duce 3 Ibs. and 1 1 Ibs., and remembering the weight of the bar, the total pressures produced would be 13 Ibs. 6 oz. and 2 1 Ibs. 6 oz. The same principles will evidently apply to the case of several weights : and the application of the rule becomes especially easy when all the weights are equal, for then the same divisions will serve for calculating the effect of each weight. 59. The principles involved in these calculations are of so much importance that we shall further examine them by a different method, which has many useful applications. EQUILIBRIUM OF A BAR SUPPORTED ON A XNIFE-EDGE. 60. The weight of the bar has hitherto somewhat com- plicated our calculations ; the results would appear more simply if we could avoid this weight ; but since we want a strong bar, its weight is not so small that we could afford to in.] EQUILIBRIUM OF A BAR. 39 overlook it altogether. By means of the arrangement of M Fig. 21, we can counterpoise the weight of the bar. To 40 EXPERIMENTAL MECHANICS. [LECT. the centre of A B a cord is attached, which, passing over a fixed pulley D, carries a hook at the other end. The bar, being a pine rod, 4 feet long and i inch square, weighs about 12 ounces ; consequently, if a weight of twelve ounces be suspended from the hook, the bar will be counterpoised, and will remain at whatever height it is placed. 61. A B is divided by lines drawn along it at distances of i* apart ; there are thus 48 of these divisions. The weights employed are furnished with rings large enough to enable them to be slipped on the bar and thus placed in any desired position. 62. Underneath the bar lies an important portion of the arrangement ; namely, the knife-edge c. This is a blunt edge of steel firmly fastened to the support which carries it. This support can be moved along underneath the bar so that the knife-edge can be placed under any of the divisions required. The bar being counterpoised, though still un- loaded with weights, may be brought down till it just touches the knife-edge ; it will then remain horizontal, and will retain this position whether the knife-edge be at either end of the bar or in any intermediate position. I shall hang weights at the extremities of the rod, and we shall find that there is for each pair of weights just one position at which, if the knife-edge be placed, it will sustain the rod horizon- tally. We shall then examine the relations between these distances and the weights that have been attached, and we shall trace the connection between the results of this method and those of the arrangement that we last used. 63. Supposing that 6 Ibs. be hung at each end of the rod, we might easily foresee that the knife-edge should be placed in the middle, and we find our anticipations verified. When the edge is exactly at the middle, the rod remains horizontal ; but if it be moved, even through a very small in.] .EQUILIBRIUM OF A BAR. 41 distance, to either side, the rod instantly descends on the other. The knife-edge is 24 inches distant from each end ; and if I multiply this number by the number of pounds in the weight, in this case 6, I find 144 for the product, and this product is the same for both ends of the bar. The importance of this remark will be seen directly. 64. If I remove one of the 6 Ib. weights and replace it by 2 Ibs., leaving the other weight and the knife-edge un- altered, the bar instantly descends on he side of the heavy weight ; but, by slipping the knife-edge along the bar, I find that when I have moved it to within a distance of 12 inches from the 6 Ibs., and therefore 36 inches from the 2 Ibs., the bar will remain horizontal. The edge must be put carefully at the right place ; a quarter of an inch to one side or the other would upset the bar. The whole load borne by the knife edge is of course 8 Ibs., being the sum of the weights. If we multiply 2, the number of pounds at one end, by 36, the distance of that end from the knife-edge, we obtain the product 72 ; and we find precisely the same product by multiplying 6, the number of pounds in the other weight, by 12, its distance from the knife-edge. To express this result concisely we shall introduce the word moment, a term of frequent use in mechanics. The 2 Ib. weight produces a force tending to pull its end of the bar downwards by making the bar turn round the knife-edge. The magnitude of th'S force, multiplied into its distance from the knife-edge, is called the moment of the force. We can express the result at which we have arrived by saying that, when the knife-edge has been so placed that the bar remains horizontal, the moments of the forces about tJie knife-edge are equal. 65. We may further illustrate this law by suspending weights of 7 Ibs. and 5 Ibs. respectively from the ends of 42 EXPERIMENTAL MECHANICS. [LECT. the bar ; it is found that the knife-edge must then be placed 20 inches from the larger weight, and, therefore, 28 inches from the smaller, but 5 x 28 = 140, and 7 x 20 = 140, thus again verifying the law of equality of the moments. From the equality of the moments we can also deduce the law for the distribution of the load given in Art. 53. Thus, taking the figures in the last experiment, we have loads of 7 Ibs. and 5 Ibs. respectively. These produce a pressure of 7 + 5 = 12 Ibs. on the knife-edge. This edge presses on the bar with an equal and opposite reaction. To ascertain the distribution of this pressure on the ends of the beam, we divide the whole beam into 12 equal parts of 4 inches each, and the 7 Ib. weight is 5 of these parts, i.e., 20 inches distant from the support. Hence the edge should be 20 inches from the greater weight, which is the condition also implied by the equality of the moments. THE COMPOSITION OF PARALLEL FORCES. 66. Having now examined the subject experimentally, we proceed to investigate what may be learned from the results we have proved. The weight of the bar being allowed for in the way we have explained, by subtracting one half of it from each of the strains indicated by the spring balance (FiG. 20), we may omit it from consideration. As the balances are pulled downwards by the bar when it is loaded, so they will react to pull the bar upwards. This will be evident if we think of a weight say 14 Ibs. suspended from one of these balances : it hangs at rest ; therefore its weight, which is constantly urging it downwards, must be counteracted by an equal force pulling it upwards. The balance of course shows 14 Ibs. ; thus the spring exerts in an upward pull a in.] PARALLEL FORCES. 43 force which is precisely equal to that by which it is itself pulled downwards. 67. Hence the springs are exerting forces at the ends of the bar in pulling them upwards, and the scales indicate their magnitudes. The bar is thus subject to three forces, viz. : the suspended weight of 20 Ibs., which acts ver- tically downwards, and the two other forces which act vertically upwards, and the united action of the three make equilibrium. 68. Let lines be drawn, representing s the forces in the manner already explained. We have then three parallel forces AP, BQ, CR acting on a rod in equilibrium (Fig. 22). The two forces AP andBQ may Q be considered as balanced by the force | CR in the position shown in the figure, but B ( the force CR would be balanced by the y equal and opposite force cs, represented i by the dotted line. Hence this last force is equivalent to AP and BQ. In other words, it must be their resultant. Here then we learn that a pair of parallel forces, acting in the same direction, can FIG. 22. be compounded into a single resultant. 69. We also see that the magnitude of the resultant is equal to the sum of the magnitudes of the forces, and further we find the position of the resultant by the following rule. Add the two forces together ; divide the distance between them into as many equal parts as are contained in the sum, measure off from the greater of these two forces as many parts as there are pounds in the smaller force, and that is the point required. This rule is very easily inferred from that which we were taught by the experiments in Art. 51. 44 EXPERIMENTAL MECHANICS. [LECT. PARALLEL FORCES ACTING IN OPPOSITE DIRECTIONS. 70. Since the forces AP, BQ, CR (Fig. 22) are in equili- brium, it follows that we may look on BQ as balancing in the position which it occupies the two forces of AP and CR in their positions. This may remind us of the numerous instances we have already met with, where one force balanced two greater forces : in the present case AP and CR are acting in opposite directions, and the force BQ which balances them is equal to their difference. A force BT equal and opposite to BQ must then be the resultant of CR and AP, since it is able to produce the same effect. Notice that in this case the resultant of the two forces is not between them, but that it lies on the side of the larger. When the forces act in the same direction, the resultant is always between them. 71. The actual position which the resultant of two oppo- site parallel forces occupies is to be found by the following rule. Divide the distance between the forces into as many equal parts as there are pounds in their difference, then measure outwards from the point of application of the larger force as many of these parts as there are pounds in the smaller force ; the point thus found determines the position of the resultant. Thus, if the forces be 14 and 20, the difference between them is 6, and therefore the distance between their directions is divided into six parts ; from the point of application of the force of 20, 14 parts are measured outwards, and thus the position of the resultant is deter- mined. Hence we have the means of compounding two parallel forces in general THE COUPLE. 72. In one case, however, two parallel forces have no resultant ; this occurs when the two forces are equal, and in in.] THE COUPLE. 45 opposite directions. A pair of forces of this kind is called a couple; there is no single force which could balance a couple, it can only be counterbalanced by another couple acting in an opposite manner. This remarkable case, may be studied by the arrangement of Fig. 23. A wooden rod, A B 48 " x o" -5 x o" - 5, has strings attached to it at points A and D, one foot distant. The string at D passes over a fixed pulley E, and at the end p a hook is attached for the purpose of receiving weights, while a similar hook depends from A ; the weight of the rod itself, which only amounts to three ounces, may be neglected, as it is very small compared with the weights which will be used. 73. Supposing 2 Ibs. to be placed at P, and i Ib. at Q, we have two parallel forces acting in opposite directions ; and since their difference is i Ib., it follows from our rule that the point F, where D F is equal to A D, is the point where the resultant is applied. You see this is easily verified, for by placing my finger over the rod at F it remains horizontal and in equilibrium; whereas, when I move my finger to one side or the other, equilibrium is impossible. If I move it nearer to B, the end A ascends. If I move it towards A, the end B ascends. 46 EXPERIMENTAL MECHANICS. [LgCT. 74. To study the case when the two forces are equal, a load of 2 Ibs. may be placed on each of the hooks p and Q. It will then be found that the finger cannot be placed in any position along the rod so as to keep it in equilibrium ; that is to say, no single force can counteract the two forces which form the couple. Let o be the point midway be- tween A and D. The forces evidently tend to raise o B and turn the part o A downwards ; but if I try to restrain o B by holding my finger above, as at the point x, instantly the rod begins to turn round x and the part from A to x descends. I find similarly that any attempt to prevent the motion by holding my finger underneath is equally un- successful. But if at the same time I press the rod down- wards at one point, and upwards at another with suitable force, I can succeed in producing equilibrium ; in this case the two pressures form a couple ; and it is this couple which neutralizes the couple produced by the weights. We learn, then, the important result that a couple can be balanced by a couple, and by a couple only. 75. The moment of a couple is the product ot one of the two equal forces into their perpendicular distance. Two couples tending to turn the body to which they are applied , in the same direction will be equivalent if their moments are equal. Two couples which tend to turn the body in opposite directions will be in equilibrium if their moments are equal. We can also compound two couples in the same or in opposite directions into a single couple of which the moment is respectively either the sum or the difference of the original moments. THE WEIGHING SCALES. 76. Another apparatus by which the nature of parallel forces may be investigated is shown in Fig. 24; it con- IIl.j THE WEIGHING SCALES. sists of a slight frame of wood ABC, 4' long. At E, a pair of steel knife-edges is clamped to the frame. The knife- edges rest on two pieces of steel, one of which is shown at o F. When the knife-edges are suitably placed the frame is balanced, so that a small piece of paper laid at A will cause that side to descend. 77. We suspend two small hooks from the points A and B : these are made of fine wire, so that their weight may be FIG. 24. left out of consideration. With this apparatus we can in the first place verify the principle of equality of moments : for example, if I place the hook A at a distance of 9" from the centre o and load it with i lb., I find that when B is laden with 0-5 lb. it must be at a distance of 18" from o in order to counterbalance A ; the moment in the one case is 9 x i, in the other 18 x 0-5, and these are obviously equal 78. Let a weight of i lb. be placed on each of the hooks, the frame will only be in equilibrium when the 48 EXPERIMENTAL MECHANICS. [LECT. hooks are at precisely the same distance from the centre. A familiar application of this principle is found in the ordinary weighing scales ; the frame, which in this case is called a beam, is sustained by two knife-edges, smaller, how- ever, than those represented in the figure. The pans p, p are suspended from the extremities of the beam, and should be at equal distances from its centre. These scale-pans must be of equal weight, and then, when equal weights are placed in them, the beam will remain horizontal. If the weight in one slightly exceed that in the other, the pan containing the heavier weight will of course descend. 79. That a pair of scales should weigh accurately, it is necessary that the weights be correct ; but even with correct weights, a balance of defective construction will give an in- accurate result. The error f equently arises from some inequality in the lengths of the arms of the beam. When this is the case, the two weights which really balance are not equal. Supposing, for instance, that with an imperfect balance I endeavour to weigh a pound of shot. If I put the weight on the short side, then the quantity of shot balanced is less than i Ib. ; while if the i Ib. weight be placed at the long side, it will require mere than i Ib. of shot to produce equilibrium. The mode of testing a pair of scales is then evident. Let weights be placed in the pans which balance each other ; if the weights be interchanged and the balance still remains horizontal, it is correct. 80. Suppose, for example, that the two arms be i o inches and 1 1 inches long, then, if i Ib. weight be placed in the pan of the lo-inch end, its moment is 10 ; and if | of i Ib. be placed in the pan belonging to the n-inch end, its moment is also 10 : hence i Ib. at the short end balances { of i Ib. at the long end ; and therefore, if the shopkeeper placed his weight in the short arm, his customers would lose T ! T in.] THE WEIGHING SCALES. 49 part of each pound for which they paid ; on the other hand, if the shopkeeper placed his i Ib. weight on the long arm, then not less than -]- Ib. would be required in the pan belonging to the short arm. Hence in this case the cus- tomer would get y 1 ^ Ib. too much. It follows, that if a shopman placed his weights and his goods alternately in the one scale and in the other he would be a loser on the whole ; for, though every second customer gets r : T Ib. less than he ought, yet the others get -^ Ib. more than they have paid for. LECTURE IV. THE FORCE OF GRA VITY. 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. INTRODUCTION. 8 1. IN the last three lectures we considered forces in the abstract ; we saw how they are to be represented by straight lines, how compounded together and how decom- posed into others ; we have explained what is meant by forces being in equilibrium, and we have shown instances where the forces lie in the same plane or in different planes, and where they intersect or are parallel to each other. These subjects are the elements of mechanics ; they form the framework which in this and subsequent lectures we shall try to present in a more attractive garb. We shall commence by studying the most remarkable force in nature, a force constantly in action, and one to which all bodies are subject, a force which distance cannot annihilate, and one the pro- perties of which have led to the most sublime discoveries of human intellect. This is the force of gravity. LECT. iv.] THE FORCE OF GRAVITY. 51 82. If I drop a stone from my hand, it falls to the ground. That which produces motion is a force : hence the stone must have been acted upon by a force which drew it to the ground. On every part of the earth's surface experi- ence shows that a body tends to fall. This fact proves that there is an attractive force in the earth tending to draw all bodies towards it. 83. Let ABCD (Fig. 25) be points from which stones are let fall, and let the circle represent the section of the earth ; let P Q R s be the points at the surface of the earth upon which the stones will drop when allowed to do so. The four stones will move in the directions of the arrows : from A to P the stone moves in an opposite direction to the motion from c to R ; from B to Q it moves from right to left, while from L) to s it moves from left to right. The movements are in different directions; but if I produce these directions, as indicated by the dotted lines, they each pass through the centre o. 52 EXPERIMENTAL MECHANICS. [LECT. 84. Hence each stone in falling moves towards the centre of the earth, and this is consequently the direction of the force. We therefore assert that the earth has an attraction for the stone, in consequence of which it tries to get as near the earth's centre as possible, and this attraction is called the force of gravitation. 85. We are so excessively familiar with the phenomenon of seeing bodies fall that it does not excite our astonishment or arouse our curiosity. A clap of thunder, which every one notices, because much less frequent, is not really more remarkable. We often look with attention at the attraction of a piece of iron by a magnet, and justly so, for the phenomenon is very interesting, and yet the falling of a stone is produced by a far grander and more important force than the force of magnetism. 86. It is gravity which causes the weight of bodies. I hold a piece of lead in my hand : gravity tends to pull it downwards, thus producing a pressure on my hand which I call weight. Gravity acts with slightly varying intensity at various parts of the earth's surface. This is due to two distinct causes, one of which may be mentioned here, while the other will be subsequently referred to. The earth is not perfectly spherical ; it is flattened a little at the poles ; con- sequently a body at the pole is nearer the general mass of the earth than a body at the equator ; therefore the body at the pole is more attracted, and seems heavier. A mass which weighs 200 Ibs. at the equator would weigh one pound more at the pole : about one-third of this increase is due to the cause here pointed out. (See Lecture XVII.) 87. Gravity is a force which attracts every particle of matter ; it acts not merely on those parts of a body which lie on the surface, but it equally affects those in the interior. This is proved by observing that a body has the same iv.] SPECIFIC GRAVITY. 53 weight, however its shape be altered : for example, suppose I take a ball of putty which weighs i lb., I shall find that its weight remains unchanged when the ball is flattened into a thin plate, though in the latter case the surface, and therefore the number of superficial particles, is larger than it was in the former. SPECIFIC GRAVITY. 88. Gravity produces different effects upon different sub- stances. This is commonly expressed by saying that some substances are heavier than others ; for example, I have here a piece of wood and a piece of lead of equal bulk. The lead is drawn to the earth with a greater force than the wood. Substances are usually termed heavy when they sink in water, and light when they float upon it. But a body sinks in water if it weigh more than an equal bulk of water, and floats if it weigh less. Hence it is natural to take water as a standard with which the weights of other substances may be compared. 89. I take a certain volume, say a cubic inch of cast iron such as this I hold in my hand, and which has been accurately shaped for the purpose. This cube is heavier than one cubic inch of water, but I shall find that a certain quantity of water is equal to it in weight ; that is to say, a certain number of cubic inches of water, and it may be fractional parts of a cubic inch, are precisely of the same weight. This number is called the specific gravity of cast iron. 90. It would be impossible to counterpoise water with the iron without holding the water in a vessel, and the weight of the vessel must then be allowed for. I adopt the following plan. I have here a number of inch cubes of wood (Fig. 26), which would each be lighter than a cubic inch of water, but I have weighted the wooden cubes by placing 54 EXPERIMENTAL MECHANICS. [LECT. grains of shot into holes bored into the wood. The weight of each cube has thus been accurately adjusted to be equal to that of a cubic inch of water. This may be tested by actual weighing. I weigh one of the cubes and find it to be 252 grains, which is well known to be the weight of a cubic inch of water. 91. But the cubes maybe shown to be identical in weight with the same bulk of water by a simpler method. One of them placed in water should have no tendency to sink, since it is not heavier than water, nor on the other hand, since it is not lighter, should it have any tendency to float. It should then remain in the water in whatever position it may be placed. It is difficult to prepare one of these cubes so accurately that this result should be attained, and it is iv.] SPECIFIC GRAVITY. 55 impossible to ensure its continuance for any time owing to changes of temperature and the absorption of water by the wood. We can, however, by a slight modification, prove that one of these cubes is at all events nearly equal in weight to the same bulk of water. In Fig. 26 is shown a tall glass jar filled with a fluid in appearance like plain water, but it is really composed in the following manner. I first poured into the jar a very weak solution of salt and water, which partially filled it ; I then poured gently upon this a little pure water, and finally filled up the jar with water containing a little spirits of wine : the salt and water is a little heavier than pure water, while the spirit and water is a little lighter. I take one of the cubes and drop it gently into the glass ; it falls through the spirit and water, and after making a few oscillations settles itself at rest in the stratum shown in the figure. This shows that our prepared cube is a little heavier than spirit and water, and a little lighter than salt and water, and hence we infer that it must at all events be very near the weight of pure water which lies between the two. We have also a number of half cubes, quarter cubes, and half-quarter cubes, which have been similarly prepared to be of equal weight with an equal bulk of water. 92. We shall now be able to measure the specific gravity of a substance. In one pan of the scales I place the inch cube of cast iron, and I find that 7^ of the wooden cubes, which we may call cubes of water, will balance it. We therefore say that the specific gravity of iron is 7^. The exact number found by more accurate methods is 7*2. It is often convenient to remember that 23 cubic inches of cast iron weigh 6 Ibs., and that therefore one cubic inch weighs very nearly ^ Ib. 93. I have also cubes of brass, lead, and ivory; by 56 EXPERIMENTAL MECHANICS. [LECT. counterpoising them with the cubes of water, we can easily find their specific gravities ; they are shown together with that of cast iron in the following table : Substance. Specific Gravity. Cast Iron 7'2 Brass 8'i Lead ii'3 Ivory I '8 94. The mode here adopted of finding specific gravities is entirely different from the far more accurate methods which are commonly used, but the explanation of the latter involve more difficult principles than those we have been considering. Our method rather offers an explanation of the nature of specific gravity than a good means of determin- ing it, though, as we have seen, it gives a result sufficiently near the truth for many purposes. THE PLUMMET AND SPIRIT-LEVEL. 95. The tendency of the earth to draw all bodies towards it is well illustrated by the useful "line and plummet.' This consists merely of a string to one end of which a leaden weight is attached. The string when at rest hangs vertically ; if the weight be drawn to one side, it will, when released, swing backwards and forwards, until it finally settles again in the vertical ; the reason is that the weight always tries to get as near the earth as it can, and this is accomplished when the string hangs vertically downwards. 96. The surface of water in equilibrium is a horizontal plane ; that is also a consequence of gravity. All the particles of water try to get as near the earth as possible, and therefore if any portion of the water were higher than the rest, it would immediately spread, as by doing so it could get lower. THE CENTRE OF GRAVITY. 57 97. Hence the surface of a fluid at rest enables us to find a perfectly horizontal plane, while the plummet gives us a perfectly vertical line : both these consequences of gravity are of the utmost practical importance. 98. The spirit-level is another common and very useful instrument which depends on gravity. It consists of a glass tube slightly curved, with its convex surface upwards, and attached to a stand with a flat base. This tube is nearly filled with spirit, but a bubble of air is allowed to remain. The tube is permanently adjusted so that, when the plate is laid on a perfectly horizontal surface, the bubble will stand in the middle : accordingly the position of the bubble gives a means of ascertaining whether a surface is level. THE CENTRE OF GRAVITY. 99. We proceed to an experiment which will give an insight into a curious property of gravity. I have here a plate of sheet iron ; it has the irregular shape shown in Fig 27. Five small holes A B c D E are punched at different positions on the margin. Attached to the framework is a small pin from which I can suspend the iron plate by one of its holes A : the plate is not sup- ported in any other way; it hangs freely from the pin, around which it can be easily turned. I find that there is one position, and one only, in which the plate will rest ; if I withdraw it from that position it returns there after a few oscillations. In order to mark this position, I suspend a line and plummet from the pin, 58 EXPERIMENTAL MECHANICS. [LECT. having rubbed the line with chalk. I allow the line to come to rest in front of the plate. I then flip the string against the plate, and thus produce a chalked mark : this of course traces out a vertical line A p on the plate. I now remove the plummet and suspend the plate from another of its holes B, and repeat the process, thus drawing a second chalked line B p across the plate, and so on with the other holes : I thus obtain five lines across the plate, repre- sented by dotted lines in the figure. It is a very remarkable circumstance that these five lines all intersect in the same point P; and if additional holes were bored in the plate, whether in the margin or not, and the chalk line drawn from each of them in the manner described, they would one and all pass through the same point. This remarkable point is called the centre of gravity of the plate, and the result at which we have arrived may be expressed by saying that the vertical line from the point of suspension always passes through the centre of gravity. 100. At the centre of gravity P a hole has been bored, and when I place the supporting pin through this hole you see that the plate will rest indifferently in all positions : this is a curious property of the centre of gravity. The centre of gravity may in this respect be contrasted with another hole Q, which is only an inch distant : when I support the plate by this hole, it has only one position of rest, viz. when the centre of gravity p is vertically beneath Q. Thus the centre of gravity differs remarkably from any other point in the plate. 1 01. We may conceive the force of gravity on the plate to act as a force applied at P. It will then be easily seen why this point remains vertically underneath the point of suspen- sion when the body is at rest. If I attached a string to the plate and pulled it, the plate would evidently place itself so iv.] STABLE AND UNSTABLE EQUILIBRIUM. 59 that the direction of the string would pass through the point of suspension ; in like manner gravity so places the plate that the direction of its force passes through the point of suspension. 102. Whatever be the form of the plate it always contains one point possessing these remarkable properties, and we may state in general that in every body, no matter what be its shape, there is a point called the centre of gravity, such that if the body be suspended from this point it will remain in equilibrium indifferently in any position, and that if the body be suspended from any other point, then it will be in equilibrium when the centre of gravity is directly underneath ' the point of suspension. In general, it will be impossible to support a body exactly at its centre of gravity, as this point is within the mass of the body, and it may also sometimes happen that the centre of gravity does not lie in the substance of the body at all, as for example in a ring, in which case the centre of gravity is at the centre of the ring. We need not, however, dwell on these exceptional cases, as sufficient illustrations of the truth of the laws mentioned will present themselves subsequently. STABLE AND UNSTABLE EQUILIBRIUM. 103. An iron rod A B, capable of revolving round an axis passing through its centre P, is shown in Fig. 28. The centre of gravity lies at the centre B, and consequently, as is easily seen, the rod will remain at rest in whatever posi- tion it be placed. But let a weight R be attached to the rod by means of a binding screw. The centre of gravity of the whole is no longer at the centre of the rod ; it has moved to a point s nearer the weight ; we may easily ascertain its position by removing the rod from its axle and then ascer- taining the point about which it will balance. This may be 60 EXPERIMENTAL MECHANICS. [LECT. done by placing the bar on a knife-edge, and moving it to and fro until the right position be secured ; mark this posi- tion on the rod, and return it to its axle, the weight being still attached. We do not now find that the rod will balance in every position. You see it will balance if the point s be directly underneath the axis, but not if it lie to one side or the other. But if s be directly over the axis, as in the figure, the rod is in a curious condition. It will, when carefully placed, remain at rest ; but if it receive the slightest displacement, it will tumble over. The rod is in equilibrium in this position, but it is what is called unstable equilibrium. If the centre of gravity be vertically below the point of suspen- sion, the rod will return again if moved away : this position is therefore called one of stable equilibrium. It is very important to notice the distinction between these two kinds of equilibrium. 104. Another way of stating the case is as follows. A body is in stable equilibrium when its centre of gravity is at the lowest point : unstable when it is at the highest. This may be very simply illustrated by an ellipse, which I hold in my hand. The centre of gravity of this figure is at its centre. The ellipse, when resting on its side, is in a posi- tion of stable equilibrium and its centre of gravity is then clearly at its lowest point. But I can also balance the ellipse on its narrow end, though if I do so the smallest touch suffices to overturn it. The ellipse is then in unstable equilibrium ; in this case, obviously, the centre of gravity is at the highest point. iv.] CENTRE OF GRAVITY. 6r 105. I have here a sphere, the centre of gravity of which is at its centre ; in whatever way the sphere is placed on a plane, its centre is at the same height, and therefore cannot be said to have any highest or lowest point ; in such a case as this the equilibrium is neutral. If the body be displaced, it will not return to its old position, as it would have done had that been a position of stable equilibrium, nor will it deviate further therefrom as if the equilibrium had been unstable : it will simply remain in the new position to which it is brought. 1 06. I try to balance an iron ring upon the end of a stick H, Fig. 29, but I cannot easily succeed in doing so. This is because its centre of gravity s is above the point of support ; but if I place the stick at F, the ring is in stable equilibrium, for now the centre of gravity is FIG. 29. below the point of support. PROPERTY OF THE CENTRE OF GRAVITY IN A REVOLVING WHEEL. 107. There are other curious consequences which follow from the properties of the centre of gravity, and we shall conclude by illustrating one of the most remarkable, which is at the same time of the utmost importance in machinery. 1 08. It is generally necessary that a machine should work as steadily as possible, and that undue vibration and shaking of the framework should be avoided : this is par- ticularly the case when any parts of the machine rotate with great velocity, as, if these be heavy, inconvenient vibration will be produced when the proper adjustments are not made. The connection between this and the centre of EXPERIMENTAL MECHANICS. [LECT. gravity will be understood by reference to the apparatus represented in the accompanying figure (Fig. 30). We have here an arrangement consisting of a large cog wheel C working into a small one B, whereby, when the handle H is turned, a velocity of rotation can be given to the iron FIG. 3 c. disk D, which weighs i4lbs, and is 18" in diameter. This disk being uniform, and being attached to the axis at its centre, it follows that its centre of gravity is also the centre of rotation. The wheels are attached to a stand, which, though massive, is still unconnected with the floor. By turning the handle I can rotate the disk very rapidly, even iv.] CENTRE OF GRAVITY. 63 as much as twelve times in a second. Still the stand re- mains quite steady, and even the shutter bell attached to it at E is silent. 109. Through one of the holes in the disk D I fasten a small iron bolt and a few washers, altogether weighing about i Ib. ; that is, only one-fourteenth of the weight of the disk. When I turn the handle slowly, the machine works as smoothly as before ; but as I increase the speed up to one revolution every two seconds, the bell begins to ring violently, and when I increase it still more, the stand quite shakes about on the floor. What is the reason of this ? By adding the bolt, I slightly altered the position of the centre of gravity of the disk, but I made no change of the axis about which the disk rotated, and consequently the disk was not on this occasion turning round its centre of gravity : this it was which caused the vibration. It is absolutely necessary that the centre of gravity of any heavy piece, rotating rapidly about an axis, should lie in the axis of rotation. The amount of vibration produced by a high velocity may be very considerable, even when a very small mass is the originating cause. no. In order that the machine may work smoothly again, it is not necessary to remove the bolt from the hole. If by any means I bring back the centre of gravity to the axis, the same end will be attained. This is very simply effected by placing a second bolt of the same size at the opposite side of the disk, the two being at equal distances from the axis; on turning the handle, the machine is seen to work as smoothly as it did in the first instance. in. The most common rotating pieces in machines are wheels of various kinds, and in these the centre of gravity is evidently identical with the centre of rotation ; but if from any cause a wheel, which is to turn rapidly, has an 64 EXPERIMENTAL MECHANICS. [LECT. iv. extra weight attached -to one part, this weight must be counterpoised by one or more on other portions of the wheel, in order to keep the centre of gravity of the whole in its proper place. Thus it is that the driving wheels of a locomotive are always weighted so as to counteract the effect of the crank and restore the centre of gravity to the axis of rotation. The cause of the vibration will be under- stood after the lecture on centrifugal force (Lect XVII.). 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. THE NATURE OF FRICTION. 1 1 2. A DISCUSSION of the force of friction is a necessary preliminary to the study of the mechanical powers which we shall presently commence. Friction renders the inquiry into the mechanical powers more difficult than it would be if this force were absent; but its effects are too important to be overlooked. 113. The nature of friction may be understood by Fig. 31, which represents a section of the top of a table of wood F.G. 31. or any other substance levelled so that c D is horizontal ; on the table rests a block A of wood or any other substance. To A a cord is attached, which, after passing over a pulley p, F 66 EXPERIMENTAL MECHANICS. [LECT. is stretched by a suspended weight B. If the magnitude of B exceeds a certain limit, then A is pulled along the table and B descends ; but if B be smaller than this limit, both A and B remain at rest. When B is not heavy enough to produce motion it is supported by the tension of the cord, which is itself neutralized by the friction produced by a certain coherence between A and the table. Friction is by this experiment proved to be a force, because it prevents the motion of B. Indeed friction is generally manifested as a force by destroying motion, though sometimes indirectly pro- ducing it. 114. The true source of the force lies in the inevitable roughness of all known surfaces, no matter how they may have been wrought. The minute asperities on one surface are detained in corresponding hollows in the other, and con- sequently force must be exerted to make one surface slide upon the other. By care in polishing the surfaces the amount of friction may be diminished ; but it can only be decreased to a certain limit, beyond which no amount of polishing seems to produce much difference. 115. The law of friction under different conditions must be inquired into, in order that we may make allowance when its effect is of importance. The discussion of the experi- ments is sometimes a little difficult, and the truths arrived at are principally numerical, but we shall find that some interesting laws of nature will appear. THE MODE OF EXPERIMENTING. 1 1 6. Friction is present between every pair of surfaces which are in contact : there is friction between two pieces of wood, and between a piece of wood and a piece of iron ; and the amount of the force depends upon the character of both surfaces. We shall only experiment upon the friction v.] THE MODE OF EXPERIMENTING. 67 of wood upon wood, as more will be learned by a careful study of a special case than by a less minute examination of a number of pairs of different substances. 117. The apparatus used is shown in Fig. 32. A plank of pine 6' x n'' x 2" is planed on its upper surface, levelled by a spirit-level, and firmly secured to the framework at a height of about 4' from the ground. On it is a pine slide 9" x 9", the grain of which is crosswise to that of the plank ; upon the slide the load A is placed. A rope is attached to the slide, which passes over a very freely mounted cast iron pulley c, 14" diameter, and carries at the other end a hook weighing one pound, from which weights B can be suspended. 1 1 8. The mode of experimenting consists in placing a cer- tain load upon A, and then ascertaining what weight applied to B will draw the loaded slide along the plane. As several trials are generally necessary to determine the power, a rope is attached to the back of the slide, and passes over the two pulleys D ; this makes it easy for the experimenter, when applying the weights at B, to draw back the slide to the end of the plane by pulling the ring E : this rope is of course left quite slack during the process of the experiment, since the slide must not be retarded. The loads placed upon A during the series of experiments ranged between one stone and eight stone. In the loads stated the weight of the slide itself, which was less than i lb., is always included. A variety of small weights were provided for the hook B ; they consisted of o-i, 0-5, i, 2, 7, and i4lbs. There is some friction to be overcome in the pulley c, but as the pulley is comparatively large its friction is small, though it was always allowed for. 119. An example of the experiments made is thus described. A weight of 56 Ibs. is placed upon the slide, and it is found 68 EXPERIMENTAL MECHANICS. [LECT. on trial that 2 gibs, on B (including the weight of the hook itself) is sufficient to start the slide ; this weight is placed THE MODE OF EXPERIMENTING. 69 upon the hook pound by pound, care being taken to make each addition gently. 120. Experiments were made in this way with various weights upon A, and the results are recorded in Table I. TABLE I. FRICTION. Smooth horizontal surface of pine 72" x 1 1" ; slide also of pine 9" x 9" ; grain crosswise ; slide is not started ; force acting on slide is gradually increased until motion commences. Number of Experiment. Load on sl.de in Ibs., including weight of slide. Force necessary to move slide, ist Series. Force necessary to move slide. 2nd Series. Mean values. j 14 5 8 6'S 2 28 15 16 IS'S 3 4 2 20 15 i7'5 4 56 29 24 265 5 70 33 31 320 6 84 43 33 3x> 7 9 8 42 38 40 x> 8 112 5 33 4i-5 In the first column a number is given to each experiment for convenience of reference. In the second column the load on the slide is stated in Ibs. In the third column is found the force necessary to overcome the friction. The fourth column records a second series of experiments per- formed in the same manner as the first series ; while the last column shows the mean of the two frictions. 121. The first remark to be made upon this table is, that the results do not appear satisfactory or concordant. Thus from 6 and 7 of the ist series it would appear that the fric- tion of 84 Ibs. was 43 Ibs., while that of 98 Ibs. was 42 Ibs., so that here the greater weight appears to have the less friction, which is evidently contrary to the whole tenor of the results, as a glance will show. Moreover the frictions in the ist and the 2nd series do not agree, being generally greater 70 EXPERIMENTAL MECHANICS. [LECT. in the former than in the latter, the discordance being espe- cially noticeable in experiment 8, where the results were 50 Ibs. and 33 Ibs. In the final column of means these irregularities are lessened, for this column shows that the friction increases with the weight, but it is sufficient to observe that as the difference of the ist and the 2nd is 9 Ibs., and that of the 2nd and the 3rd is only 2 Ibs., the discovery of any law from these results is hopeless. 122. But is friction so capricious that it is amenable to no better law than these experiments appear to indicate ? We must look a little more closely into the matter. When two pieces of wood have remained in contact and at rest for some time, a second force besides friction resists their separation : the wood is compressible, the surfaces become closely approximated, and the coherence due to this cause must be overcome before motion commences. The initial coherence is uncertain ; it depends probably on a multitude of minute circumstances which it is impossible to estimate, and its presence has vitiated the results which we have found so unsatisfactory. 123. We can remove these irregularites by starting the slide at the commencement. This may be conveniently effected by the screw shown at F in Fig. 32 ; a string attached to its end is fastened to the slide, and by giving the handle of the screw a few turns the slide begins to creep. A body once set in motion will continue to move with the same velocity unless acted upon by a force ; hence the weight at B just overcomes the friction when the slide moves uniformly after receiving a start : this velocity was in one case of average speed measured to be 1 6 inches per minute. 124. Indeed in no case can the slide commence to move unless the force exceed the friction. The amount of this V-] THE MODE OF EXPERIMENTING. excess is indeterminate. It is certainly greater between wooden surfaces than between less compressible surfaces like those of metals or glass. In the latter case, when the force exceeds the friction by a small amount, the slide starts off with an excessively slow motion ; with wood the force must exceed the friction by a larger amount before the slide commences to move, but the motion is then comparatively rapid. 125. If the power be too small, the load either does not continue moving after the start, or it stops irregularly. If the power be too great, the load is drawn with an accelerated velocity. The correct amount is easily recognised by the uniformity of the movement, and even when the slide is heavily laden, a few tenths of a pound on the power hook cause an appreciable difference of velocity. 126. The accuracy with which the friction can be measured may be appreciated by inspecting Table II. TABLE II. FRICTION. Smooth horizontal surface of pine 72" x 1 1" ; slide also of pine 9" x 9" ; grain crosswise ; slide started ; force applied is sufficient to maintain uniform motion of the slide. Number of Load on slide in Ibs. , including Force necessary to maintain motion. Force necessary to maintain motion. Mean Expenment. weight of slide. ist Series. 2nd Series. I 14 4 - 9 4'9 4'9 2 28 8-5 8-6 -5 3 42 12-6 12-4 I2'5 4 56 163 16-2 162 5 70 197 200 I 9 -8 6 84 23 7 23-0 23 "4 7 9 8 26-5 26' I 263 8 112 297 29-9 29-8 127. Two series of experiments to determine the power necessary to maintain the motion have been recorded. 72 EXPERIMENTAL MECHANICS. [LECT. Thus, in experiment 7, the load on the slide being 98 Ibs., it was found that 26-5 Ibs. was sufficient to sustain the motion, and a second trial being made independently, the power found was 26'! Ibs. : a mean of the two values, 26-3 Ibs., is adopted as being near the truth. The greatest difference between the two series, amounting to 07 lb., is found in experiment 6 ; a third value was therefore ob- tained for the friction of 84 Ibs. : this amounted to 23-5 Ibs., which is intermediate between the two former results, and 23 '4 Ibs., a mean of the three, is adopted as the final result. 128. The close accordance of the experiments in this table shows that the means of the fifth column are probably very near the true values of the friction for the correspond- ing loads upon the slide. 129. The mean frictions must, however, be slightly di- minished before we can assert that they represent only the friction of the wood upon the wood. The pulley over which the rope passes turns round its axle with a small amount of friction, which must also be overcome by the power. The mode of estimating this amount, which in these experiments never exceeds o'5 lb., need not now be discussed. The corrected values of the friction are shown in the third column of Table III. Thus, for example, the 4-9 Ibs. of friction in experiment i consists of 47, the true friction of the wood, and 0-2, which is the friction of the pulley ; and 26-3 of experiment 7 is similarly composed of 25-8 and 0*5. It is the corrected frictions which will be employed in our subsequent calculations. FRICTION IS PROPORTIONAL TO THE PRESSURE. 130. Having ascertained the values of the force of friction for eight different weights, we proceed to inquire into the v.] FRICTION PROPORTIONAL TO PRESSURE. 73 laws which may be founded on our results. It is evident that the friction increases with the load, of which it is always greater than a fourth, and less than a third. It is natural to surmise that the friction (f) is really a constant fraction of -the load (1?) in other words, that F = kR, where k is a constant number. 131. To test this supposition we must try to determine k ; it may be ascertained by dividing any value of F by the corresponding value of R. If this be done, we shall find that each of the experiments yields a different quotient ; the first gives 0*336, and the last 0*262, while the other experi- ments give results between these extreme values. These numbers are tolerably close together, but there is still sufficient discrepancy to show that it is not strictly true to assert that the friction is proportional to the load. 132. But the law that the friction varies proportionally to the pressure is so approximately true as to be sufficient for most practical purposes, and the question then arises, which of the different values of k shall we adopt ? By a method which is described in the Appendix we can deter- mine a value for k which, while it does not represent any one cf the experiments precisely, yet represents them collec- tively better than it is possible for any other value to do. The number thus found is 0*27. It is intermediate be- tween the two values already stated to be extreme. The character of this result is determined by an inspection of Table III. The fourth column of this table has been calculated from the formula F = 0*27 R. Thus, for example, in experiment 5, the friction of a load of 70 Ibs. is 19-4 Ibs., and the product of 70 and 0-27 is 18-9, which is 0*5 Ib. less than the true amount. In the last column of this table the discrepancies between the observed and the calculated values are recorded, 74 EXPERIMENTAL MECHANICS. [LECT. for facility of comparison. It will be observed that the greatest difference is under i Ib. TABLE III. FRICTION. Friction of pine upon pine ; the mean values of the friction given in Table II. (corrected for the friction of the pulley) compared with the formula F = o'2"j R. Number of Experiment. R. Total load on slide in Ibs. Corrected mean value of friction. F. Calculated value of friction. Discrepancies between the observed and calculated frictions. I H 47 3-8 -0'9 2 28 8'2 7-6 -0-6 3 42 1 2 '2 ii'3 -0-9 4 56 15-8 -07 5 7 19-4 18-9 -'5 6 84 23-0 227 -0-3 7 98 2 5 -8 26-5 + 07 8 112 29 "3 30-2 + 0-9 133. Hence the law F = 0-27 R represents the experi- ments with tolerable accuracy; and the numerical ratio 0*27 is called the coefficient of friction. We may apply this law to ascertain the friction in any case where the load lies between 14 Ibs. and 112 Ibs. ; for example, if the load be 63 Ibs., the friction is 63 x 0*27 = 17-0. 134. The coefficient of friction would have been slightly different had the grain of the slide been parallel to that of the plank ; and it of course varies with the nature of the surfaces. Experimenters have given tables of the coefficients of friction of various substances, wood, stone, metals, c. The use of these coefficients depends upon the assumption of the ordinary law of friction, namely, that the friction is proportional to the pressure : this law is accurate enough for most purposes, especially when used for loads that lie be- tween the extreme weights employed in calculating the value of the coefficient which is employed. v.] A MORE ACCURATE LAW OF FRICTION. 75 A MORE ACCURATE LAW OF FRICTION. 135. In making one of our measurements with care, it is unusual to have an error of more than a few tenths of i Ib. and it is hardly possible that any of the mean frictions we have found should be in error to so great an extent as o'5 Ib. But with the value of the coefficient of friction which is used in Table III., the discrepancies amount sometimes to 0-9 Ibs. With any other numerical coefficient than 0^27, the discrepancies would have been even still more serious. As these are too great to be attributed to errors of experiment, we have to infer that the law of friction which has been assumed cannot be strictly true. The signs of the discrepancies indicate that the law gives frictions which for small loads are too small, and for large loads are too great. 136. We are therefore led to inquire whether some other relation between ^and-/? may not represent the experiments with greater fidelity than the common law of friction. If we diminished the coefficient by a small amount, and then added a constant quantity to the product of the coefficient and the load, the effect of this change would be that for small loads the calculated values would be increased, while for large loads they would be diminished. This is the kind of change -which we have indicated to be necessary for re- conciliation between the observed and calculated values. 137. We therefore infer that a relation of the form F = x + y R will probably express a more correct law, provided we can find x and y. One equation between x and y is obtained by introducing any value of R with the corresponding value of F, and a second equation can be found by taking any other similar pair. From these two equations the values of x and of y may be deduced by elementary algebra, but the best formula will be obtained 76 EXPERIMENTAL MECHANICS. [LECT. by combining together all the pairs of corresponding values. For this reason the method described in the Appendix must be used, which, as it is founded on all the experiments, must give a thoroughly representative result. The formula thus determined, is f= i '44 4- 0*252 ^?. This formula is compared with the experiments in Table IV. TABLE IV. FRICTION. Friction of pine upon pine ; the mean values of the friction given in Table II. (corrected for the friction of the pulley) compared with the formula F = I -44 + 0*252 Jf. Number of Experiment. R. Total load on slide in Ibs. Corrected mean value of friction. F. Calculated value of friction. Discrepancies between the observed and calculated frictions j H 47 S' 1-0-3 2 28 8-2 8-5 + 0-3 3 42 1 2 '2 I2'O -O'2 4 56 IS '8 I 5 -6 -0'2 | e 19-4 23-0 I9-I 22-6 -03 -0- 4 7 9 8 25-8 26-1 + 0'3 8 112 29-3 297 + 0'4 The fourth column contains the calculated values : thus, for example, in experiment 4, where the load is 56 Ibs., the calculated value is i -44 + 0-252 x 56=15-6; the difference o'2 between this and the observed value 15 '8 is shown in the last column. 138. It will be noticed that the greatest discrepancy in this column is 0-4 Ibs., and that therefore the formula repre- sents the experiments with considerable accuracy. It is undoubtedly nearer the truth than the former law (Art. 132) ; in fact, the differences arc now such as might really belong to errors unavoidable in making the experiments. v.] COEFFICIENT VARIES WITH WEIGHTS. 77 139. This formula maybe used for calculating the friction for any load between 14 Ibs. and 112 Ibs. Thus, if the load be 63 Ibs., the friction is 1-44 + 0-252 x 63 := 17-3 Ibs., which does not differ much from 17-0 Ibs., the value found by the more ordinary law. We must, however, be cautious not to apply this formula to weights which do not lie between the limits of the greatest and least weight used in those experiments by which the numerical values in the for- mula have been determined ; for example, to take an extreme case, if R = o, the formula would indicate that the friction was i '44, which is evidently absurd; here the formula errs in excess, while if the load were very large it is certain the formula would err in defect. THE COEFFICIENT VARIES WITH THE WEIGHTS USED. 140. In a subsequent lecture we shall employ as an inclined plane the plank we have been examining, and we shall require to use the knowledge of its friction which we are now acquiring. The weights which we shall then employ range from 7 Ibs. to 56 Ibs. Assuming the ordinary law of friction, we have found that 0-27 is the best value of its coefficient when the loads range between 14 Ibs. and 112 Ibs. Suppose we only consider loads up to 56 Ibs., we find that the coefficient 0*288 will best represent the experiments within this range, though for 112 Ibs. it would give an error of nearly 3 Ibs. The results calculated by the formula F = 0-288 J? are shown in Table V., where the greatest differ- ence is 0-7 Ib. 141. But we can replace the common law of friction by the more accurate law of Art. 137, and the formula computed so as to best harmonise the experiments up to 56 Ibs., disregarding the higher loads, is F 0-9 + 0-266 . This EXPERIMENTAL MECHANICS. [LECT. TABLE V. FRICTION. Friction of pine upon pine ; the mean values of the friction given in Table II. (corrected for the friction of the pulley) compared with the formula F = 0-288 R Number of Experiment. R. Total load on slide in Ibs. Corrected mean value of friction. F. Calculated value of friction. Discrepancies between the observed and calculated frictions I 2 14 28 47 8'2 4'0 8-1 -07 -O'l 3 4 42 56 I2'2 158 16-1 -O'l formula is obtained by the method referred to in Art. 137. We find that it represents the experiments better than that used in Table V. Between the limits named, this formula is also more accurate than that of Table IV. It is compared with the experiments in Table VI., and it will be noticed that it represents them with great precision, as no discrepancy exceeds o"i. TABLE VI. FRICTION. Friction of pine upon pine ; the mean values of the friction given in Table II. (corrected for the friction of the pulley) compared with the formula F = 0-9 + 0-266 R. Number of Experiment. R. Total Load on slide in Ibs. Corrected mean value of friction. F. Calculated value of friction. Discrepancies between the observed and calculated friction^ I 2 14 28 13 4-6 -o-i 3 4 42 56 1 2 '2 15-8 Is* -O'l O'O THE ANGLE OF FRICTION. 142. There is another mode of examining the action of friction besides that we have been considering. The appa- ratus for this purpose is shown in Fig. 33, in which B c represents the plank of pine we have already used. . It is THE ANGLE OF FRICTION. 79 8o EXPERIMENTAL MECHANICS. [LECT. now mounted so as to be capable of turning about one end B ; the end c is suspended from one hook of the chain from the " epicycloidal" pulley-block E. This block is very con- venient for the purpose. By its means the inclination of the plank can be adjusted with the greatest nicety, as the raising chain G is held in one hand and the lowering chain F in the other. Another great convenience of this block is that the load does not run down when the lifting chain is left free. The plank is clamped to the hinge about which it turns. The frames by which both the hinge and the block are supported are weighted in order to secure steadiness. The inclination of the plane is easily ascertained by measuring the difference in height of its two ends above the floor, and then making a drawing on the proper scale. The starting- screw D, whose use has been already mentioned, is also fastened to the frame-work in the position shown in the figure. 143. Suppose the slide A be weighted and placed upon the inclined plane B c ; if the end c be only slightly elevated, the slide remains at rest ; the reason being that the friction between the slide and the plane neutralizes the force of gravity. But suppose, by means of the pulley-block, c be gradually raised ; an elevation is at last reached at which the slide starts off, and runs with an accelerating velocity to the bottom of the plane. The angle of elevation of the plane when this occurs is called the angle of statical friction. 144. The weights with which the slide was laden in these experiments were 14 Ibs., 56 Ibs., and 112 Ibs., and the results are given in Table VII. We see that a load of 56 Ibs. started when the plane reached an inclination of 20'! in the first series, and of i7'2 in the second, the mean value i8'6 being given in the fifth column. These means for the three different weights v.] THE ANGLE OF FRICTION. Si agree so closely that we assert the remarkable law that the angle of friction does not depend upon the magnitude of the load. TABLE VII. ANGLE OF STATICAL FRICTION. A smooth plane of pine 72" x n" carries a loaded slide of pine 9" x 9"; one end of the plane is gradually elevated until the slide starts off. Number of Experiment. Total load en the sl.de in Ibs. Angle of elevation. is: Series. Angle of elevation. 2nd Series. Mean values of the angles. I H 1 9 '5 i 9 -5 2 56 20 'I if -2 i8-6 3 112 20- 3 1 8 -9 19-6 145. We might, however, proceed differently in deter- mining the angle of friction, by giving the load a start, and ascertaining if the motion will continue. To do so requires the aid of an assistant, who will start the load with the help of the screw, while the elevation of the plane is being slowly increased. The result of these experiments is given in Table VIII. TABLE VIII. ANGLE OF FRICTION. A smooth plane of pine 72" x n" carries a loaded slide of pine 9" x 9" ; one end of the plane is gradually elevated until the slide, having received a start, moves off uniformly. Number of Experiment. Total load on the slide in Ibs. Angle of inclination. I 2 3 H 56 112 14 '3 i3-o I3"o tVe see from this table also that the angle of friction is independent of the load, but the angle is in this case less by 5 or 6 than was found necessary to impart motion when a start was not given. G 8^ EXPERIMENTAL MECHANICS. [LECT. 146. It is commonly stated that the coefficient of friction equals the tangent of the angle of friction, and this can be proved to be true when the ordinary law of friction is assumed. But as we have seen that the law of friction is only approximately correct, we need not expect to find this other law completely verified. 147. When the slide is started, the mean value of the angle of friction is i3'4. The tangent of this angle is 0-24 : this is about n per cent, less than the coefficient of friction 0*27, which we have already determined. The mean value of the angle of friction when the slide is not started is i9'2, and its tangent is 0-35. The experiments of Table I. are, as already pointed out, rather unsatis- factory, but we refer to them here to show that, so far as they go, the coefficient of friction is in no sense equal to the tangent of the angle of friction. If we adopt the mean values given in the last column of Table I., the best coefficient of friction which can be deduced is 0-41. Whether, therefore, the slide be started or not started, the tangent of the angle of friction is smaller than the corre- sponding coefficient of friction. When the slide is started, the tangent is about 1 1 per cent, less than the coefficient ; and when the slide is not started, it is about 14 per cent, less. There are doubtless many cases in which these differ- ences are sufficiently small to be neglected, and in which, therefore, the law may be received as true. ANOTHER LAW OF FRICTION. 148. The area of the wooden slide is 9" x 9", but we would have found that the friction under a given load was practically the same whatever were the area of the slide, so long as its material remained unaltered. This follows as a consequence of v.j CONCLUDING REMARKS. 83 the approximate law that the friction is proportional to the pressure. Suppose that the weight were 100 Ibs., and the area of the slide 100 inches, there would then be a pressure cf i Ib. per square inch over the surface of the slide, and therefore the friction to be overcome on each square inch would be 0-27 Ib., or for the whole slide 27 Ibs. If, how- ever, the slide had only an area of 50 square inches, the load would produce a pressure of 2 Ibs., per square inch ; the friction would therefore be 2X0-27= 0-54 Ib. for each square inch, and the total friction would be 50X0-54 = 27 Ibs., the same as before : hence the total friction is independent of the extent of surface. This would remain equally true even though the weight were not, as we have supposed, uniformly distributed over the surface of the slide. CONCLUDING REMARKS. 149. The importance of friction in mechanics arises from its universal presence. We often recognize it as a destroyer or impeder of motion, as a waster of our energy, and as a source of loss or inconvenience. But, on the other hand, friction is often indirectly the means of producing motion, and of this we have a splendid example in the locomotive engine. - The engine being very heavy, the wheels are pressed closely to the rails ; there is friction enough to prevent the wheels slipping, consequently when the engines force the wheels to turn round they must roll onwards. The coefficient of friction of wrought iron upon wrought iron is about 0-2. Suppose a locomotive weigh 30 tons, and the share of this weight borne by the driving wheels be 10 tons, the friction between the driving wheels and the rails is 2 tons. This is the greatest force the engine can exert on a level line. A force of 10 Ibs. for every ton G 2 84 EXPERIMENTAL MECHANICS. [LECT. v. weight of the train is known to be sufficient to sustain the motion, consequently the engine we have supposed should draw along the level a load of 448 tons. 150. But we need not invoke the steam engine to show the use of friction. We could not exist without it. In the first place we could not move about, for walking is only possible on account of the friction between the soles of our boots and the ground ; nor if we were once in motion could we stop without coming into collision with some other object, or grasping something to hold on by. Objects could only be handled with difficulty, nails would not remain in wood, and screws would be equally useless. Buildings could hardly be erected, nay, even hills and mountains would gradually dis- appear, and finally dry land would be immersed beneath the level of the sea. Friction is, so far as we are concerned, quite as essential a law of nature as the law of gravitation. We must not seek to evade it in our mechanical discussions because it makes them a little more difficult. Friction obeys laws ; its action is not vague or uncertain. When inconvenient it can be diminished, when useful it can be increased ; and in our lectures on the mechanical powers, to which we now proceed, we shall have opportunities of describing machines which have been devised in obedience to its laws. 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. INTRODUCTION. 151. THE pulley forms a good introduction to the im- portant subject of the mechanical powers. But before entering on the discussions of the next few chapters, it will be necessary for us to explain what is meant in mechanics by "work," and by "energy, "which is the capacity for performing work, and we shall therefore include a short outline of this subject in the present lecture. 152. The pulley is a machine which is employed for the purpose of changing the direction of a force. We frequently wish to apply a force in a different direction from that in which it is convenient to exert it, and the pulley enables us to do so. We are not now speaking of these arrangements for increasing power in which pulleys play an important part ; these will be considered in the next lecture : we at present refer only to change of direction. In fact, as we shall shortly 86 EXPERIMENTAL MECHANICS. [LECT. see, some force is even wasted when the single fixed pulley is used, so that this machine certainly cannot be called a mechanical power. 153. The occasions upon which a single fixed pulley is used are numerous and familiar. Let us suppose a sack of corn has to be elevated from the lower to one of the upper ttories of a building. It may of course be raised by a man who carries it, but he has to lift his own weight in addition to that of the sack, and therefore the quantity of exertion used is greater than absolutely necessary. But supposing there be a pulley at the top of the building over which a rope passes ; then, if a man attach one end of the rope to the sack and pull the other, he raises the sack without raising his own weight. The pulley has thus provided the means by which the downward force has been changed in direction to an upward force. 154. The weights, ropes, and pulleys which are used in our windows for counterpoising the weight of the sash afford a very familiar instance of how a pulley changes the direction of a force. Here the downward force of the weight is changed by means of the pulley into an upward force, which nearly counterbalances the weight of the sash. FRICTION BETWEEN A ROPE AND AN IRON BAR. 155. Every one is familiar with the ordinary form of the pulley ; it consists of a wheel capable of turning freely on its axle, and it has a groove in its circumference in which the rope lies. But why is it necessary to give the pulley this form ? Why could not the direction of the rope be changed by simply passing it over a bar, as well as by the more com- plicated pulley? We shall best answer this question by actually trying the experiment, which we can do by means of the apparatus of Fig. 34 (see page 90).- In this are shown vi.] A ROPE AND AN IRON BAR. 87 two iron studs, G, H, o"-6 diameter, and about 8" apart ; over these passes a rope, which has a hook at each end. If I suspend a weight of 14 Ibs. from one hook A, and pull the hook B, I can by exerting sufficient force raise the weight on A, but with this arrangement I am conscious of having to exert a very much larger force than would have been necessary to raise 14 Ibs. by merely lifting it. 156. In order to study the question exactly, we shall ascertain what weight suspended from the hook B will suffice to raise A. I find that in order to raise 14 Ibs. on A no less than 47 Ibs. is necessary on B, consequently there is an enor- mous loss of force : more than two-thirds of the force which is exerted is expended uselessly. If instead of the 14 Ibs. weight I substitute any other weight, I find the same result, viz. that more than three times its amount is necessary to raise it by means of the rope passing over the studs. If a labourer, in raising a sack, were to pass a rope over two bars such as these, then for every stone the sack weighed he would have to exert a force of more than three stones, and there would be a very extravagant loss of power. 157. Whence arises this loss? The rope in moving slides over the -surface of the iron studs. Although these are quite smooth and polished, yet when there is a strain on the rope it presse~s closely upon them, and there is a certain amount of force necessary to make the rope slide along the iron. In other words, when I am trying to raise up 14 Ibs. with this contrivance, I not only have its weight opposed to me, but also another force due to the sliding of the rope on the iron : this force is due to friction. Were it not for friction, a force of 14 Ibs. on one hook would exactly balance 14 Ibs. on the other, and the slightest addition to either weight would make it descend and raise the other. If, then, we are obliged to change the direction of a force, we must devise some means 88 EXPERIMENTAL MECHANICS. [LECT. of doing so which does not require so great a sacrifice as the arrangement we have just used. THE USE OF THE PULLEY. 158. We shall next inquire how it is that we are enabled to obviate friction by means of a pulley. It is evident we must provide an arrangement in which the rope shall not be required to slide upon an iron surface. This end is attained by the pulley, of which we may take i, Fig. 34, as an example. This represents a cast iron wheel 14" in diameter, with a V~ sna P e d groove in its circumference to receive the rope : this wheel turns on a f -inch wrought iron axle, which is well oiled. The rope used is about o"'2$ in diameter. 159. From the hooks E, F at each end of the rope a i4lb. weight is suspended. These equal weights balance each other. According to our former experiment with the studs, it would be necessary for me to treble the weight on one of these hooks in order to raise the other, but now I find that an additional 0-5 Ib. placed on either hook causes it to descend and make the other ascend. This is a great improvement ; 0-5 Ib. now accomplishes what 33 Ibs. was before required for. We have avoided a great deal of friction, but we have not got rid of it altogether, for 0-25 Ib. is incompetent, when added to either weight, to make that weight descend. 160. To what is the improvement due ? When the weight descends the rope does not slide upon the wheel, but it causes the wheel to revolve with it, consequently there is little or no friction at the circumference of the pulley ; the friction is transferred to the axle. We still have some resistance to overcome, but for smooth oiled iron axles the friction is very small, hence the advantage of the pulley. vi.] LARGE AND SMALL PULLEYS. 89 There is in every pulley a small loss of power from the force expended in bending the rope ; this need not concern us at present, for with the pliable plaited rope that we have employed the effect is inappreciable, but with large strong ropes the loss becomes of importance. The amount of loss by using different kinds of ropes has been determined by careful experiments. LARGE AND SMALL PULLEYS. 161. There is often a considerable advantage obtained by using large rather than small pulleys. The amount of force necessary to overcome friction varies inversely as the size of the pulley. We shall demonstrate this by actual experiment with the apparatus of Fig. 34. A small pulley K is attached to the large pulley i ; they are iri fact one piece, and turn together on the same axle. Hence if we first determine the friction with the rope over the large pulley, and then with the rope over the small pulley, any difference can only be due to the difference in size, as all the other circumstances are the same. 162. In making the experiments we must attend to the following point. The pulleys and the socket on which they are mounted weigh several pounds, and consequently there is friction on the axle arising from the weight of the pulleys, quite independently of any weights that may be placed on the hooks. We must then, if possible, evade the friction of the pulley itself, so that the amount of friction which is observed will be entirely due to the weights raised. This can be easily done. The rope and hooks being on the large pulley i, I find that 0*16 Ib. attached to one of the hooks, E, is sufficient to overcome the friction of the pulley, and to make that hook descend and raise F. If therefore we leave 0-16 Ib. on E, we may consider the EXPERIMENTAL MECHANICS. [LECT. friction due to the weight of the pulley, rope, and hooks as neutralized. 163. I now place a stone weight on each of the hooks E and F. The amount necessary to make the hook E and its load descend is 0-28 Ib. This does not of course include the weight of 0-16 Ib. already referred to. We see therefore that with the large pulley the amount of friction to be overcome in raising one stone is 0-28 Ib. 164. Let us now perform precisely the same experiment with the small pulley. I transfer the same rope and hooks VI.] LARGE AND SMALL PULLEYS. 91 to K, and I find that 0-16 Ib. is not now sufficient to over- come the friction of the pulley, but I add on weights until c will just descend, which occurs when the load reaches 0-95 Ib. This weight is to be left on c as a counterpoise, for the reasons already pointed out. I place a stone weight on c and another on D, and you see that c will descend when it receives an additional load of 1*35 Ibs. ; this is therefore the amount of friction to be overcome when a stone weight is raised over the pulley K. 165. Let us compare these results with the dimensions of the pulleys. The proper way to measure the effective circumference of a pulley when carrying a certain rope is to measure the length of that rope which will just embrace it. The length measured in this way will of course depend to a certain extent upon the size of the rope. I find that the circumferences of the two pulleys are 43"'o and g"$. The ratio of these is 4*5 ; the corresponding resistances from friction we have seen to be o'28 Ib. and 1*35 Ibs. The larger of these quantities is 4 '8 times the smaller. This number is very close to 4*5 ; we must not, as already explained, expect perfect accuracy in experiments in friction. In the present case the agreement is within the i-i6th of the whole, and we may regard it as a proof of the law that the friction of a pulley is inversely proportional to its circumference. 1 66. It is easy to see the reason why friction should diminish when the size of the pulley is increased. The friction acts at the circumference of the axle about which the wheel turns ; it is there present as a force tending to retard motion. Now the larger the wheel the greater will be the distance from the axis at which the force acts which overcomes the friction, and therefore the less need be the magnitude of the force. You will perhaps understand 9 2 EXPERIMENTAL MECHANICS. [LECT. this better after the principle of the lever has been discussed. 167. We may deduce from these considerations the prac- tical maxim that large pulleys are economical of power. This rule is well known to engineers ; large pulleys should be used, not only for diminishing friction, but also to avoid loss of power by excessive bending of the rope. A rope is bent gradually around the circumference of a large pulley with far less force than is necessary to accommodate it to a smaller pulley : the rope also is apt to become injured by excessive bending. In coal pits the trucks laden with coal are hoisted to the surface by means of wire ropes which pass from the pit over a pulley into the engine-house : this pulley is of very large dimensions, for the reasons we have pointed out. THE LAW OF- FRICTION IN THE PULLEY. 168. I have here a wooden pulley 3^-5 in diameter ; the hole is lined with brass, and the pulley turns very freely on an iron spindle. I place the rope and hooks upon the groove. Brass rubbing on iron has but little friction, and when 7 Ibs. is placed on each hook, 0*5 Ib. added to either will make it descend and raise up the other. Let 14 Ibs. be placed on each hook, 0-5 Ib. is no longer sufficient ; i Ib. is required : hence when the weight is doubled the friction is also doubled. Repeating the experiment with 21 Ibs. and 28 Ibs. on each side, the corresponding weights necessary to overcome friction are 1*5 Ibs. and 2 Ibs. In the four experiments the weights used are in the proportion i, 2, 3, 4 ; and the forces necessary to overcome friction, 0^5 Ib., i Ib., 1-5 Ibs., and 2 Ibs., are in the same proportion. Hence the friction is proportional to the load. WHEELS. 169. The wheel is one of the most simple and effective vi.] WHEELS. 93 devices for overcoming friction. A sleigh is an admirable vehicle on a smooth surface such as ice, but it is totally unadapted for use on common roads ; the reason being that the amount of friction between the sleigh and the road is so great that to move the sleigh the horse would have to exert a force which would be very great compared with the load he was drawing. But a vehicle properly mounted on wheels moves with the greatest ease along the road, for the circum- ference of the wheel does not slide, and consequently there is no friction between the wheel and the road ; the wheel however turns on its axle, therefore there is sliding, and consequently friction, at the axle, but the axle and the wheel are properly fitted to each other, and the surfaces are lubricated with oil, so that the friction is extremely small. 170. With large wheels the amount of friction on the axle is less than with small wheels ; other advantages of large wheels are that they do not sink much into depres- sions in the roads, and that they have also an increased facility in surmounting the innumerable small obstacles from which even the best road is not free. 171. When it is desired to make a pulley turn with extremely small friction, its axle, instead of revolving in fixed bfarings, is mounted upon what are called friction wheels. A set of friction wheels is shown in the apparatus of Fig. 66 : as the axle revolves, the friction between the axles and the wheels causes the latter to turn round with a comparatively slow motion; thus all the friction is trans- ferred to the axles of the four friction wheels ; these revolve in their bearings with extreme slowness, and consequently the pulley is but little affected by friction. The amount of friction in a pulley so mounted may be understood from the following experiment. A silk cord is placed on the pulley, 94 EXPERIMENTAL MECHANICS. [LECT. and i Ib. weight is attached to each of its ends : these of course balance. A number of fine wire hooks, each weigh- ing o-ooi Ib., are prepared, and it is found that when a weight of 0-004 Ib. is attached to either side it is sufficient to overcome friction and set the weights in motion. ENERGY. 172. In connection with the subject of friction, and also as introductory to the mechanical powers, the notion of "work," or as it is more properly called "energy," is of great importance. The meaning of this word as employed in mechanics will require a little consideration. 173. In ordinary language, whatever a man does that can cause fatigue, whether of body or mind, is called work. In mechanics, we mean by energy that particular kind of work which is directly or indirectly equivalent to raising weights. 174. Suppose a weight is lying on the floor and a stool is standing beside it : if a man raise the weight and place it upon the stool, the exertion that he expends is energy in the sense in which the word is used in mechanics. The amount of exertion necessary to place the weight upon the stool de- pends upon two things, the magnitude of the weight and the height of the stool. It is clear that both these things must be taken into account, for although we know the weight which is raised, we cannot tell the amount of exer- tion that will be required until we know the height through which it is to be raised ; and if we know the height, we can- not appreciate the quantity of exertion until we know the weight. 175. The following plan has been adopted for expressing quantities of energy. The small amount of exertion necessary to raise i Ib. avoirdupois through one British vi.] ENERGY. 95 foot is taken as a standard, compared with which all other quantities of energy are estimated. This quantity of exer- tion is called in mechanics the unit of energy, and some- times also the "foot-pound." 176. If a weight of i Ib. has to be raised through a height of 2 feet, or a weight of 2 Ibs. through a height of i foot, it will be necessary to expend twice as much energy as would have raised a weight of i Ib. through i foot, that is, 2 foot- pounds. If a weight of 5 Ibs. had to be raised from the floor up to a stool 3 feet high, how many units of energy would be re- quired? To raise 5 Ibs. through i foot requires 5 foot- pounds, and the process must be again repeated twice before the weight arrive at the top of the stool. For the whole operation 15 foot-pounds will therefore be neces- sary. If 100 Ibs. be raised through 20 feet, too foot-pounds of energy is required for the first foot, the same for the second, third, &c., up to the twentieth, making a total of 2,000 foot-pounds. Here is a practical question for the sake of illustration. Which would it be preferable to hoist, by a rope passing over a single fixed pulley, a trunk weighing 40 Ibs. to a height of 20 feet, or a trunk weighing 50 Ibs. to a height of 1 5 feet ? We shall find how much energy would be necessary in each case : 40 times 20 is 800 ; therefore in the first case the energy would be 800 foot-pounds. But 50 times 15 is 750 ; therefore the amount of work, in the second case, is only 750 Ibs. Hence it is less exertion to carry 50 Ibs. up 15 feet than 40 Ibs. up 20 feet. 177. The rate of working of every source of energy, whether it lie in the muscles of men or other animals, in water-wheels, steam-engines, or other prime movers, is to 96 EXPERIMENTAL MECHANICS. [LECT. be measured by the number of foot-pounds produced in the unit of time. The power of a steam-engine is denned by its equivalent in horse-power. For example, it is meant that a steam- engine of 3 horse-power, could, when working for an hour, do as much work as 3 horses could do when working for the same time. The power of a horse is, however, an un- certain quantity, differing in different animals and not quite uniform in the same individual; accordingly the selection of this measure for the efficiency of the steam- engine is inconvenient. We replace it by a convenient standard horse-power, which is, however, a good deal larger than that continuously obtainable from any ordinary horse. A one horse-power steam-engine is capable of accomplishing 33,000 foot-pounds per minute. 178. We shall illustrate the numerical calculation of horse-power by an example : if a mine be 1,000 feet deep, how much water per minute would a 50 horse-power engine be capable of raising to the surface? The engine would yield 50 x 33,000 units of work per minute, but the weight has to be raised 1,000 feet, consequently the number of pounds of water raised per minute is 5 X 33.0QO = Ij650 . 1,000 179. We shall apply the principle of work to the con- sideration of the pulley already described (p. 90). In order to raise the weight of 14 Ibs., it is necessary that the rope to which the power is applied should be pulled downwards by a force of 15 Ibs., the extra pound being on account of the friction. To fix our ideas, we shall suppose the 1 4 Ibs. to be raised i foot ; to lift this load directly, with- out the intervention of the pulley, 14 foot-pounds would be necessary, but when it is raised by means of the pulley, 15, vi.] ENERGY. 97 foot-pounds are necessary. Hence there is an absolute loss of ^-th of the energy when the pulley is used. If a steam- engine of i horse-power were employed in raising weights by a rope passing over a pulley similar to that on which we have experimented, only -rfths of the work would be use- fully employed ; but we find 33,000 x = 30,800. The engine would therefore perform 30,800 foot-pounds of useful work per minute. 1 80. The effect of friction on a pulley, or on any other machine, is always to waste energy. To perform a piece of work directly requires a certain number of foot-pounds, while to do it by a machine requires more, on account of the loss by friction. This may at first sight appear somewhat paradoxical, as it is well known that, by levers, pulleys, &c., an enormous mechanical advantage may be gained. This subject will be fully explained in the next and following lectures, which relate to the mechanical powers. 181. We shall conclude with a few observations on a point of the greatest importance. We have seen a case where 15 foot-pounds of energy only accomplished 14 foot- pounds of- work, and thus i foot-pound appeared to be lost. We say that this was expended upon the friction ; but what is the friction ? The axle is gradually worn away by rub- bing in its bearings, and, if it be not properly oiled, it becomes heated. The amount of energy that seems to dis- appear is partly expended in grinding down the axle, and is partly transformed into heat ; it is thus not really lost, but unfortunately assumes a form which we do not require and in which it is rather injurious than otherwise. Indeed we know that energy cannot be destroyed, however it may be H 98 EXPERIMENTAL MECHANICS. [LECT. vi. transformed; if it disappear in one shape, it is only to reappear in another. A so-called loss of energy by friction only means a diversion of a part of the work to some pur- pose other than that which we wish to accomplish. It has long been known that matter is indestructible : it is now equally certain that the same may be asserted of energy. LECTURE VII. THE PULLEY-BLOCK. Introduction. The Single Moveablc Pulley. The Three-sheave Pulley-block. The Differential Pulley-block. The Epicycloidal Pulley-block. INTRODUCTION. 182. IN the first lecture I showed how a large weight could be raised by a smaller weight, and I stated that this subject would again occupy our attention. I now fulfil this promise. The questions to be discussed involve the most advantageous methods of employing a small force to overcome" a greater. Here is a subject of practical impor- tance. A man of average strength cannot raise more than a hundredweight without great exertion, yet the weights which it is necessary to lift and move about often weigh many hundredweights, or even many tons. It is not always practicable to employ numerous hands for the purpose, nor is a steam-engine or other great source of power at all times available. But what are called the mechanical powers enable the forces at our disposal to be greatly increased. One man, by their aid, can exert as much force as several H 2 loo EXPERIMENTAL MECHANICS. [LECT. could without such assistance; and when they are employed to augment the power of several men or of a steam-engine, gigantic weights, amounting sometimes to hundreds of tons, can be managed with facility. 183. In the various arts we find innumerable cases where great resistances have to be overcome ; we also find a cor- responding number and variety of devices contrived by human skill to conquer them. The girders of an iron bridge have to be lifted up to their piers ; the boilers and engines of an ocean steamer have to be placed in position ; a great casting has to be raised from its mould ; a railway locomotive has to be placed on the deck of a vessel for transit ; a weighty anchor has to be lifted from the bottom of the sea ; an iron plate has to be rolled or cut or punched : for all of these cases suitable arrangements must be devised in order that the requisite power may be obtained. 184. We know but little of the means which the ancients employed in raising the vast stones of those buildings whick travellers in the East have described to us. It is sometimes thought that a large number of men could have transported these stones without the aid of appliances which we would now use for a similar purpose. But it is more likely that some of the mechanical powers were used, as, with a multi- tude of men, it is difficult fo ensure the proper application of their united strength. In Easter Island, hundreds of miles distant from civilised land, and now inhabited by savages, vast idols of stone have been found in the hills which must have been raised by human labour. It is useless to speculate on the extinct race by whom this work was achieved, or on the means they employed. 185. The mechanical powers are usually enumerated as follows : The pulley, the lever, the wheel and axle, the wedge, the inclined plane, the screw. These different powers are so VIL] THE SINGLE MOVEABLE PULLEY. 101 frequently used in combination that the distinctions cannot be always maintained. The classification will, however, suffice to give a general notion of the subject. 186. Many of the most valuable mechanical powers are machines in which ropes or chains play an important part. Pulleys are usually employed wherever it is necessary to change the direction of a rope or chain which is transmitting power. In the present lecture we shall examine the most important mechanical powers that are produced by the combination of pulleys. THE SINGLE MOVEABLE PULLEY. 187. We commence wit a the most simple case, that of the single moveable pulley (Fig. 35). The rope is firmly secured at one end A; it then passes down under the move- able pulley B, and upwards over a fixed pulley. To the free end, which depends from the fixed pulley, the power is applied while the load to be raised is suspended from the moveable pulley. We shall first study the relation between the power and the load in a simple way, and then we shall describe a few exact experiments. 1 88. When the load is raised the moveable pulley itself must of course be also raised, and a part of the power is expended* for this purpose. But we can eliminate the weight of the moveable pulley, so far as our calculations are con- cerned, by first attaching to the power end of the rope a sufficient weight to lift up the moveable pulley when not carrying a load. The weight necessary for doing this is found by trial to be a little over 1-5 Ibs. So that when a load is being raised we must reduce the apparent power by 1-5 Ibs. to obtain the power really effective. 189. Let us suspend 14 Ibs. from the load hook at B, and ascertain what power will raise the load. We leave the weight EXPERIMENTAL MECHANICS. [LECT. of the moveable pulley and i '5 Ibs. of the power at c out of consideration. I then find by experiment that 7 Ibs. of effec- tive power is not sufficient to raise the load, but if one pound more be added, the power descends, and the load is raised. FlQ. 35. Here, then, is a remarkable result ; a weight of 8 Ibs. has overcome 14 Ibs. In this we have the first application of the mechanical powers to increase our available forces. 190. Let us examine the reason of this mechanical advantage. If the load be raised one foot, it is plain that the power must descend two feet : for in order to raise the vii.] THE SINGLE MOVEABLE PULLEY. 103 load the two parts of the rope descending to the moveable pulley must each be shortened one foot, and this can only be done by the power descending two feet. Hence when the load of 14 Ibs. is lifted by the machine, for every foot it is raised the power must descend two feet : this simple point leads to a conception of the greatest importance, on which depends the efficiency of the pulley. In the study of the mechanical powers it is essential to examine the number of feet through which the power must act in order to raise the load one foot : this number we shall always call the velocity ratio. 191. To raise 14 Ibs. one foot requires 14 foot-pounds of energy. Hence, were there no such, thing as friction, 7 Ibs. on the power hook would be sufficient to raise the load ; because 7 Ibs. descending through two feet yields 14 foot- pounds. But there is a loss of energy on account of friction, and a power of 7 Ibs. is not sufficient : 8 Ibs. are necessary. Eight Ibs. in descending two feet performs 1 6 foot-pounds ; of these only 14 are utilised on the load, the remainder being the quantity of energy that has been diverted by friction. We learn, then, that in the moveable pulley the quantity of energy employed is really greater than that which would lift the weight directly, but that the actual force which has to be exerted is less. 192. Suppose that 28 Ibs. be placed on the load hook, a few trials assure us that a power of 16 Ibs. (but not less) will be sufficient for motion ; that is to say, when the load is doubled, we find, as we might have expected, that the power must be doubled also. It is easily seen that the loss of energy by friction then amounts to 4 foot-pounds. We thus verify, in the case of the moveable pulley, the approximate law that the friction is proportional to the load. 193. By means of a moveable pulley a man is able to raise a weight nearly double as great as he could lift 104 EXPERIMENTAL MECHANICS. [LECT. directly. From a series of careful experiments it has been found that when a man is employed in the particular exer- tion necessary for raising weights over a pulley, he is able to work most efficiently when the pull he is required to make is about 40 Ibs. A man could, of course, exert greater force than this, but in an ordinary day's work he is able to perform more foot-pounds when the pull is 40 Ibs. than when it is larger or smaller. If therefore the weights to be lifted amount to about So Ibs., energy may really be econo- mized by the use of the single moveable pulley, although by so doing a greater quantity of energy would be actually expended than would have been necessary to raise the weights directly. 194. Some experiments on larger loads have been tried with the moveable pulley we have just described ; the results are recorded in Table IX. TABLE IX. SINGLE MOVEABLE PULLEY. Moveable pulley of cast iron 3" "25 diameter, groove o" "6 wide, wrought iron axle o"'6 diameter ; fixed pulley of cast iron 5" diameter, groove o"'4 wide, wrought iron axle o""6 diameter, axles oiled ; flexible plaited rope o"'25 diameter ; velocity ratio 2, mechanical efficiency I '8, useful effect 90 per cent. ; formula P = 2'2i + 0-5453 A". Discrepancies be- Number of Experiment. K. Load in Ibs. Observed power in Ibs. Calculated power i.n Ibs. tween observed and calculated powers. 1 28 i7'5 175 O'O 2 57 33'5 33 '3 - 0-2 3 ^5 485 48-6 + O'l 4 "3 64 o 638 - O'2 5 142 80-0 796 - 0'4 6 170 94-5 94-9 + o'4 I ic,8 226 110-5 125-5 IIO'2 I25-5 - 03 O'O The dimensions of the pulleys are precisely stated because, for pulleys of different construction, the numerical VIL] THE SINGLE MOVEABLE PULLEY. 105 coefficients would not necessarily be the same. An attentive study of this table will, however, show the general character of the relation between the power and the load in all arrangements of this class. The table consists of five columns. The first contains merely the numbers of the experiments for convenience of reference. In the second column, headed ^?, the loads, expressed in pounds, which are raised in each experiment, are given ; that is, the weight attached to the hook, not including the weight of the lower pulley. The weight of this pulley is not included in the stated loads. In the third column the powers are recorded, which were found to be sufficient to raise the corresponding loads in the second column. Thus, in experiment 7, it is found that a power of 110-5 Ibs. will be sufficient to raise a load of 198 Ibs. The third column has thus been determined by gradually increasing the power until motion begins. 195. From an examination of the columns showing the power and the load, we see that the power always amounts to more than half the load. The excess is partly due to a small portion of the power (about 1-5 Ibs.) being employed in raising the lower block, and partly to friction. For example, in experiment 7, if there had been no friction and if the knver block were without weight, a power of 99 Ibs. would have been sufficient ; but, owing to the presence of these disturbing causes, 110-5 Ibs. are necessary: of this amount 1-5 Ibs. is due to the weight of the pulley, 10 Ibs. is the force of friction, and the remaining 99 Ibs. raises the load. 196. By a calculation based on this table we have ascertained a certain relation between the power and the load ; they are connected by the formula which may be enunciated as follows The power is found by multiplying the weight of the load 106 EXPERIMENTAL MECHANICS. [LECT. into 0-5453, and adding 2*21 to the product. Calling P the power and R the load, we may express the relation thus : P = 2-2i -f 0-5453 R. For example, in experiment 5, the product of 142 and 0-5453 is 77-43, to which, when 2'2i is added, we find for P 79-64, very nearly the same ar, 80 Ibs., the observed value of the power. In the fourth column the values of P calculated by means of this formula are given, and in the last we exhibit the discrepancies between the observed and the calculated values for the sake of comparison. It will be seen that the discrepancy in no case amounts to 0-5 lb., consequently the formula expresses the experiments very well. The mode of deducing it is given in the Appendix. 197. The quantity 2 '21 is partly that portion of the power expended in overcoming the weight of the moveable pulley, and partly arises from friction. 198. We can readily calculate from the formula how much power will be required to raise a given weight ; for example, suppose 200 Ibs. be attached to the moveable pulley, we find that in Ibs. must be applied as the power. But in order to raise 200 Ibs. one foot, the power exerted must act over two feet ; hence the number of foot-pounds required is 2 x in = 222. The quantity of energy that is lost is 22 foot-pounds. Out of every 222 foot-pounds applied, 200 are usefully employed ; that is to say, about 90 per cent, of the applied energy is utilized, while the remaining 10 per cent, is lost by friction. THE THREE-SHEAVE PULLEY-BLOCK. 199. The next arrangement we shall employ is a pair of pulley-blocks s T, Fig. 35, each containing three sheaves, as the small pulleys are termed. A rope is fastened to the upper block, s ; it then passes down to the "lower block T vii.] THE THREE-SHEAVE PULLEY-BLOCK. 107 under one sheave, up again to the upper block and over a sheave, and so on, as shown in the figure. To the end of the rope from the last of the upper sheaves the power H is applied, and the load G is suspended from the hook attached to the lower block. When the rope is pulled, it gradually raises the lower block and to raise the load one foot, each of the six parts of the rope from the upper block to the lower block must be shortened one foot, and therefore the power must have pulled out six feet of rope. Hence, for every foot that the load is raised the power must have acted through six feet ; that is to say, the Telocity ratio is 6. 200. If there were no friction, the power would only be one-sixth of the load. This follows at once from the prin- ciples already explained. Suppose the load be 60 Ibs., then to raise it one foot would require 60 foot-pounds ; and the power must therefore exert 60 foot-pounds ; but the power moves over six feet, therefore a power of 10 Ibs. would be sufficient. Owing, however, to friction, some energy is lost, and we must have recourse to experiment in order to test the real efficiency of the machine. The single moveable pulley nearly doubled our power ; we shall prove that the three-sheave pulley-block will quadruple it. In this case we deal with larger weights, with reference to which we may leave the weight cf the lower block out of consideration. 20 1. Let us first attach i cwt. to the load hook ; we find that 29 Ibs. on the power hook is the smallest weight that can produce motion : this is only i Ib. more than one-quarter of the load raised. If 2 cwt. be the load, we find that 56 Ibs. will just raise it : this time the power is exactly one- puarter of the load. The experiment has been tried of placing 4 cwt. on the hook ; it is then found that 109 Ibs. will raise it, which is only 3 Ibs. short of i cwt. These experiments demonstrate that for a three-sheave pulley- io8 EXPERIMENTAL MECHANICS. [LECT. block of this construction we may safely apply th rule, that the power is one-quarter of the load. 202. We are thus enabled to see how much of our ex- ertion in raising weights must be expended in merely over- coming friction, and how much may be utilized. Suppose for example that we have to raise a weight of 100 Ibs. one foot by means of the pulley-block ; the power we must apply is 25 Ibs., and six feet of rope must be drawn out from between the pulleys: therefore the power exerts 150 foot-pounds of energy. Of these only 100 foot-pounds are usefully employed, and thus 50 foot-pounds, one-third of the whole, have been expended on friction. Here we see that notwithstanding a small force overcomes a large one, there is an actual loss of energy in the machine. The real advantage of course is that by the pulley-block I can raise a greater weight than I could move without assistance, but I do not create energy ; I merely modify it, and lose by the process. 203. The result of another series of experiments made with this pair of pulley-blocks is given in Talle X. TABLE X. THREE-SHEAVE PULLEY-BLOCKS. Sheaves cast iron 2" '5 diameter; plaited rope o"'25 diameter; velocity ratio 6 ; mechanical advantage 4 ; useful effect 67 per cent. ; formula /'= 2-36 + 0-238 A'. Number of Experiment. R. Load in Ibs. Observed power in Ibs. p. Calculated pjwer in Ibs. Discrepancies between observed and calculated power-. I 57 I5'5 I5'9 + 0-4 2 114 29'5 29'5 O 3 171 4T5 43'i - 04 4 228 56-0 566 + O6 281 70 -o 69-2 - 0-8 6 338 83-0 82-8 - 0-2 7 395 97 - o 964 -0-6 8 452 109 - o 109-9 + 0-9 vii.] THE THREE SHEAVE PULLEY-BLOCK. 109 204. This table contains five columns ; the weights raised (shown in the second column) range up to somewhat over 4 cwt. The observed values of the power are given in the third column ; each of these is generally about one-quarter of the corresponding value of the load. There is, however, a more accurate rule for finding the power ; it is as follows. 205. To find the power necessary to raise a given load, multiply the loads in Ibs. by 0*238, and add 2-36 Ibs. to the product. We may express the rule by the formula P= 2-36 + 0-238 R. 206. To find the power which would raise 228 Ibs. : the product of 228 and 0-238 is 54-26; adding 2-36, we find 56-6 Ibs. for the power required ; the actual observed power is 56 Ibs., so that the rule is accurate to within about half a pound. In the fourth column will be found the values of P calculated by means of this rule. In the fifth column, the discrepancies between the observed and the calculated values of the powers are given, and it will be seen that the diffe- rence in no case reaches i Ib. Of course it will be understood that this formula is only reliable for loads which lie between those employed in the first and last of the experiments. We can calculate the power for any load between 57 Ibs. and 452 Ibs., but for loads much larger than 452 or less than 57 it would probably be better to use the simple fourth of the load rather than the power computed by the formula. 207. I will next perform an experiment with the three- sheave pulley-block, which will give an insight into the exact amount of friction without calculation by the help of the velocity ratio. We first counterpoise the weight of the lower block by attaching weights to the power. It is found that about r6 Ibs. is sufficient for this purpose. I attach a 56 Ib. weight as a load, and find that 13-1 Ibs. ii> sufficient power for motion. This amount is partly com i io EXPERIMENTAL MECHANICS. [LECT. posed of the force necessary to raise the load if there were no friction, and the rest is due to the friction. I next gradually remove the power weights : when I have taken off a pound, you see the power and the load balance each other; but when I have reduced the power so low as 5^5 Ibs. (not including the counterpoise for the lower block), the load is just able to overhaul the power, and run down. We have therefore proved that a power of 13-1 Ibs. or greater raises 56 Ibs. , that any power between 13-1 Ibs. and 5-5 Ibs. balances 56 Ibs., and that any power less than 5*5 Ibs. is raised by 56 Ibs. When the power is raised, the force of friction, together with the power, must be overcome by the load. Let us call X the real power that would be necessary to balance 56 Ibs. in a perfectly frictionless machine, and Y the force of friction. We shall be able to determine X and F by the experiments just performed. When the load is raised a power equal to X + Y must be applied, and therefore X + Y= 13'!. On the other hand, when the power is raised, the force X is just sufficient to overcome both the friction Y and the weight 5-5 ; therefore X= Y + 5-5. Solving this pair of equations, we find that X = 9-3 and K= 3-8. Hence we infer that the power in the frictionless machine would be 9*3 ; but this is exactly what would have been deduced from the velocity ratio, for 56 + 6 9-3 Ibs. In this result we find a perfect accordance between theory and experiment. THE DIFFERENTIAL PULLEY-BLOCK. 208. By increasing the number of sheaves in a pair of pulley-blocks the power may be increased ; but the length of rope (or chain) requisite for several sheaves becomes a practical inconvenience. There are also other reasons vii.] THE DIFFERENTIAL PULLEY-BLOCK. in which make the differential pulley-block, which we shall now consider, more convenient for many purposes than the common pulley blocks when a considerable augmentation of power is required. 209. The principle of the differential pulley is very ancient, and in modern times it has been embodied in a machine of practical utility. The object is to secure, that while the power moves over a considerable distance, the load shall only be raised a short distance. When this has been attained, we then know by the principle of energy that we have gained a mechanical advantage. 210. Let us consider the means by which this is effected in that ingenious contrivance, Weston's differential pulley- block. The principle of this machine will be understood from Fig. 36 and Fig. 37. It consists of three parts,- moveable pulley, and an endless chain. We shall briefly describe them. The upper block p is furnished with a hook for attachment to a support. The sheave it contains resembles two sheaves,~one a little smaller than the other, fastened together : they are in fact one piece. The grooves are provided with ridges, adapted to prevent the chain from slipping. The lower pulley Q consists of one sheave, which is also furnished with a groove -an upper pulley-block, a hook, 112 EXPERIMENTAL MECHANICS. [LECT. which the load is attached. The endless chain performs a part that will be understood from the sketch of the principle in Fig. 36. The chain passes from the hand at A up to L over the larger groove in the upper pulley, then downwards at B, under the lower pulley, up again at c, over the smaller groove in the upper pulley at A, and then back again by D to the hand at A. When the hand pulls the chain downwards, the two grooves of the upper pulley begin to turn together in the direction shown by the arrows on the chain. The large groove is therefore winding up the chain, while the smaller groove is lowering. 211. In the pulley which has been employed in the experiments to be described, the effective circumference of the large groove is found to be 1 i"'84, while that of the small groove is io"-^6. When the upper pulley has made one revolution, the large groove must have drawn up u"'84 of chain, since the chain cannot slip on account of the ridges ; but in the same time the small groove has lowered io"'36 of chain : hence when the upper pulley has revolved once, the chain between the two must have been shortened by the difference between n"'84 and io"-^6, that is by i"'48 ; but this can only have taken place by raising the moveable pulley through half i*'48, that is, through a space o"'74. The power has then acted through n' /< 84, and has raised the resistance o // 74. The power has therefore moved through a space 16 times greater than that through which the load moves. In fact, it is easy to verify by actual trial that the power must be moved through 16 feet in order that the load may be raised i foot. We express this by saying that the velocity ratio is 16. 212. By applying power to the chain at D proceeding from the smaller groove, the chain is lowered by the large groove faster than it is raised by the small one, and the lower vii.] THE DIFFERENTIAL PULLEY-BLOCK. 113 pulley descends. The load is thus raised or lowered by simply pulling one chain A or the other D. 213. We shall next consider the me- chanical efficiency of the differential pulley-block. The block (Fig. 37) which we shall use is intended to be worked by one man, and will raise any weight not exceeding a quarter of a ton. We have already learned that with this block the power must act through six- teen feet for the load to be raised one foot. Hence, were it not for friction, the power need only be the sixteenth part of the load. A few trials will show us that the real efficiency is not so large, and that in fact more than half the work exerted is merely expended upon overcoming friction. This will lead afterwards to a result of considerable practical import- ance. 214. Placing upon the load-hook a weight of 200 Ibs., I find that 38 Ibs. attached to a hook fastened on the power- chain -is sufficient to raise the load ; that is to say, the power is about one-sixth of the load. If I make the load 400 Ibs. I find the requisite power to be 64 Ibs., which is only about 3 Ibs. less than one-sixth of 400 Ibs. We may safely adopt the practical rule, that with this differential pulley-block a man would be able to raise a. weight six times as great as he could raise without such assistance. 215. A series of experiments carefully tried with different loads have given the results shown in Table XI. FIG. 37. EXPERIMENTAL MECHANICS. [LECT. TABLE XL THE DIFFERENTIAL PULLEY-BLOCK. Circumference of large groove u"'84, of small groove io"'36 ; velocity ratio 1 6 ; mechanical efficiency 6 '07 ; useful effect 38 per cent ; formula P = 3*87 + 0-1508 R. Number cf Experiment. /?. Load in Ibs. Observed power in Ibs. p. Calculated power in Ibs. Difference of the observed and calculated values. j 56 IO 12-3 + 2-T. 2 112 20 20-8 + 0'8 3 168 31 29-2 - 1-8 4 224 38 377 - 0-3 280 48 46-1 - I-Q I 336 392 448 g 72 54-6 63-1 71'S + 0-6 - 0-9 -0-5 9 504 80 80-0 O'O ,0 86 88-4 + 2-4 The first column contains the numbers of the experiments, the second the weights raised, the third the observed values of the corresponding powers. From these the following rule for finding the power has been obtained : 216. To find the power, multiply the load by 0-1508, and add 3-87 Ibs. to the product; this rule may be expressed by the formula P = 3-87 + 0-1508 ^?. (See Appendix.) 217. The calculated values of the powers are given in the fourth column, and the differences between the observed and calculated values in the last column. The differences do not in any case amount to 2-5 Ibs., and considering that the loads raised are up to a quarter of a ton, the formula represents the experiments with satisfactory precision. 218. Suppose for example 280 Ibs. is to be raised; the product of 280 and 0-1508 is 42-22, to which, when 3-87 is vii.] THE DIFFERENTIAL PULLEY-BLOCK. 115 added, we find 46-09 to be the requisite power. The mechanical efficiency found by dividing 46*09 into 280 is 6*07. 219. To raise 280 Ibs. one foot 280 foot-pounds of energy would be necessary, but in the differential pulley- block 46-09 Ibs. must be exerted for a distance of 16 feet in order to accomplish this object. The product of 46-09 and 1 6 is 737-4. Hence the differential pulley-block requires 737-4 foot-pounds of. energy to be applied in order to yield 280 useful foot-pounds ; but 280 is only 38 per cent, of 737-4, and therefore with a load of 280 Ibs. only 38 per cent, of the energy applied to a differen- tial pulley-block is utilized. In general, we may state that not more than about 40 per cent, is profitably used, and that the remainder is expended in overcoming friction. 220. It is a remarkable and useful property of the differential pulley, that a weight which has been hoisted will remain suspended when the hand is removed, even though the chain be not secured in any manner. The pulleys we have previously considered do not possess this convenient property. The weight raised by the three-sheave pulley-block, for example, will run down unless the free end of" the rope be properly secured. The difference in this respect between these two mechanical powers is not a consequence of any special mechanism; it is simply caused by the excessive friction in the differential pulley- block. 221. The reason why the load does not run down in the differential pulley may be thus explained. Let us suppose that a weight of 400 Ibs. is to be raised one foot by the differential pulley-block ; 400 units of work are necessary, and therefore 1,000 units of work must be applied to the I 2 u6 EXPERIMENTAL MECHANICS. [LECT. power chain to produce the 400 units (since only 40 per cent, is utilized). The friction will thus have consumed 600 units of work when the load has been raised one foot. If the power-weight be removed, the pressure supported by the upper pulley-block is diminished. In fact, since the power- weight is about th of the load, the pressure on the axle when the power-weight has been removed is only 4ths of its previous value. The friction is nearly proportional to that pressure : hence when the power has been removed the friction on the upper axle is -Jyths of its previous value, while the friction on the lower pulley remains unaltered. We may therefore assume that the total friction is at least |ths of what it was before the power-weight was removed. Will friction allow the load to descend ? 600 foot-pounds of work were required to overcome the friction in the ascent: at least f x 600 = 514 foot-pounds would be necessary to overcome friction in the descent. But where is this energy to come from ? The load in its descent could only yield 400 units, and thus descent by the mere weight of the load is impossible. To enable the load to descend we have actually to aid the movement by pulling the chain D (Figs. 36 and 37), which proceeds from the small groove in the upper pulley. 222. The principle which we have here established extends to other mechanical powers, and may be stated generally. Whenever more than half the applied energy is consumed by friction, the load will remain without running down when the machine is left free. THE EPICYCLOIDAL PULLEY-BLOCK. 223. We shall conclude this lecture with some experi- ments upon a useful mechanical power introduced by Mr. Eade under the name of the epicycloidal pulley-block. It vii.] THE EPICYCLOIDAL PULLEY-BLOCK. 117 is shown in Fig. 33, and also in Fig. 49. In this machine there are two chains : one a slight endless chain to which the power is applied ; the other a stout chain which has a hook at each end, from either of which the load may be suspended. Each of these chains passes over a sheave in the block : these sheaves are connected by an ingenious piece of mechanism which we need not here describe. This mechanism is so contrived that, when the power causes the sheave to revolve over which the slight chain passes, the sheave which carries the large chain is also made to revolve, but very slowly. 224. By actual trial it is ascertained that the power must be exerted through twelve feet and a half in order to raise the load one foot; the velocity ratio of the machine is therefore 12-5. 225. If the machine were frictionless, its mechanical efficiency would be of course equal to its velocity ratio; owing to the presence of friction the mechanical efficiency is less than the velocity ratio, and it will be necessary to make experiments to determine the exact value. I attach to the load-hook a weight of 280 Ibs., and insert a few small hooks into the links of the power chain in order to receive weights : 56 Ibs. is sufficient to produce motion-, hence the mechanical efficiency is 5. Had there been no friction a power of 56 Ibs. would have been capable of overcoming a load of 12-5x56 = 700 Ibs. Thus 700 units of energy must be applied to the machine in order to perform 280 units of work. In other words, only 40 per cent, of the applied energy is utilized. 226. An extended series of experiments upon the epicycloidal pulley-block is recorded in Table XII. Ii8 EXPERIMENTAL MECHANICS. [LECT. vii. TABLE XII. THE EPICYCLOIDAL PULLEY-BLOCK. Size adapted for lifting weights up to 5 cwt. ; velocity ratio 12-5 ; mechanical efficiency 5 ; useful effect 40 per cent. ; calculated formula P = 5-8 + 0-185 # Number of Experiment. R. Loads in Ibs. Observed power in Ibs. P. Calculated power in Ibs. Difference of the observed and calculated values. I 56 '5 16-2 + 1-2 2 112 27 26-5 ~ '5 3 168 40 - 3'i 4 224 47 47 -2 + O'2 280 56 57-6 + 1-6 6 336 66 680 + 2'O 7 392 78 78-3 + O*3 8 448 88 88-6 + o'6 9 504 100 99-0 - I'D 10 560 IIO 109-4 -0-6 The fourth column shows the calculated values of the powers derived from the formula. It will be seen by the last column that the formula represents the experiments with but little error. 227. Since 60 per cent, of energy is consumed by friction, this machine, like the differential pulley-block, sustains its load when the chains are free. The differential pulley- block gives a mechanical efficiency of 6, while the epicy- cloidal pulley-block has only a mechanical efficiency of 5, and so far the former machine has the advantage ; on the other hand, that the epicycloidal pulley contains but one block, and that its lifting chain has two hooks, are practical conveniences strongly in its favour. LECTURE VIII. THE LEVER. The Lever of the First Order. The Lever of the Second Order. The Shears. The Lever of the Third Order. THE LEVER OF THE FIRST ORDER. 228. THERE are many cases in which a machine for over- coming great resistance is necessary where pulleys would be quite inapplicable. To meet these various demands a correspondingly various number of contrivances has been devised. Amongst these the lever in several different forms holds "an important place. 229. The lever of the first order will be understood by reference to Fig. 38. It consists of a straight rod, to one end of which the power is applied by means of the weight c. At another point B the load is raised, while at A the rod is supported by what is called the fulcrum. In the case represented in the figure the rod is of iron, i" x i" in section and 6' long; it weighs 19 Ibs. The power is pro- duced by a 56 Ib. weight : the fulcrum consists of a moderately sharp steel edge firmly secured to the framework. EXPERIMENTAL MECHANICS. [LECT. The load in this case is replaced by a spring balance H, and the hook of the balance is attached to the frame. The spring is strained by the action of the lever, and the index FIG. 38. records the magnitude of the force produced at the short end. This is the lever with which we shall commence our experiments. VIIL] THE LEVER OF THE FIRST ORDER. 121 230. In examining the relation between the power and the load, the question is a little complicated by the weight of the lever itself (19 Ibs.), but we shall be able to evade the difficulty by means similar to those employed on a former occasion (Art. 60) ; we can counterpoise the weight of the iron bar. This is easily done by applying a hook to the middle of the bar at D, thence carrying a rope over a pulley F, and suspending a weight G of 19 Ibs. from its free extremity. Thus the bar is balanced, and we may leave its weight out of consideration. 231. We might also adopt another plan analogous to that of Art. 51, which is however not so convenient. The weight of the bar produces a certain strain upon the spring balance. I may first read off the strain produced by the bar alone, and then apply the weight c and read again. The observed strain is due both to the weight c and to the weight of the bar. If I subtract the known effect of the bar, the remainder is the effect of c. It is, however, less complicated to counterpoise the bar, and then the strains indicated by the balance are entirely due to the power. 232. The lever is 6' long ; the point B is 6" from the end, and B c is 5' long. B c is divided into 5 equal portions of i'; A is at one of these divisions, i' distant from B, and c is 5' dis- tant, frem B in the figure ; but c is capable of being placed at any position, by simply sliding its ring along the bar. 233. The mode of experimenting is as follows : The weight is placed on the bar at the position c : a strain is immediately produced upon H ; the spring stretches a little, and the bar becomes inclined. It may be noticed that the hook of the spring balance passes through the eye of a wire- strainer, so that by a few turns of the nut upon the strainer the lever can be restored to the horizontal position. 234. The power of 56 Ibs. being 4' from the fulcrum, 122 EXPERIMENTAL MECHANICS. [LECT. while the load is i' from the fulcrum, it is found that the strain indicated by the balance is 224 Ibs. ; that is, four times the amount of the power. If the weight be moved, so as to be 3' from the fulcrum, the strain is observed to be 168 Ibs.; and whatever be the distance of the power from the fulcrum, we find that the strain produced is obtained by multiplying the magnitude of the power in pounds by the distance expressed in feet, and fractional parts of a foot. This law may be expressed more generally by stating that tfie pouter is to the load as the distance of the load from the fulcrum is to the distance of the power from the fulcrum. 235. We can verify this law under varied circumstances. I move the steel edge which forms the fulcrum of the lever until the edge is 2' from B, and secure it in that position. I place the weight c at a distance of 3' from the fulcrum. I now find that the strain on the balance is 84 Ibs. ; but 84 is to 56 as 3 is to 2, and therefore the law is also verified in this instance. 236. There is another aspect in which we may express the relation between the power and the load. The law in this form is thus stated : The power multiplied by its distance from the fulcrum is equal to the load multiplied by its distance from the fulcrum. Thus, in the case we have just considered, the product of 56 and 3 is 168, and this is equal to the product of 84 and 2. The distances from the fulcrum are commonly called the arms of the lever, and the rule is expressed by stating that The power multiplied into its arm is equal to the load multiplied into its arm : hence the load may be found by dividing the product of the power and the power arm by the load arm. This simple law gives a very convenient method of calculating the load, when we know the power and the distances of the power and the load from the fulcrum. VIIL] THE LEVER OF THE FIRST ORDER. 123 237. When the power arm is longer than the load arm, the load is greater than the power ; but when the power arm is shorter than the load arm, the power is greater than the load. We may regard the strain on the balance as a power which supports the weight, just as we regard the weight to be a power producing the strain on the balance. We see, then, that for the lever of the first order to be efficient as a mechanical power it is necessary that the power arm be longer than the load arm. 238. The lever is an extremely simple mechanical power ; it has only one moving part. Friction produces but little effect upon it, so .that the laws which we have given may be actually applied in practice, without making any allow- ance for friction. In this we notice a marked difference between the lever and the pulley-blocks already de- scribed. 239. In the lever of the first order we find an excellent machine for augmenting power. A power of 14 Ibs. can by its means overcome a resistance of a hundredweight, if the power be eight times as far from the fulcrum as the load is from the fulcrum. This principle it is which gives utility to the crowbar. The end of the bar is placed under a heavy stone, -which it is required to raise ; a support near that end serves as a fulcrum, and then a comparatively small force exerted at the power end will suffice to elevate the stone. 240. The applications of the lever are innumerable. It is used not only for increasing power, but for modifying and transforming it in various ways. The lever is also used in weighing-machines, the principles of which will be readily understood, for they are consequences of the law we have explained. Into these various appliances it is not our intention to enter at present ; the great majority of them 124 EXPERIMENTAL MECHANICS. [LECT. may, when met with, be easily understood by the principle we have laid down. THE LEVER OF THE SECOND ORDER. 241. In the lever of the second order the power is at one end, the fulcrum at the other end, and the load lies between the two : this lever therefore differs from the lever of the first order, in which the fulcrum lies between the two forces. The relation between the power and the load in the lever of the second order may be studied by the arrangement in Fig. 39. 242. The bar A c is the same rod of iron 72" x i" x i" which was used in the former experiment. The fulcrum A is a steel edge on which the bar rests ; the power consists of a spring balance H, in the hook of which the end c of the bar rests ; the spring balance is sustained by a wire-strainer, by turning the nut of which the bar may be adjusted hori- zontally. The part of the bar between the fulcrum A and the power c is divided into five portions, each i' long, and the points A and c are each 6" distant from the extremities of the bar. The load employed is 56 Ibs. ; through the ring of this weight the bar passes, and thus the bar sup- ports the load. The bar is counterpoised by the weight of 19 Ibs. at G, in the manner already explained (Art. 231). 243. The mode of experimenting is as follows : Let the weight B be placed i' from the fulcrum; the strain shown by the spring balance is about 1 1 Ibs. If we calculate the value of the power by the rule already given, we should have found the same result. The product of the load by its distance from the fulcrum is 56, the distance of the power from the fulcrum is 5 ; hence the value of the power should be 56 -f- 5 = 1 1 -2. 244. If the weight be placed 2' from the fulcrum the vin.] THE LEVER OF THE SECOND ORDER. 125 strain is about 22-5 Ibs. and it is easy to ascertain that this is the same amount as would have been found by the appli- cation of the rule. A similar result would have been FIG. 39. obtained if the 56 Ib. weight had been placed upon any other part of the bar ; and hence we may regard the rule proved for the lever of the second order as well as for the 126 EXPERIMENTAL MECHANICS. [LECT. lever of the first order : that, the power multiplied by its distance from the fulcrum is equal to the load mul- tiplied by its distance from the fulcrum. In the present case the load is uniformly 56 Ibs., while the power by which it is sustained is always less than 56 Ibs. FIG. 40. 245. The lever of the second order is frequently applied to practical purposes ; one of the most instructive of these applications is illustrated in the shears shown in Fig. 40. The shears consist of two levers of the second order, which by their united action enable a man to exert a greatly increased force, sufficient, for example, to cut with ease a rod of iron 0^-25 squarg. The mode of action is simple. The first lever A F has a handle at one end F, which is 22" distant from the other end A, where the fulcrum is placed. vili.] THE LEVER OF THE SECOND ORDER. 127 At a point B on this lever, i"-8 distant from the fulcrum A, a short link B c is attached ; the end of the link c is jointed to a second lever c D ; this second lever is 8" long ; it forms one edge of the cutting shears, the other edge being fixed to the framework. 246. I place a rod of iron o" - 25 square between the jaws of the shears in the position E, the distance D E being 3 //g 5, and proceed to cut the iron by applying pressure to the handle. Let us calculate the amount by which the levers increase the power exerted upon F. Suppose for example that I press downwards on the handle with a force of 10 Ibs., what is the magnitude of the pressure upon the piece of iron ? The effect of each lever is to be calculated separately. We may ascertain the power exerted at B by the rule of moments already explained ; the product of the power and its arm is 22 x 10=220: this divided by the number of inches, 1*8 in the line A B, gives a quotient 122, and this quotient is the number of pounds pressure which is exerted by means of the link upon the second lever. We proceed in the same manner to find the magnitude of the pressure upon the iron at E. The product of 122 and 8 is 976. This is divided by 3-5, and the quotient found is 279. Hence the exertion of a pressure of 10 Ibs. at F produces a pressure of 279 Ibs. at E. In round numbers, we may say that the pressure is magnified 28-fold by means of this combination of levers of the second order. 247. A pressure of 10 Ibs. is not sufficient to shear across the bar of iron, even though it be magnified to 279 Ibs. I therefore suspend weights from F, and gradually increase the load until the bar is cut. I find at the first trial that 112 Ibs. is sufficient, and a second trial with the same bar gives 114 Ibs.; 113 Ibs., the mean between these results, may be considered an adequate force. This is the load on 128 EXPERIMENTAL MECHANICS. [LECT. F; the real pressure on the bar is 113 x 27'9=3i53 Ibs. : thus the actual pressure which was necessary to cut the bar amounted to more than a ton. 248. We can calculate from this experiment the amount of force necessary to shear across a bar one square inch in section. We may reasonably suppose that the necessary power is proportional to the section, and therefore the power will bear to 3153 Ibs. the proportion which a square of one inch bears to the square of a quarter inch ; but this ratio is 16: hence the force is 16x3153 Ibs., equal to about 22-5 tons. 249. It is noticeable that 22-5 tons is nearly the force which would suffice to tear the bar in sunder by actual tension. We shall subsequently return to the subject of shearing iron in the lecture upon Inertia (Lecture XVI.). THE LEVER OF THE THIRD ORDER. 250. The lever of the third order may be easily under- stood from Fig. 39, of which we have already made use. In the lever of the third order the fulcrum is at one end, the load is at the other end, while the power lies between the two. In this case, then, the power is represented by the 56 Ib. weight, while the load is indicated by the spring balance. The power always exceeds the load, and con- sequently this lever is to be used where speed is to be gained instead of power. Thus, for example; when the power, 56 Ibs., is 2' distant from the fulcrum, the load indicated by the spring balance is about 23 Ibs. 251. The treadle of a grindstone is often a lever of the third order. The fulcrum is at one end, the load is at the other end, and the foot has only to move through a small distance. vin.] THE LEVER OF THE THIRD ORDER. 129 252. The principles which have been discussed in Lecture III. with respect to parallel forces explain the laws now laid down for levers of different orders, and will also enable us to express these laws more concisely. 253. A comparison between Figs. 20 and 39 shows that the only difference between the contrivances is that in Fig. 20 we have a spring balance c in the same place as the steel edge A in Fig 39. We may in Fig. 20 regard one spring balance as the power, the other as the fulcrum, and the weight as the load. Nor is there any essential difference between the apparatus of Fig. 38 and that of Fig. 20. In Fig. 38 the bar is pulled down by a force at each end, one a weight, the other a spring balance, while it is supported by the upward pressure of the steel edge. In Fig. 20 the bar is being pulled upwards by a force at each end, and downwards by the weight. The two cases are substantially the same. In each of them we find a bar acted upon by a pair of parallel forces applied at its extremities, and retained in equilibrium by a third force. 254. We may therefore apply to the lever the principles of parallel forces already explained. We showed that two parallel forces acting upon a bar could be compounded into a resultant, applied at a certain point of the bar. We have defined the moment of a force (Art. 64), and proved that the moments of two parallel forces about the point of application of their resultant are equal. 255. In the lever of the first order there are two parallel forces, one at each end ; these are compounded into a resultant, and it is necessary that this resultant be applied exactly over the steel edge or fulcrum in order that the bar may be maintained at rest. In the levers of the second and third orders, the power and the load are two parallel forces acting in opposite directions ; their resultant, therefore, does 130 EXPERIMENTAL MECHANICS. [LECT. vin. not lie between the forces, but is applied on the side of the greater, and at the point where the steel edge supports the bar. In all cases the moment of one of the forces about the fulcrum must be equal to that of the other. From the equality of moments it follows that the product of the power and the distance of the power from the fulcrum equals the product of the load, and the distance of the load from the fulcrum : this principle suffices to demonstrate the rules already given. 256. The laws governing the lever may be deduced from the principle of work ; the load, if nearer than the power to the fulcrum, is moved through a smaller distance than the power. Thus, for example, in the lever of the first order : if the load be 1 2 times as far as the power from the fulcrum, then for every inch the load moves it can be demonstrated that the power must move 12 inches. The number of units of work applied at one end of a machine is equal to the number yielded at the other, always excepting the loss due to friction, which is, however, so small in the lever that we may neglect it. If then a power of i Ib. be applied to move the power end through 12 inches, one unit of work will have been put into the machine. Hence one unit of work must be done on the load, but the load only moves through T ^ of a foot, and therefore a load of 12 Ibs. could be overcome : this is the same result as would be given by the rule (Art. 236). 257. To conclude : we have first determined by actual experiment the relation between the power and the load in the lever; we have seen that the law thus obtained harmonizes with the principle of the composition of parallel forces ; and, finally, we have shown how the same result can be deduced from the fertile and important principle of work. 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. THE INCLINED PLANE WITHOUT FRICTION. 258. THE mechanical powers now to be considered are often used for other purposes beside those of raising great weights. For example : the parts of a structure have to be forcibly drawn together, a powerful compression has to be exerted, a mass of timber or other material has to be riven asunder by splitting. For purposes of this kind the inclined plane in its various forms, and the screw, are of the greatest use. The screw also, in the form of the screw- jack, is sometimes used in raising weights. It is principally convenient when the weight is enormously great, and the distance through which it has to be raised comparatively small. 259. We shall commence with the study of the inclined plane. The apparatus used is shown in Fig. 41. A B is a plate of glass 4' long, mounted on a frame and turning K 2 132 EXPERIMENTAL MECHANICS. [LECT. round a hinge at A ; B D is a circular arc, with its centre at A, by which the glass may be supported ; D c is a vertical rod, to which the pulley c is clamped. This pulley can be moved up and down, to be accommodated to the position of A B ; the pulley is made of brass, and turns very freely. A little truck R is adapted to run on the plane of glass. The truck is laden to weigh i lb., and this weight is unaltered throughout the experiments ; the wheels are very free, so that the truck runs with but little friction. 260. But the friction, though small, is appreciable, and it will be necessary to measure the amount and then endeavour to counteract its effect upon the motion. The silk cord attached to the truck is very fine, and its weight is neglected. A series of weights is provided ; they are made from pieces of brass wire, and weigh o'i lb. and o'oi lb. : these can easily be hooked into the loop on the cord at P. We first make the plane A B horizontal, and bring down the pulley c so that the cord shall be parallel to the plane ; we find that a force must be applied by the cord in order to draw the truck along the plane : this force is of course the friction, and 1 by a suitable weight at P the friction may be said to be counterbalanced. But we cannot expect that the friction will be the same when the plane is horizontal as when the plane is inclined. We must therefore examine this question by a method analogous to that used in Art. 207. 261. Let the plane be elevated until B E, the elevation of B above A D, is 20" ; let c be properly adjusted : it is found that when P is 0-45 lb. R is just pulled up ; and on the IX.] INCLINED PLANE WITHOUT FRICTION. 133 other hand, when P is only 0-40 Ib. the truck descends and raises P ; and when P has any value intermediate between these two, the truck remains in equilibrium. Let us denote the force of gravity acting down the plane by R, and it follows that R must be 0*425 Ib., and the friction 0^025 Ib. For when P raises R, it must overcome fric- tion as well as R; therefore the power must be 0*025 + 0-425 = 0-45. On the other hand, when R raises P, it must also overcome the friction 0-025, therefore P can only be 0-425 0-025 = '4 > an< ^ R i g trm s found to be a mean between the greatest and least values of P consistent with equilibrium. If the plane be raised so that the height B E is 33", the greatest and least values of P are o'66 and 0-71 ; therefore R is 0-685 an d the friction 0*025, the same as before. Finally, making the height B E only 2'', the friction is found to be 0-020, which is not much less than the previous determinations. These experiments show that we may consider this very small friction to be practically con- stant at these inclinations. (Were the friction large, other methods are necessary, see Art. 265.) As in the experi- ments R is always raised we shall give P the permanent load of 0-025 lb- thus sufficiently counteracting friction, which we may therefore dismiss from consideration. It is hardly 'necessary to remark that, in afterwards recording the weights placed at P, this counterpoise is not to be included. 262. We have now the means of studying the relation between the power and the load in the frictionless inclined plane. The incline being set at different elevations, we shall observe the force necessary to draw up the constant load of i Ib. Our course will be guided by first making use of the principle of energy. Suppose B E to be 2' ; when the truck has been moved from the bottom of the plane to the top, it will have been raised vertically through a height of 2', 134 EXPERIMENTAL MECHANICS. [LECT. and two units of energy must have been consumed. But the plane being 4' long, the force which draws up the truck need only be 0*5 lb., for 0*5 Ib. acting over 4' pro- duces two units of work. In general, if / be the length of the plane and h its height, R the load, and P the power, the number of units of energy necessary to raise the load is R h, and the number of units expended in pulling it up the plane is PI : hence R h = PI, and consequently P : h : : R : I ; that is, the power is to the height of the plane as the load is to its length. In the present case R = i lb., / = 48" ; therefore P = 0-0208 h, where h is the height of the plane in inches, and P the power in pounds. 263. We compare the powers calculated by this formula with the actual observed values : the result is given in Table XIII. TABLE XIII. INCLINED PLANE. Glass Plane 48" long, truck I lb. in weight, friction counterpoised ; formula />=O'O2o8 x h". Number of Experiment. Height of plane. Observed power in Ibs. P. Calculated power in Ibs. Difference of the observed and cal- culated powers. I 2" 0-04 0-04 O'OO 2 4" 0-08 0'08 O'OO 3 6" 0-13 O'I2 -O'OI 4 8" O'i6 0-17 + O'OI 5 10" 0-21 0'2I O'OO 6 15" 0-31 0-31 O'OO 7 20" 0-42 0-42 O'OO 8 33" 071 O'69 -0'02 Thus for example, in experiment 6, where the height B E is 15", it is observed that the power necessary to draw the truck is o'3i lb. The truck is placed in the middle of the plane, and the power is adjusted so as to be sufficient to ix.] INCLINED PLANE WITH FRICTION. 135 draw the truck to the top with certainty ; the necessary power calculated by the formula is also 0-31 Ibs., so that the theory is verified. 264. The fifth column of the table shows the difference between the observed and the calculated powers. The very slight differences, in no case exceeding the fiftieth part of a pound, may be referred to the inevitable errors of experiment. THE INCLINED PLANE WITH FRICTION. 265. The friction of the truck upon the glass plate is always very small, and is shown to have but little varia- tion at those inclinations of the plane which we used. But when the friction is large, we shall not be justified in neglecting its changes at different elevations, and we must adopt more rigorous methods. For this inquiry we shall use the pine plank and slide already described in Art. 1*7. We do not in this case attempt to diminish friction by the aid of wheels, and consequently it will be of considerable amount. 266. In another respect the experiments of Table XIII. are also in contrast with those now to be described. In the former the load was constant, while the elevation was changed. In the latter the elevation remains constant while a succession of different loads are tried. We shall find in this inquiry also that when the proper allowance has been made for friction, the theoretical law connecting the power and the load is fully verified. 267. The apparatus used is shown in Fig. 33 ; the plane, is, however, secured at one inclination, and the pulley c shown in Fig. 32 is adjusted to the apparatus, so that the rope from the pulley to the slide is parallel to the incline. The elevation of the plane in the position adopted is i7'2, so 136 EXPERIMENTAL MECHANICS. [LECT. that its length, base, and height are in the proportions of the numbers j, 0*955, and o'2()6. Weights ranging from 7 Ibs. to 56 Ibs. are placed upon the slide, and the power is found which, when the slide is started by the screw, will draw it steadily up the plane. The requisite power consists of two parts, that which is necessary to overcome gravity acting down the plane, and that which is necessary to over come friction. 268. The forces are shown in Fig. 42. R G, the force of gravity, is resolved into R L and R M ; R L is evidently the com- ponent acting down the plane, and R M the pressure against the plane; the triangle GLR is similar to A B c, hence if R be the load, the force R L acting down the plane must be 0-296 R, and the pressure upon the plane 0*955 R. 269. We shall first make a calculation with the ordinary law that the friction is proportional to the pressure. The pressure upon the plane A B, to which the friction is pro- portional, is not the weight of the load. The pressure is that component (R M) of the load which is perpendicular to the plane A B. When the weights do not extend beyond 56 Ibs., the best value for the coefficient of friction is 0*288 (Art. 141) : hence the amount of friction upon the plane is 0-288 x 0-955 & = ' 2 75 & This force must be overcome in addition to 0-296 R (the component of gravity acting down the plane) : hence the expression for the power is 0-275 R + 0-296 R = 0-571 R. INCLINED PLANE WITH FRICTION. 137 270. The values of the observed powers compared with the powers calculated from the expression 0-571 R are shown in Table XIV. TABLE XIV. INCLINED PLANE. Smooth plane of pine 72" x n" ; angle of inclination I7'2 ; slide of pine, grain crosswise ; slide started ; formula P=o~$Jj R. Number of Experiment. R. Total load on slide in Ibs. Power in Ibs. 1 P. which just Calculated value draws up slide. of the power. Difference of the observed and cal- culated powers. I 7 4-6 40 -0-6 2 H 8-3 8-0 -0'3 3 21 I2'3 12-0 -0-3 4 28 i6'5 i6'o -0'5 5 35 2O '0 20 x> O'O 5 42 24'2 24-0 -O'2 7 49 28-0 28-0 O'O 8 56 SI'S 32-0 + 0-2 271. Thus for example, in experiment 6, a load of 42 Ibs. was raised by a force of 24*2 Ibs., while the cal- culated value is 24-0 Ibs. ; the difference, 0*2 Ibs., is shown in the last column. 272. The calculated values are found to agree tolerably well with the observed values, but the presence of the large differences in No. i and No. 4 leads us to inquire whether by employing the more accurate law of friction (Art. 141) a better result may not be obtained. In Table VI. we have shown that the friction for weights not exceeding 56 Ibs. is expressed by the formula F= 0-9 + 0-266 X pressure, but the pressure is in this case =0*955 ^> and hence the friction is 0-9 + 0-2547?. To this must be added 0-296 J? s the component of the force 138 EXPERIMENTAL MECHANICS. [LECT. of gravity which must be overcome, and hence the total force necessary is 0-9 + 0-55^. The powers calculated from this expression are compared with those actually observed in Table XV. TABLE XV. INCLINED PLANE. Smooth plane of pine 72" x n" ; angle of inclination I7'2 ; slide of pine, grain crosswise ; slide started ; formula P 0-9 + 0-55 R. Number of Experiment. R. Total load on slide in Ibs. Power in Ibs. which just draws up slide. p. Calculated value of the power. Difference of the observed and cal- culated powers. j 7 4'6 47 + OT 2 H 8'3 8-6 + 0-3 3 21 12-3 12-5 -t-O'2 4 28 i6'5 16-3 -0-2 5 35 20'0 2O'I + 0'I 6 42 24-2 24-0 -O'2 7 49 28-0 2 7 -8 -O*2 8 56 3i-8 317 -O'l For example : in experiment 5, a load of 35 Ibs. is found to be raised by a power of 20-0 Ibs., while the calculated power is 0*9 + 0^55 x 35 = 20*1 Ibs. 273. The calculated values of the powers are shown by this table to agree extremely well with the observed values, the greatest difference being only 0.3 Ib. Hence there can be no doubt that the principles on which the formula has been calculated are correct. This table may therefore be regarded as verifying both the law of friction, and the rule laid down for the relation between the power and the load in the inclined plane. 274. The inclined plane is properly styled a mechanical power. For let the weight be 30 Ibs., we calculate by the formula that 17*4 Ibs. would be sufficient to raise it, so that, ix.] THE SCREW. 139 notwithstanding the loss by friction, we have here a smaller force overcoming a larger one, which is the essential feature of a mechanical power. The mechanical efficiency is 304-17 -4=172. 275. The velocity ratio in the inclined plane is the ratio of the distance through which the power moves to the height through which the weight is raised, that is i -H 0*296 = 3-38. To raise 30 Ibs. one foot, a force of 17-4 Ibs. must therefore be exerted through 3-38 feet. The number of units of work expended is thus 17 -4x3 '38 = 58*8. Of this 30 units, equivalent to 51 per cent., are utilized. The remaining 28-8 units, or 49 per cent, are absorbed by friction. 276. We have pointed out in Art. 222 that a machine in which less than half the energy is lost by friction will permit the load to run down when free : this is the case in the present instance ; hence the weight will run down the plane unless specially restrained. That it should do so agrees with Art. 147, for it was there shown that at about i3'4, and still more at any greater inclination, the slide would descend when started. THE SCREW. 277. The inclined plane as a mechanical power is often used in^the form of a wedge or in the still more disguised form of a screw. A wedge is an inclined plane which is forced under the load ; it is usually moved by means of a hammer, so that the efficiency of the wedge is augmented by the dynamical effect of a blow. 278. The screw is one of the most useful mechanical powers which we possess. Its form may be traced by wrapping a wedge-shaped piece of paper around a cylinder, and then cutting a groove in the cylinder along the spiral line indicated by the margin of the paper. Such a groove 140 EXPERIMENTAL MECHANICS. [LECT. is a screw. In order that the screw may be used it must revolve in a nut which is made from a hollow cylinder, the internal diameter of which is equal to that of the cylinder from which the screw is cut. The nut contains a spiral ridge, which fits into the corresponding thread in the screw ; when the nut is turned round, it moves back- wards or forwards according to the direction of the rotation. Large screws of the better class, such as those upon which we shall first make experiments, are always turned in a lathe, and are thus formed with extreme accuracy. Small screws are made in a simpler manner by means of dies and other contrivances. 279. A characteristic feature of a screw is the inclination of the thread to the axis. This is most conveniently de- scribed by the number of complete turns which the thread makes in a specified length of the screw, usually an inch. For example : a screw is said to have ten threads to the inch when it requires 10 revolutions of the nut in order to move it one inch. The shape of the thread itself varies with the purposes for which the screw is intended; the section may be square or triangular, or, as is generally the case in small screws, of a rounded form. 280. There is so much friction in the screw that ex- periments are necessary for the determination of the law connecting the power and the load. 281. We shall commence with an examination of the screw by the apparatus shown in Fig. 43. The nut A is secured upon a stout frame ; to the end of the screw hooks are attached, in order to receive the load, which in this apparatus does not exceed 224 Ibs. ; at the top of the screw is an arm E by which the screw is turned ; to the end of the arm a rope is attached, which passing over a pulley D, carries a hook for receiving the power c. IX.] THE SCREW. 141 282. We first apply the principle of work to this screw, and calculate the relation between the power and the load as it would be found if friction were absent. The diameter of the circle described by the end of the arm is 20" -5 ; its circumference is therefore 64" -4. The screw contains three threads in the inch, hence in order to raise the load i" the FIG. 43. power moves 3 x 64" - 4= 193" very nearly; therefore the velocity ratio is 193, and were the screw capable of working without friction, 193 would represent the mechanical effi- ciency. In actually performing the experiments the arm E is placed at right angles to the rope leading to the pulley, and the power hook is weighted until, with a slight start, 142 EXPERIMENTAL MECHANICS. [LECT. the arm is steadily drawn ; but the power will only move the arm a few inches, for when the cord ceases to be perpen- dicular to the arm the power acts with diminished efficiency; consequently the load is only raised in each experiment through a small fraction of an inch, but quite sufficient for our purpose. TABLE XVI. THE SCREW. Wrought iron screw, square thread, diameter i"'25, with 3 threads to the inch, length of arm io"*25 ; nut of cast iron, bearing surfaces oiled, velocity ratio 193, useful effect 36 per cent., mechanical efficiency 70 ; formula P= Number of Experiment. R. Load in Ibs. Observed power in Ibs. P. Calculated power of Ibs. Difference of the observed and cal- culated powers. I 28 0'4 0'4 O'O 2 56 0-8 0-8 O'O 3 84 12 I'2 O'O 4 112 1-6 1-6 O'O 1 140 [68 2'O 2'4 2'O 2"4 O'O o-o 7 196 27 2-8 + 0-1 8 224 3'3 3'2 -O'l 283. The results of the experiments are shown in Table XVI. If the motion had not been aided by a start the powers would have been greater. Thus in experiment 6, 2-4 Ibs. is the power with a start, when without a start 3-2 Ibs. was found to be necessary. The experiments have all been aided by a start, and the results recorded have been corrected for the friction of the pulley over which the rope passes : this correction is very small, in no case exceeding 0-2 Ib. The fourth column contains the values of the powers computed by the formula P= 0-0143 & This formula has been deduced from the observations in the IX.] THE SCREW. 143 manner described in the Appendix. The fifth column proves that the experiments are truly represented by the formula : in each of the experiments 7 and 8, the difference between the calculated and observed values amounts to o - i lb., but this is quite inconsiderable in comparison with the weights we are employing. 284. In order to lift 100 Ibs. the expression 0*0143 -^ shows that i '43 Ibs. would be necessary: hence the mechanical efficiency of the screw is IOO-T 1-43 =70. Thus this screw is vastly more powerful than any of the pulley systems which we have discussed. A machine so capable, so com- pact, and so strong as the screw, is invaluable for innumerable purposes in the Arts, as well as in multitudes of appliances in daily use. 285. It is evident, however, that the distance through which the screw can raise a weight must be limited by the length cf the screw itself, and that in the length of lift the screw cannot compete with many of the other contrivances used in raising weights. 286. We have seen that the velocity ratio is 193 ; there- fore, to raise 100 Ibs. i foot, we find that 1-43 x 193 = 276 units of energy must be expended : of this only 100 units, or 36 per cent., is usefully employed ; the rest being consumed in overcoming the friction of the screw. Thus nearly two-thirds of the energy applied to such a screw is wasted. Hence we find that friction does not permit the load to run down, since less than fifty per cent, of the applied energy is usefully employed (Art. 222). This is one of the valuable properties which the screw possesses. 287. We may contrast the screw with the pulley block (Art. 199). They are both powerful machines : the latter is bulky and economical of power, the former is compact and wasteful of power; the latter is adapted for raising ix.] THE SCREW-JACK. 145 weights through considerable distances, and the former for exerting pressures through short distances. THE SCREW-JACK. 288. The importance of the screw as a mechanical power justifies us in examining another of its useful forms, the screw-jack. This machine is used for exerting great pres- sures, such for example as starting a ship which is reluctant to be launched, or replacing a locomotive upon the line from which its wheels have slipped. These machines vary in form, as well as in the weights for which they are adapted ; one of them is shown at D in Fig. 44, and a description of its details is given in Table XVII. We shall determine the powers to be applied to this machine for overcoming resistances not exceeding half a ton. 289. To employ weights so large as half a ton would be inconvenient if not actually impossible in the lecture-room, but the required pressures can be produced by means of a lever. In Fig. 44 is shown a stout wooden bar 16' long. It is prevented from bending by means of a chain ; at E the lever is attached to a hinge, about which it turns freely ; at A a tray is placed for the purpose of receiving weights. The screw-jack is 2' distant from E, consequently the bar is a lever of-the second order, and any weight placed in the tray exerts a pressure eightfold greater upon the top of the screw-jack. Thus each stone in the tray produces a pres- sure of i cwt. at the point D. The weight of the lever and the tray is counterpoised by the weight c, so that until the tray receives a load there is no pressure upon the top of the screw-jack, and thus we may omit the lever itself from con- sideration. The screw-jack is furnished with an arm D G ; at the extremity G of this arm a rope is attached, which passes over a pulley and supports the power B. L EXPERIMENTAL MECHANICS. [LECT. 290. The velocity ratio for this screw-jack with an arm of 33'', is found to be 414, by the method already described (Art. 283). 291. To determine its mechanical efficiency we must resort to experiment. The result is given in Table XVII. TABLE XVII. THE SCREW-JACK. Wrought iron screw, square thread, diameter 2", pitch 2 threads to the inch, arm 33" ; nut brass, bearing surfaces oiled ; velocity ratio 414 ; useful effect, 28 per cent.; mechanical efficiency 116; formula P= o "66 -t- o '0075 'ft. Number of Experiment. R. Load in Ibs. Observed power in Ibs. P. Calculated power in Ibs. Difference of the observed and cal- culated powers. I 112 I '4 i '5 + I 2 224 2 '2 2-3 + I 3 336 3 "3 3-2 -0 I 4 448 4'i 4-0 -0 I 560 5' 4'9 -0 I 5 672 57 57 OX) 7 784 6-5 6-5 O'O 8 8 9 6 7 '4 7 '4 O'O 9 I008 8'i 8-2 + 0'I 10 1 1 20 9-0 9-1 + O'I 292. It may be seen from the column of differences how closely the experiments are represented by the formula. The power which is required to raise a given weight, say 600 Ibs., may be calculated by this formula ; it is o 66 + 0-0075 x 600 = 5-16. Hence the mechanical efficiency of the screw-jack is 6004-5-16 = 116. Thus the screw is very powerful, increasing the force applied to it more than a hundredfold. In order to raise 600 Ibs. one foot, a quantity of work represented by 5-16X414=2136 units must be ex- IX.] THE SCREW-JACK. 147 pended; of this only 600, or 28 per cent., is utilized, so that nearly three-quarters of the energy applied is expended upon friction. 293. This screw does not let the load run down, since less than 50 per cent, of energy is utilised ; to lower the weight the lever has actually to be pressed backwards. 294. The details of an experiment on this subject will be instructive, and afford a confirmation of the principles laid down. In experiment 10 we find that 9*0 Ibs. suffice to raise 1,120 Ibs.; now by moving the pulley to the other side of the lever, and placing the rope perpendicularly to the lever, I find that to produce motion the other way that is, of course to lower the screw a force of 3-4 Ibs. must be applied. Hence, even with the assistance of the load, a force of 3 '4 Ibs. is necessary to overcome friction. This will enable us to determine the amount of friction in the same manner as we determined the friction in the pulley- block (Art. 207). Let x be the force usefully employed in raising, and Y the force of friction, which acts equally in either direction against the production of motion ; then to raise the load the power applied must be sufficient to over- come both x and Y, and therefore we have x+Y=9 - o. When the weight is to be lowered the force x of course aids in the- lowering, but x alone is not sufficient to overcome the friction; it requires the addition of 3-4 Ibs., and we have therefore x-r-3'4=:Y r and hence x = 2'8, Y=6'2. That is, 2-8 is the amount of force which with a friction- less screw would have been sufficient to raise half a ton. But in the frictionless screw the power is found by dividing the load by the velocity ratio. In this case 1120-^414=27, which is within o'i Ib. of the value of x. The agreement of these results is satisfactory. L 2 148 EXPERIMENTAL MECHANICS. [LECT. ix. THE SCREW BOLT AND NUT. 295. One of the most useful applications of the screw is met with in the common bolt and nut, shown in Fig. 45. It consists of a wrought-iron rod with a head at one end and a screw on the other, upon which the nut works. Bolts in many different sizes and forms represent the stitches by which machines and frames are most readily united. There are several reasons why the bolt is so convenient. It draws the parts into close contact with tremendous force ; it is itself so strong that the parts united practically form one piece. It can be adjusted quickly, and removed as readily. The same bolt by the use of washers can be applied to pieces of very different sizes. No skilled hand is required to use the simple tool that turns the nut. Adding to this that bolts are cheap and durable, we shall easily understand why they are so extensively used. 296. We must remark in conclusion that the bolt owes its utility to friction ; screws of this kind do not overhaul, hence when the nut is screwed home it does not recoil. If it were not that more than half the power applied to a screw is consumed in friction, the bolt and the nut would either be rendered useless, or at least would require to be furnished with some complicated apparatus for preventing the motion of the nut. FIG. 45. 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. INTRODUCTION. 297. THE mechanical powers discussed in these lec- tures may be grouped into two classes, the first where ropes or chains are used, and the second where ropes or chains are absent. Belonging to that class in which ropes are not employed, we have the screw discussed in the last lecture-, and the lever discussed in Lecture VIII. ; while among those machines in which ropes or chains form an essential part of the apparatus, the pulley and the wheel and axle hold a prominent place. We have already examined several forms of the pulley, and we now proceed to the not less important subject of the wheel and axle. 298. Where great resistances have to be overcome, but where the distance through which the resistance must be urged is short, the lever or the screw is generally found to be the most appropriate means of increasing power. When, 150 EXPERIMENTAL MECHANICS. [LECT. however, the resistance has to be moved a considerable distance, the aid of the pulley, or the wheel and axle, or sometimes of both combined, is called in. The wheel and axle is the form of mechanical power which is generally used when the distance is considerable through which a weight must be raised, or through which some resistance must be overcome. 299. The wheel and axle assumes very many forms cor- responding to the various purposes to which it is applied. x.] THE WHEEL AND AXLE. 151 The general form of the arrangement will be understood from Fig. 46. It consists of an iron axle B, mounted in bearings, so as to be capable of turning freely ; to this axle a rope is fastened, and at the extremity of the rope is a weight D, which is gradually raised as the axle revolves. Attached to the axle, and turning with it, is a wheel A with hooks in its circumference, upon which lies a rope ; one end of this rope is attached to the circumference of the wheel, and the other supports a weight E. This latter weight may be called the power, while the weight D suspended from the axle is the load. When the power is sufficiently large, E descends, making the wheel to revolve ; the wheel causes the axle to revolve, and thus the rope is wound up and the load D is raised. 300. When compared with the differential pulley as a means of raising a weight, this arrangement appears rather bulky and otherwise inconvenient, but, as we shall presently learn, it is a far more economical means of applying energy. In its practical application, moreover, the arrange- ment is simplified in various ways, two of which may be mentioned. 301. The capstan is essentially a wheel and axle ; the power is not in this case applied by means of a rope, but by direct pressure on the part of the men working it ; nor is there actually a wheel employed, for the pressure is applied to what would be the extremities of the spokes of the wheel if a wheel existed. 302. In the ordinary winch, the power of the labourer is directly applied to the handle which moves round in the circumference of a circle. 303. There are innumerable other applications of the principle which are constantly met with, and which can be easily understood with a little attention. These we shall 152 EXPERIMENTAL MECHANICS. [LECT. not stop to describe, but we pass on at once to the important question of the relation between the power and the load. EXPERIMENTS UPON THE WHEEL AND AXLE. 304. We shall commence a series of experiments upon the wheel A and axle B of Fig. 46. We shall first determine the velocity ratio, and then ascertain the mechanical effi- ciency by actual experiment The wheel is of wood ; it is about 30" in diameter. The string to which the power is attached is coiled round a series of hooks, placed near the margin of the wheel ; the effective circumference is thus a little less than the real circumference. I measure a single coil of the string and find the length to be 88"'5. This length, therefore, we shall adopt for the effective circum- ference of the wheel. The axle is o"75 in diameter, but its effective circumference is larger than the circle of which this length is the diameter. 305. The proper mode of finding the effective circum- ference of the axle in a case where the rope bears a considerable proportion to the axle is as follows. Attach a weight to the extremity of the rope sufficient to stretch it thoroughly. Make the wheel and axle revolve suppose 20 times, and measure the height through which the weight is lifted ; then the one-twentieth part of that height is the effective circumference of the axle. By this means I find the circumference of the axle we are using to be a'^Sy. 306. We can now ascertain the velocity ratio in this machine. When the wheel and axle have made one com- plete revolution the power has been lowered through a distance of 88"'5, and the load has been raised through 2"~&-j. This is evident because the wheel and axle are x.] THE WHEEL AND AXLE. 153 attached together, and therefore each completes one revolu- tion in the same time ; hence the ratio of the distance which the power moves over to that through which the load is raised is 88"'5 -r- 2" - 87 =31 very nearly. We shall there- fore suppose the velocity ratio to be 31. Thus this wheel and axle has a far higher velocity ratio than any of the systems of pulleys which we have been considering. 307. Were friction absent the velocity ratio of 31 would necessarily express the mechanical efficiency of this wheel and axle ; owing to the presence of friction the real efficiency is less than this how much less, we must ascertain by experiment. I attach a load of 56 Ibs. to the hook which is borne by the rope descending from the axle : this load is shown at D in Fig. 46. I find that a power of 2*6 Ibs. applied at E is just sufficient to raise D. We infer from this result that the mechanical efficiency of this machine is 56 -f 2-6 = 21-5. I add a second 56 Ib. weight to the load, and I find that a power of 5-0 Ibs. raises the load of 112 Ibs. The mechanical efficiency in this case is 112 ^5 = 22*5. We adopt the mean value 22. Hence the mechanical efficiency is reduced by friction from 31 to 22. 308. We may compute from this result the number of units of energy which are utilized out of every 100 units appliedr Let us suppose a load of 100 Ibs. is to be raised one foot ; a force of 100 -r 22 = 4-6 Ibs. will suffice to raise this load. This force must be exerted through a space of 31', and consequently 31 x 4-6 = 143 units of energy must be expended ; of thij amount 100 units are usefully em- ployed, and therefore the percentage of energy utilized is 100 -r- 143 x 100 = 70. It follows that 30 per cent, of the applied energy is consumed in overcoming friction. 309. We can see the reason why the wheel and axle overhauls that is, runs down of its own accord when 154 EXPERIMENTAL MECHANICS. [LECT. allowed to do so ; it is because less than half the applied energy is expended upon friction. 310. A series of experiments which have been carefully made with this wheel and axle are recorded in Table XVIII. TABLE XVIII. WHEEL AND AXLE. Wheel of wood ; axle of iron, in oiled brass bearings ; weight of wheel and axle together, 16-5 Ibs. ; effective circumference of wheel, 88"'5 ; effective circumference of axle, 2" "87 ; velocity ratio, 31 ; mechanical efficiency, 22 ; useful effect, 70 per cent. ; formula, P 0-204 + 0-0426 R. Number of Experiment. R. Load in Ibs. Observed power in Ibs. P. Calculated Difference of the observed and calculated values. I 28 I'4 I "4 O'O 2 42 2'O 2-0 O'O 3 56 2'6 2'6 O'O 4 5 11 3'2 37 % O'O + O'l 6 9 4'4 4'4 O'O 7 112 S'o S'o O'O By 26-0 O'O 330. The large amount of friction present in this con- trivance is the consequence of winding the rope directly upon the axle instead of upon a barrel, as already pointed out in Art. 3 1 9. We might place barrels upon these axles and demonstrate the truth of this statement ; but we need not delay to do so, as we use the barrel in the machines which we shall next describe. THE CRANE. 331. We have already explained (Arf. 38) the construction of the lifting crane, so far as its framework is concerned. We now examine the mechanism by which the load is raised. We shall employ for this purpose the model which is repre- X.] THE CRANE. 163 sented in Fig. 48. The jib is supported by a wooden bar as a tie, and the crane is steadied by means of the weights placed at H : some such counterpoise is necessary, for otherwise the machine would tumble over when a load is suspended from the hook. 332. The load is supported by a rope or chain which passes over the pulley E and thence to the barrel D, upon which it is to be wound. This barrel receives its motion from a large wheel A, which contains 200 teeth. The wheel A is turned by the pinion B which contains 25 teeth. In the actual use of the crane, the axle which carries this pinion would be turned round by means of a handle ; but for the purpose of experiments upon the relation of the power to the load, the handle would be inconvenient, and therefore we have placed upon the axle of the pinion a wheel c containing a groove in its circumference. Around this groove a string is wrapped, so that when a weight G is suspended from the string it will cause the wheel to revolve. This weight G will constitute the power by which the load may be raised. 333. Let us compute the velocity ratio of this machine before commencing experiments upon its mechanical efficiency. The effective circumference of the barrel D is found by trial to be i4*'9. Since there are 200 teeth on A and 25 on B, it follows that the pinion B must revolve eight times to produce one revolution of the barrel. Hence the wheel c at the circumference of which the power is applied must also revolve eight times for one revolution of the barrel The effective circumference of c is 43'' ; the power must therefore have been applied through 8 x 43"=344", in order to raise the load i5'"9. The velocity ratio is 344-:- 14-9 = 23 very nearly. We can easily verify this value of the velocity ratio by actually raising the load i', when it appears that the M 2 LECT. X.] THE CRANE. 165 number of revolutions of the wheel B is such that the power must have moved 23'. 334. The mechanical efficiency is to be found as usual by trial. 561bs. placed at F is raised by 3*1 Ibs. at G; hence the mechanical efficiency deduced from this experiment is 56-r3'i = i8. The percentage of useful effect is easily shown to be 78 by the method of Art. 323. Here, then, we have a machine possessing very considerable efficiency, and being at the same time economical of energy. TABLE XXI. THE CRANE. Circumference of wheel to which the power is applied, 43" ; train of wheels, 25 -f- 200; circumference of drum on which rope is wound, I4"'9; velocity ratio, 23 ; mechanical efficiency, 18 ; useful effect, 78 per cent. ; formula, P = 0*0556 A'. Difference of Number of Kxperiment. R. Load in Ibs, Observed power in Ibs. Calculated power in Ibs. the observed and calculated values. I H 0'9 0-8 -OT 2 28 1-6 1-6 OX) 3 42 2'4 2-3 -OT 4 56 3'i 3'i O'O 5 70 3'8 3 '9 + 0'I 6 84 4'5 47 + 0-2 I _ 9 8 112 H 11 + 0-2 + 0'0 335. A series of experiments made with this crane is recorded in Table XXL, and a comparison of the calculated and observed values will show that the formula/' =0-05 56 R represents the experiments with considerable accuracy. 336. It may be noticed that in this formula the term independent of Jv>, which we frequently meet with in the ex- pression of the relation between the power and the load, is absent. The probable explanation is to be found in the fact that some minute irregularity in the form of the 166 EXPERIMENTAL MECHANICS. [LECT. barrel or of the wheel has been constantly acting like a small weight in favour of the power. In each experiment the motion is always started from the same position of the wheels, and hence any irregularity will be constantly acting in favour of the power or against it ; here the former appears to have happened. In other cases doubtless the latter has occurred; the difference is, however, of extremely small amount. The friction of the machine itself when without a load is another source for the production of the constant term ; it has happened in the present case that this friction has been almost exactly balanced by the accidental influence referred to. 337. In cranes it is usual to provide means of adding a second train of wheels, when the load is very heavy. In another model we applied the power to an axle with a pinion of 25 teeth, gearing into a wheel of 200 teeth ; on the axle of the wheel with 200 teeth is a pinion of 30 teeth, which gears into a wheel of 180 teeth ; the barrel is on the axle of the last wheel. A series of experiments with this crane is shown in Table XXII. TABLE XXII. THE CRANE FOR HEAVY LOADS. Circumference of wheel to which power is applied, 43"; train of wheels, 25 -r 200 x 30 -j- 1 80 ; circumference of drum on which rope is wound, 14" '9 ; velocity ratio, 137 ; mechanical efficiency, 87 ; useful effect, 63 per cent. ; formula, P = 0-185 + 0-00782 R. Number of Experiment. R. Load in Ibs. Observed power in Ibs. P. Calculated power in Ibs. Difference of the observed and calculated values. I '4 0-30 0-29 -O'OI 2 28 0-40 O'4O O'OO 3 42 0-50 0-51 + O'OI 4 56 0'60 O-62 + O'O2 I i 075 0-8 5 073 084 -0'02 -O'OI I 9 8 112 0'95 1-05 25 O'OO + 0'OI x.] THE CRANE. 167 The velocity ratio is now 137, and the mechanical efficiency is 87 ; one man could therefore raise a ton with ease by applying a power of 26 Ibs. to a crane of this kind. CONCLUSION. 338. It will be useful to contrast the wheel and axle on which we have experimented (Art. 304) with the differential pulley (Art. 209). The velocity ratio of the former machine is nearly double that of the latter, and its mechanical efficiency is nearly four times as great. Less than half the applied power is wasted in the wheel and axle, while more than half is wasted in the differential pulley. This makes the wheel and axle both a more powerful machine, and a more economical machine than the differential pulley. On the other hand, the greater compactness of the latter, its facility of application, and the practical conveniences arising from the property of not allowing the load to run down, 'do often more than compensate for the superior mechanical advantage of the wheel and axle. 339. We may also contrast the wheel and axle with the screw (Art. 277). The screw is remarkable among the mechanical powers for its very high velocity ratio, and its excessive friction. Thus we have seen in Art. 291 how the velocity ratio of a screw-jack with an arm attached amounted to 414, while its mechanical efficiency was little more than one-fourth as great. No single wheel and axle could conveniently be made to give a mechanical efficiency of 116 ; but from Art. 337 we could easily design a combina- tion of wheels and axles to yield an efficiency of quite this amount. The friction in the wheel and axle is very much less than in the screw, and consequently energy is saved by the use of the former machine. 168 EXPERIMENTAL MECHANICS. [LECT. x. 340. In practice, however, it generally happens that economy of energy does not weigh much in the selection of a mechanical power for any purpose, as there are always other considerations of greater consequence. 341. For example, let us take the case of a lifting crane employed in loading or unloading a vessel, and inquire why it is that a train of wheels is generally used for the purpose of producing the requisite power. The answer is simple, the train of wheels is convenient, for by their aid any length of chain can be wound upon the barrel ; whereas if a screw were used, we should require a screw as long as the greatest height of lift. This screw would be incon- venient, and indeed impracticable, and the additional cir- cumstance that a train of wheels is more economical of energy than a screw has no influence in the matter. 342. On the other hand, suppose that a very heavy load has to be overcome for a short distance, as for example in starting a ship launch, a screw-jack is evidently the proper machine to employ; it is easily applied, and has a high mechanical efficiency. The want of economy of energy is of no consequence in such an operation. 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. INTRODUCTION 343. IN the lectures on the mechanical powers which have been just completed, we have seen how great weights may be raised or other large resistances overcome. We are now to consider the important subject of the applica- tion of mechanical principles to structures. These are fixtures, while machines are adapted for motion ; a roof or a bridge is a structure, but a crane or a screw-jack is a machine. Structures are employed for supporting weights, and the mechanical powers give the means of raising them. 344. A structure has to support both its own weight and also any load that is to be placed upon it. Thus a railway bridge must at all times sustain what is called the permanent load, and frequently, of course, the weight of one or more trains. The problem which the engineer solves is to design a bridge which shall be sufficiently strong, and, at the same I7o EXPERIMENTAL MECHANICS. [LECT. time, economical ; his skill is shown by the manner in which he can attain these two ends in the same structure. 345. In the four lectures of the course which will be devoted to this subject it will only be possible to give a slight sketch, and therefore but few details can be introduced. An extended account of the properties of different materials used in structures would be beyond our scope, but there are some general principles relating to the strength of materials which may be discussed. Timber, as a building material, has, in modern times, been replaced to a great extent by iron in large structures, but timber is more capable than iron of being experimented upon in the lecture room. The ele- mentary laws which we shall demonstrate with reference to the strength of timber, are also, substantially the same as the corresponding laws for the strength of iron or any other material. Hence we shall commence the study of structures by two lectures on timber. The laws which we shall prove experimentally will afterwards be applied to a few simple cases of bridges and other actual structures. THE GENERAL PROPERTIES OF TIMBER. 346. The uses of timber in the arts are as various as its qualities. Some woods are useful for their beauty, and others for their strength or durability under different circumstances. We shall only employ " pine " in our experimental inquiries. This wood is selected because it is so well known and so much used. A knowledge of the properties of pine would probably be more useful than a knowledge of the properties of any other wood, and at the same time it must be remem- bered that the laws which we shall establish by means of slips of pine may be generally applied. 347. A transverse section of a tree shows a number of rings, each of which represents the growth of wood in one Xl.] GENERAL PROPERTIES OF TIMBER. 171 year. The age of the tree may sometimes be approximately found by counting the number of distinguishable rings. The outer rings are the newer portions of the wood. 348. When a tree is felled it contains a large quantity of sap, which must be allowed to evaporate before the wood is fit for use. With this object the timber is stored in suitable yards for two or more years according to the purposes for which it is intended ; sometimes the process of seasoning, as it is called, is hastened by other means. Wood, when seasoning, contracts ; hence blocks of timber are often found split from the circumference to the centre, for the outer rings, being newer and containing more sap, contract more than the inner rings. For the same reason a plank is found to warp when the wood is not thoroughly seasoned. The side of the plank which was farthest from the centre of the tree contracts more than the other side, and becomes concave. This can be easily verified by looking at the edge of the plank, for we there see the rings of which it is com- posed. 349. Timber may be softened by steaming. I have here a rod of pine, 24" x o" - 5 Xo"'5, and here a second rod cut from the same piece and of the same size, which has been exposed to steam of boiling water for more than an hour : securing these at one end to a firm stand, I bend them down together, and you see that after the dry rod has broken the steamed rod can be bent much farther before it gives way. This property of wood is utilized in shaping the timbers of wooden ships. We shall be. ; able to understand the action of steam if we reflect that wood is composed of a number of fibres ranged side by side and united together. A rope is composed of a number of fibres laid to gether and twisted, but the fibres are not coherent as they are in wood. Hence we find that a rod of wood is stiff, while 172 EXPERIMENTAL MECHANICS. [LECT. a rope is flexible. The steam finds its way into the interstices between the fibres of the wood ; it softens their connections, and increases the pliability of the fibres themselves, and thus, the operation of steaming tends to soften a piece of timber and render it tractable. 350. The structure of wood is exhibited by the following simple experiment : Here are two pieces of pine, each 9" x i"x i". One of them I easily snap across with a blow, while my blows are unable to break the other. The differ- ence is merely that one of these pieces is cut against the grain, while the other is with it. In the first case I have only to separate the connection between the fibres, which is quite easy. In the other case I would have to tear asunder the fibres themselves, which is vastly more difficult. To a certain ex- tent the grained structure is also found in wrought iron, but the contrast between the strength of iron with the grain and against the grain is not so marked as it is in wood. RESISTANCE TO EXTENSION. 351. It will be necessary to explain a little more definitely what is meant by the strength of timber. We may conceive a rod to be broken in three different ways. In the first place the rod may be taken by a force at each end and torn asunder by pulling, as a thread may be broken. To do this requires very great power, and the strength of the material with reference to such a mode of destroying it is called its resistance to extension. In the second place, it may be broken by longitudinal pressure at each end, as a pillar may be crushed by the superincumbent weight being too large ; the strength that relates to this form of force is called resist- ance to compression : finally, the rod may be broken by a force applied transversely. The strength of pine with reference to these different applications of force will be XL] RESISTANCE TO EXTENSION. 173 considered successively. The rods that are to be used have been cut from the same piece of timber, which has been selected on account of its straightness of grain and freedom from knots. They are of different rectangular sections, FIG. 49. i"Xo'"5 and o"'5 xo"'5 being generally used, but sometimes i x i ' is employed. 352. With reference to the strength of timber in its capacity to resist extension, we can do but little in the lecture room. I have here a pine rod A B, of dimensions 48' x o'"5 x o" - 5, Fig. 49. Each end of this rod is firmly secured 174 EXPERIMENTAL MECHANICS. [LECT. between two cheeks of iron, which are bolted together : the rod is suspended by its upper extremity from the hook of the epicyloidal pulley-block (Art. 213), which is itself sup- ported by a tripod ; hooks are attached to the lower end of the rod for carrying the weights. By placing 3 cwt. on these hooks and pulling the hand chain of the pulley- block, I find that I can raise the weight safely, and therefore the rod will resist at all events a tension of 3 cwt. From experiments which have been made on the subject, it is ascertained that about a ton would be necessary to tear such a rod asunder ; hence we see that pine is enormously strong in resisting a force of extension. The tensile strength of the rod does not depend upon its length, but upon the area of the cross section. That of the rod we have used is one- fourth of a square inch, and the breaking weight of a rod one square inch in section is about four tons. 353. A rod of any material generally elongates to some extent under the action of a suspended weight ; and we shall ascertain whether this occurs perceptibly in wood. Before the rod was strained I had marked two points upon it exactly 2 feet apart. When the rod supports 3 cwt. I find that the distance between the two points has not appreciably altered, though by more delicate measurement I have no doubt we should find that the distance had elongated to an insignifi- cant extent. 354. Let us contrast the resistance of a rod of timber to extension with the effect upon a rope under the same cir- cumstances. I have here a rope about o".25 diameter; it is suspended from a point, and bears a 14 Ib. weight in order to be completely stretched. I mark points upon the rope 2 apart. I now change the stone weight for a weight of i cwt, and on measurement I find that the two points which before were 2' apart, are now 2' 2'; thus the XL] RESISTANCE TO COMPRESSION. 175 rope has stretched at the rate of an inch per foot for a strain of i cwt., while the timber did not stretch perceptibly for a strain of 3 cwt. 355. We have already explained in Art, 37 the meaning of the word " lie." The material suitable for a tie should be capable of offering great resistance, not only to actual rupture by tension, but even to appreciable elongation. These qualities we have found to be possessed by wood. They are, however, possessed in a much higher degree by wrought iron, which possesses other advantages in durability and facility of attachment. RESISTANCE TO COMPRESSION. 356. We proceed to examine into the capability of timber to resist forces of longitudinal compression, either as a pillar or in any other form of " strut," such for instance, as the jib of the crane represented in Fig. 17. The use of timber as a strut depends in a great degree upon the coherence of the fibres to each other, as well as upon their actual rigidity. The action of timber in resisting forces of compression is thus very different from its action when resisting forces of extension ; we can examine, by actual ex- periment, the strength of timber under the former conditions, as the weights which it will be necessary to employ are within the capabilities of our lecture-room apparatus. 357. The apparatus is shown in Fig. 50. It consists of a lever of the second order, 10' long, the mechanical advan- tage of which is threefold ; the resistance of the pillar D E to crushing is the load to be overcome, and the power consists of weights, to receive which the tray B is used ; every pound placed in the tray produces a compressive force of 3 Ibs. on the pillar at D. The fulcrum is at A and guides at G. The lever and the tray would somewhat complicate our 176 EXPERIMENTAL MECHANICS. [LECT. calculations unless their weights were counterpoised. A cord attached to the extremity of the lever passes over a pulley F ; at the other end of this cord, sufficient weights c are attached to neutralize the weight of the apparatus. In fact, the lever and tray now swing as if they had no weight, and we may therefore leave them out of consideration. The pillar to be experimented upon is fitted at its lower FIG. so. end E into a hole in a cast-iron bracket : this bracket can be adjusted so as to take in pieces of different lengths ; the upper end of the pillar passes through a hole in a second piece of cast-iron, which is bolted to the lever : thus our little experimental column is secured at each end, and the risk of slipping is avoided. The stands are heavily weighted to secure the stability of the arrangement. 358. The first experiment we shall make with this xi.] RESISTANCE TO COMPRESSION. 177 apparatus is upon a pine rod 40" long and o"'5 square ; the lower bracket is so placed that the lever is horizontal when just resting upon the top of the rod. Weights placed in the tray produce a pressure three times as great down the rod, the effect of which will first be to bend the rod, and, when the deflection has reached a certain amount, to break it across. I place 28 Ibs. in the tray: this produces a pressure of 84 Ibs. upon the rod, but the rod still remains perfectly straight, so that it bears this pressure easily. When the pressure is increased to 96 Ibs. a very slight amount of deflection may be seen. When the strain reaches 114 Ibs. the rod begins to bend into a curved form, though the deflection of the middle of the rod from its original position is still less than o'-z^. Gradually augmenting the pressure, I find that when it reaches 132 Ibs. the deviation has reached o"'5 ; and finally, when 48 Ibs. is placed in the tray, that is, when the rod is subjected to 144 Ibs., it breaks across the middle. Hence we see that this rod sustained a load of 96 Ibs. without sensibly bending, but that fracture ensued when the load was increased about half as much again. Another experiment \vith a similar rod gave a slightly less value (132 Ibs.) for the breaking load. If I add these results together, and divide the sum by 2, I find 138 Ibs. as the mean value of the breaking load, and this is a sufficiently exact determination. 359. Let us next try the resistance of a shorter rod of the same section. I place a piece of pine 20" long and o"*5 square in the apparatus, firmly securing each end as in the former case. The lower bracket is adjusted so as to make the lever horizontal ; the counterpoise, of course, remains the same, and weights are placed in the tray as before. No deflection is noticed when the rod supports N i;8 EXPERIMENTAL MECHANICS. [LECT. 126 Ibs. ; a very slight amount of bending is noticeable with 1 86 Ibs. ; with 228 Ibs., the amount by which the centre of the rod has deviated laterally from its original position is about o"'2 ; and finally, when the load reaches 294 Ibs., the rod breaks. Fracture first occurs in the middle, but is immediately followed by other fractures near where the ends of the rod are secured. 360. Hence the breaking load of a rod 20" long is more than double the breaking load of a rod of 40" long the same section ; from this we learn that the sections being equal, short pillars are stronger than long pillars. It has been ascertained by experiment that the strength of a square pillar to resist compression is proportional to the square of its sectional area. Hence a rod of pine, 40" long and i" square, having four times the section of the rod of the same length we have experimented on, would be sixteen times as strong, and consequently its breaking weight would amount to nearly a ton. The strength of a rod used as a tie depends only on its section, while the strength of a rod used as a strut depends on its length as well as on its section. CONDITION OF A BEAM STRAINED BY A TRANSVERSE FORCE. 361. We next come to the important practical subject of the strength of timber when supporting a transverse strain ; that is, when used as a beam. The nature of a transverse strain may be understood from Fig. 51, which represents a small beam, strained by a load at its centre. Fig. 52 shows two supports 40" apart, across which a rod of pine 48" x i" x i" is laid ; at the middle of this rod a hook is placed, from which a tray for the reception of weights is suspended. A rod thus supported, and bearing weights, is XL] TRANSVERSE STRAIN. 179 said to be strained transversely. A rafter of a roof, the flooring of a room, a gangway from the wharf to a ship, many forms of bridge, and innumerable other examples, might be given of beams strained in this manner. To this important subject we shall devote the remainder of this lecture and the whole of the next. 362. The first point to be noticed is the deflection of the beam from which a weight is suspended. The beam is at first horizontal ; but as the weight in the tray is augmented, the beam gradually curves downwards until, when the weight reaches a certain amount, the beam breaks across in the middle and the tray falls. For convenience in recording the experiments the tray chain and hooks have been adjusted to weigh exactly 14 Ibs. (Fig. 52). A B is a cord which is kept stretched by the little weights D : this cord gives a rough measure of the deflection of the beam from its horizontal position when strained by a load in the tray. In order to observe the deflection N 2 i8o EXPERIMENTAL MECHANICS. [LECT. accurately an instrument is used called the cathetometer (G). It consists of a small telescope, always directed horizontally, though capable of being moved up and down a vertical triangular pillar ; on one of the sides of the pillar a scale is engraved, so that the height of the telescope in any FIG. 52. position can be accurately determined. The cathetometer is levelled by means of the screws H H, so that the tri- angular pillar on which the telescope slides is accurately vertical: the dotted line shows the direction of the visual ray when the centre c of the beam is seen by the observer through the telescope. XL] TRANSVERSE STRAIN. 181 Inside the telescope and at its focus a line of spider's web is fixed horizontally ; on the bar to be observed, and near its middle point c, a cross of two fine lines is marked. The tray being removed, the beam becomes horizontal ; the telescope of the cathetometer is then directed towards the beam, so that the lines marked upon it can be seen dis- tinctly. By means of a screw the telescope may be raised or lowered until the spider's web inside the telescope is observed to pass through the image of the intersection of the lines. The scale then indicates precisely how high the telescope is on the pillar. 363. While I look through the telescope my assistant suspends the tray from the beam. Instantly I see the cross descend in the field of view. I lower the telescope until the spider's web again passes through the image of the intersection of the lines, and then by looking at the scale I see that the telescope has been moved down o"*i9, that is, about one-fifth of an inch : this is, therefore, the distance by which the cross lines on the beam, and therefore the centre of the beam itself, must have descended. Indeed, even a simpler apparatus would be competent to measure the amount of deflection with some degree of precision. By placing successively one stone after another upon the tray, the beam is seen to deflect more and more, until even without the telescope you see the beam has deviated from the horizontal. 364. By carefully observing with the telescope, and measuring in the way already described, the deflections shown in Table XXIII. were determined. The scale along the vertical pillar was read after the spider's web had been adjusted for each increase in the weight. The movement from the original position is recorded as the deflection for each load. 1 82 EXPERIMENTAL MECHANICS. [LECT. TABLE XXIII. DEFLECTION OF A BEAM. A rod of pine 48" x i" x i" ; resting freely on supports 40" apart ; and laden in the middle. Number of Experiment. Magnitude of load. Deflection.' I 14 o"'I9 2 28 o"'37 3 42 o""55 4 56 ",' 74 s ?''?3 7 98 i'-|S 8 112 I *6i 9 126 i"'95 10 I 4 2"'37 365. The first column records the number of the experi- ment. The second represents the load, and the third contains the corresponding deflections. It will be seen that up to 98 Ibs. the deflection is about o"*2 for every stone weight, but afterwards the deflection increases more rapidly. When the weight reaches 140 Ibs. the deflection at first indicated is 2"'37 ; but gradually the cross lines are seen to descend in the field of the telescope, showing that the beam is yielding and finally it breaks across. This experiment teaches us that a beam is at first deflected by an amount proportional to the weight it supports ; but that when two- thirds of the breaking weight is reached, the beam is deflected more rapidly. 366. It is a question of the utmost importance to ascertain the greatest load a beam can sustain without injury to its strength. This subject is to be studied by examining the effect of different deflections upon the fibres of a beam. A beam is always deflected whatever be the load it supports ; XI.] TRANSVERSE STRAIN. 183 thus by looking through the telescope of the cathetometer I can detect an increase of deflection when a single pound is placed in the tray : hence whenever a beam is loaded we must have some deflection. An experiment will show what amount of deflection may be experienced without producing any permanently injurious effect. 367. A pine rod 40" x i x i" is freely supported at each end, the distances between the supports being 38", and the tray is suspended from its middle point. A fine pair of cross lines is marked upon the beam, and the telescope of the cathetometer is adjusted so that the spider's line exactly passes through the image of the intersection. 14 Ibs. being placed in the tray, the cross is seen to descend ; the weight being removed, the cross returns precisely to its original position with reference to the spider's line : hence, after this amount of deflection, the beam has clearly returned to its initial condition, and is evidently just as good as it was before. The tray next received 56 Ibs. ; the beam was, of course, considerably deflected, but when the weight was removed the cross again returned, at all events, to within o" - oi of where the spider's line was left to indicate its former position. We may consider that the beam is in this case also restored to its original condition, even though it has b~brne a strain which, including the tray, amounted to 70 Ibs. But when the beam has been made to carry 84 Ibs. for a few seconds, the cross does not completely return on the removal of the load from the tray, but it shows that the beam has now received a permanent deflection of o"'O3. This is still more apparent after the beam has carried 98 Ibs., for when this load is removed the centre of the beam is permanently deflected by o"'i3. Here, then, we may infer that the fibres of the beam are beginning to be strained beyond their powers of resistance, and this is 1 84 EXPERIMENTAL MECHANICS. [LECT. verified when we find that with 28 additional pounds in the tray a collapse ensues. 368. Reasoning from this experiment, we might infer that the elasticity of a beam is not affected by a weight which is less than half that which would break it, and that, therefore, it may bear without injury a weight not exceeding this amount. As, however, in our experiments the weight was only applied once, and then but for a short time, we cannot be sure that a longer-continued or more frequent application of the same load might not prove injurious ; hence, to be on the safe side, we assume that one-third of the breaking weight of a beam is the greatest load it should be made to bear in any structure. In many cases it is found desirable to make the beam much stronger than this ratio would indicate. 369. We next consider the condition of the fibres of a beam when strained by a transverse force. It is evident that since the fracture commences at the lower surface of the beam, the fibres there must be in a state of tension, while those at the concave upper surface of the beam are compressed together. This condition of the fibres may be proved by the following experiment. 370. I take two pine rods, each 48" x i" x i", perfectly similar in all respects, cut from the same piece of timber, and therefore probably of very nearly identical strength. With a fine tenon saw I cut each of the rods half through at its middle point. I now place one of these beams on the supports 40" apart, with the cut side of the beam upwards. I suspend from it the tray, which I gradually load with weights until the beam breaks, which it does when the total weight is 8 1 Ibs. If I were to place the second beam on the same supports with the cut upwards, then there can be no doubt that it XL] TRANSVERSE STRAIN. 185 would require as nearly as possible the same weight to break it. I place it, however, with the cut downwards, I suspend the tray, and find that the beam breaks with a load of 3 1 Ibs. This is less than half the weight that would have been required if the cut had been upwards. 371. What is the cause of this difference? The fibres being compressed together on the upper surface, a cut has no tendency to open there ; and if the cut could be made with an extremely fine saw, so as to remove but little material, the beam would be substantially the same as if it had not been tampered with. On the other hand, the fibres at the lower surface are in a state of tension ; therefore when the cut is below it yawns open, and the beam is greatly weakened. It is, in fact, no stronger than a beam of 48"xo"'5Xi", placed with its shortest dimension vertical. If we remember that an entire beam of the same size required about 140 Ibs. to break it (Art. 366), we see that the strength of a beam is reduced to one-fourth by being cut half-way through and having the cut underneath. 372. We may learn from this the practical consequence that the sounder side of a beam should always be placed downwards. Any flaw on the lower surface will seriously weaken the beam : so that the most knotty face of the wood should certainly be placed uppermost. If a portion of the actual substance of a beam be removed for example, if a notch be cut out of it this will be almost equally injurious on either side of the beam. 373. We may illustrate the condition of the upper surface of the beam by a further experiment. I make two cuts o"*5 deep in the middle of a pine rod 48" x i"X i". These cuts are o" - 5 apart, and slightly inclined ; the piece between them being removed, a wedge is shaped to fit tightly into the space ; the wedge is long enough to project a little on one 1 86 EXPERIMENTAL MECHANICS. [LECT. side. If the wedge be uppermost when the beam is placed on the supports, the beam will be in the same condition as if it had two fine cuts on the upper surface. I now load the beam with the tray in the usual manner, and I find it to bear 70 Ibs. securely. On examining the beam, which has curved down considerably, I find that the wedge is held in very tightly by the pressure of the fibres upon it, but, by a sharp tap at the end, I knock out the wedge, and instantly the load of 70 Ibs. breaks the beam ; the reason is simple the piece being removed, there is no longer any resistance to the compressive strain of the upper fibres, and con- sequently the beam gives way. 374. The collapse of a beam by a transverse strain commences by fracture of the fibres on the lower surface, followed by a rupture of all fibres up to a considerable depth. Here we see that by a transverse force the fibres in a beam of 48" x i" X i" have been broken by a strain of 140 Ibs. (Art. 366) ; but we have already stated (Art. 353) that to tear such a rod across by a direct pull at each end a force of about four tons is necessary. The breaking strain of the fibres must be a certain definite quantity, yet we find that to overcome it in one way four tons is necessary, while by another mode of applying the strain 1 40 Ibs. is sufficient. 375. To explain this discrepancy we may refer to the experiment of Art. 28, wherein a piece of string was broken by the transverse pull of a piece of thread in illustration of the fact that one force may be resolved into two others, each of them very much greater than itself. A similar resolution of force occurs in the transverse deflection of the beam, and the force of 140 Ibs. is changed into two other forces, each of them enormously greater and sufficiently strong to rupture the fibres. We need not suppose that XL] TRANSVERSE STRAIN. 187 the force thus developed is so great as four tons, because that is the amount required to tear across a square inch of fibres simultaneously, whereas in the transverse fracture the fibres appear to be broken row after row; the fracture is thus only gradual, nor does it extend through the entire depth of the beam. 376. We shall conclude this lecture with one more remark, on the condition of a beam when strained by a transverse force. We have seen that the fibres on the upper surface are compressed, while those on the lower surface are extended ; but what is the condition of the fibres in the interior? There can be no doubt that the following is the state of the case : The fibres immediately beneath the upper surface are in compression ; at a greater depth the amount of compression diminishes until at the middle of the beam the fibres are in their natural condition ; on approaching the lower surface the fibres commence to be strained in extension, and the amount of the extension gradually increases until it reaches a maximum at the lower surface. 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. A BEAM FREE AT THE ENDS AND LOADED IN THE MIDDLE. 377. IN the preceding lecture we have examined some general circumstances in connection with the condition of a beam acted on by a transverse force ; we proceed in the present to inquire more particularly into the strength under these conditions. We shall, as before, use for our experiments rods of pine only, as we wish rather to illustrate the general laws than to determine the strength of different materials. The strength of a beam depends upon its length, breadth, and thickness ; we must endeavour to distinguish the effects of each of these elements on the capacity of the beam to sustain its load. We shall only employ beams of rectangular section ; this being generally the form in which beams of wood are used. Beams of iron, when large, are usually not rect- angular, as the material can be more effectively disposed LECT. xil.] STRENGTH OF A BEAM. 189 in sections of a different form. It is important to distinguish between the stiffness of a beam in its capacity to resist flexure, and the strength of a beam in its capacity to resist fracture. Thus the stiffest beam which can be made from the cylindrical trunk of a tree i' in diameter is 6" broad and io"'5 deep, while the strongest beam is 7" broad and 9"- 7 5 deep. We are now discussing the strength (not the stiffness) of beams. 378. We shall commence the inquiry by making a number of experiments : these we shall record in a table, and then we shall endeavour to see what we can learn from an examination of this table. I have here ten pieces of pine, of lengths varying from i' to 4', and of three different sections, viz. i" x i", i" X o"'5, and o"'5 X o"'5. I have arranged four different stands, on which we can break these pieces : on the first stand the distance between the points of support is 40", and on the other stands the distances are 30," 20", and 10" respectively ; the pieces being 4', 3', 2', and i' long, will just be conveniently held on the supports. 379. The mode of breaking is as follows: The beam being laid upon the supports, an S hook is placed at its middle point, and from this S hook the tray is suspended. Weights a"re then carefully added to the tray until the beam breaks ; the load in the tray, together with the weight of the tray, is recorded in the table as the breaking load. 380. In order to guard as much as possible against error, I have here another set of ten pieces of pine, duplicates of the former. I shall also break these ; and whenever I find any difference between the breaking loads of two similar beams, I shall record in the table the mean between the two loads. The results are shown in Table XXIV. 190 EXPERIMENTAL MECHANICS. [LECT. TABLE XXIV. STRENGTH OF A BEAM. Slips of pine (cut from the same piece) supported freely at each end ; the length recorded is the distance between the points of support ; the load is suspended from the centre of the beam, and gradually increased until the beam breaks ; Formula, P = 6080 area of section x depth span No.of Dimensions. Mean of the p. Difference of Ex- observations Calculated the observed peri- ment. Span. Breadth. Depth. ofthebreak- ng load in Ibs. breaking load in Ibs. and calculated values. I 4 o"-o l"'O l"'O 152 152 O'O 2 40" -o o"-s l"'O 77 7 6 I'O 3 4 o"-o i"-o o"- 5 38 38 O'O 4 40" -o o"'5 o"'5 19 19 O'O 30" -o r 'O "'S 59 SI -8-0 6 7 30" -o 23"'O "'5 i"-o ,7 5 o"'5 25 74 9 O'O + 2'0 8 20" -0 o"'5 o'"5 36 38 + 2'0 9 10 IO"'O IO"'O IX) "'5 o"-s o"- 5 a 152 7 6 -2'0 + 8-0 381. In the first column is a series of figures for con- venience of reference. The next three columns are occupied with the dimensions of the beams. By span is meant the distance between the points of support ; the real length is of course greater; the depth is that dimension of the beam which is vertical. The fifth column gives the mean of t\vo observations of the breaking load. Thus for example, in experiment No. 5 the two beams used were each 36" x i"xo"'5, they were placed on points of support 30" distant, so the span recorded is 30" : one of the beams xii.] STRENGTH OF A BEAM. 191 was broken by a load of 58 Ibs., and the second by a load of 60 Ibs. ; the mean between the two, 59 Ibs., is recorded as the mean breaking load. In this manner the column of breaking loads has been found. The meaning of the two last columns of the table will be explained presently. 382. We shall endeavour to elicit from these observations the laws which connect the breaking load with the span, breadth, and depth of the beam. 383. Let us first examine the effect of the span ; for this purpose we bring together the observations upon beams of the same section, but of different spans. Sections of o"'5 x o" - 5 will be convenient for this purpose ; Nos. 4, 6, 8, and 10 are experiments upon beams of this section. Let us first compare 4 and 8. Here we have two beams of the same section, and the span of one (40") is double that of the other (20"). When we examine the breaking weights we find that they are 19 Ibs. and 36 Ibs. ; the former of these numbers is rather more than half of the latter. In fact, had the breaking load of 40" been fib. less, 18-25 Ibs., and had that of 20" been | Ib. more, 36-5 Ibs., one of the breaking loads would have been exactly half the other. 384. We must not look for perfect numerical accuracy in these experiments ; we must only expect to meet with approximation, because the laws for which we are in search are in reality only approximate laws. Wood itself is variable in quality, even when cut from the same piece : parts near the circumference are different in strength from those nearer the centre ; in a young tree they are generally weaker, and in an old tree generally stronger. Minute differences in the grain, greater or less perfectness in the seasoning, these are also among the circumstances which prevent one piece of timber from being identical with another. We shall, however, generally find that the 192 EXPERIMENTAL MECHANICS. [LECT. effect of these differences is small, but occasionally this is not the case, and in trying many experiments upon the breaking of timber, discrepancies occasionally appear for which it is difficult to account. 385. But you will find, I think, that, making reasonable allowances for such difficulties as do occur, the laws on the whole represent the experiments very closely. 386. We shall, then, assume that the breaking weight of a bar of 40" is half that of a bar of 20" of the same section, and we ask, Is this generally true? is it true that the breaking weight is inversely proportional to the span? In order to test this hypothesis, we can calculate the breaking weight of a bar of 30" (No. 6), and then compare the result with the observed value ; if the supposition be true, the breaking weight should be given by the proportion 30" : 40" :: 19 : Answer. The answer is 25-3 Ibs. ; on reference to the table we find 25 Ibs. to be the observed value, hence our hypothesis is verified for this bar. 387. Let us test the law also for the 10" bar, No. 10 10" : 40" :: 19 : Answer. The answer in this case is 76, whereas the observed value is 68, or 8 Ibs. less ; this does not agree very well with the theory, but still the difference, though 8 Ibs., is only about ii or 12 per cent, of the whole, and we shall still retain the law, for certainly there is no other that can express the result as well. 388. But the table will supply another verification. In experiment No. 3 a 40" bar, i" broad, and o"'5 deep, broke with 38 Ibs. ; and in experiment No. 7 a 20" bar of XIL] STRENGTH OF A BEAM. 193 the same section broke with 74 Ibs. ; but this is so nearly double the breaking weight of the 40" bar, as to be an additional illustration of the law, that for a given section the breaking load varies inversely as the span. ' 389. We next inquire as to the effect of the breadth of the beam upon its strength ? For this purpose we compare experiments Nos. 3 and 4 : we there find that a bar 4o"xi"Xo"'5 is broken by a load of 38 Ibs., while a bar just half the breadth is broken by 19 Ibs. We might have anticipated this result, for it is evident that the bar of No. 3 must have the same strength as two bars similar to that of No. 4 placed side by side. 390. This view is confirmed by a comparison of Nos. 7 and 8, where we find that a 20" bar takes twice the load to break it that is required for a bar of half its breadth. The law is not quite so well verified by Nos. 5 and 6, for half the breaking weight of No. 5, namely 29-5 Ibs., is more than 25, the observed breaking weight of No. 6 : a similar remark may be made about Nos. 9 and TO. 391. Supposing we had a beam of 40" span, 2" broad, and o'"5 deep, we can easily see that it is equivalent to two bars like that of No. 3 placed side by side ; and we infer generally that the strength of a bar is proportional to its breadth ; or to speak- more definitely, if hvo beams have the same span and depth, the ratio of their breaking loads is the same as the ratio of their breadths. 392. We next examine the effect of the depth of a beam upon its strength. In experimenting upon a beam placed edgewise, a precaution must be observed, which would not be necessary if the same beam were to be broken flatwise. When the load is suspended, the beam, if merely laid edgewise on the supports, would almost certainly turn over ; it is therefore necessary to place its extremities in recesses in O 194 EXPERIMENTAL MECHANICS. [LECT. the supports, which will obviate the possibility of this occurrence; at the same time the ends must not be prevented from bending upwards, for we are at present discussing a beam free at each end, and the case where the ends are not free will be subsequently considered. 393. Let us first compare together experiments Nos. 2 and 3 ; here we have two bars of the same dimensions, the section in each being i"-o x o'"5, but the first bar is broken edgewise, and the second flatwise. The first breaks with 77 Ibs., and the second with 38 Ibs. ; hence the same bar is twice as strong placed edgewise as flatwise when one dimension of the section is twice as great as the other. We may generalize this law, and assert that the strength of a rectangular beam broken edgewise is to the strength of a beam of like span and section broken flatwise, as the greater dimen- sion of the section is to the lesser dimension. 394. The strength of a beam 4o"Xo'"5x"i is four times as great as the strength of 4o"Xo'"5Xo"-5, though the quantity of wood is only twice as great in one as in the other. In general we may state that if a beam were bisected by a longitudinal cut, the strength of the beam would be halved when the cut was horizontal, and unaltered when the cut was vertical; thus, for example, two beams of experiment No. 4, placed one on the top of the other, would break with about 40 Ibs., whereas if the same rods were in one piece, the breaking load would be nearly 80 Ibs. 395. This may be illustrated in a different manner. I have here two beams of 4o"Xi"Xo'"5 superposed; they form one beam, equivalent to that of No. i in bulk, but I find that they break with 80 Ibs., thus showing that the two are only twice as strong as one. 396. I take two similar bars, and, instead of laying them loosely one on the other, I unite them tightly with iron xii.] STRENGTH OF A BEAM. 195 clamps like those represented in Fig. 56. I now find that the bars thus fastened together require iO4lbs. for fracture. We can readily understand this increase of strength. As soon as the bars begin to bend under the action of the weight, the surfaces which are in contact move slightly one upon the other in order to accommodate themselves to the change of form. By clamping I greatly impede this motion hence the beams deflect less, and require a greater load before they collapse ; the case is therefore to some extent approximated to the state of things when the two rods form one solid piece, in which case a load of 152 Ibs. would be required to produce fracture. 397. We shall be able by a little consideration to under- stand the reason why a bar is stronger edgewise than flatwise. Suppose I try to break a bar across my knee by pulling the ends held one in each hand, what is it that resists the breaking? It is chiefly the tenacity of the fibres on the convex surface of the bar. If the bar be edgewise, these fibres are further away from my knee and therefore resist with a greater moment than when the bar is flatwise : nor is the case different when the bar is supported at each end, and the load placed in the centre ; for then the reactions of the supports correspond to the forces with which I pulled the ends" of the bar. 398. We can now calculate the strength of any rectangular beam of pine: Let us suppose it to be 12' long, 5" broad, and 7" deep. This is five times as strong as a beam i" broad and 7" deep for we may conceive the original beam to consist of 5 of these beams placed side by side (Art 391); the beam i" broad and 7" deep, is 7 times as strong as a beam 7" broad, i" deep (Art. 393). Hence the original beam must be 35 times as strong as a beam 7" broad, i" deep ; but the o 2 196 EXPERIMENTAL MECHANICS. [LECT. beam 7" broad and i" deep is seven times stronger than a beam the section of which is i" X i", hence the original beam is 245 times as strong as a beam 12' long and i" x i" in section ; of which we can calculate the strength, by Art. 388, from the proportion 144" : 40" : : 152 : Answer. The answer is 42*2 Ibs., and thus the breaking load of the original beam is about 10,300 Ibs. 399. It will be useful to deduce the general expression for the breaking load of a beam /' span, b" broad, and d" deep, supported freely at the ends and laden in the centre. Let us suppose a bar /" long, and i" x i" in section. The breaking load is found by the proportion / : 40 : : 152 : Answer; and the result obtained is A beam which is d" broad, /" span, and i" deep, would be just as strong as d of the beams /" X i" X i placed side by side ; of which the collective strength would be If such a beam, instead of resting flatwise, were placed edge- wise, its strength would be increased in the ratio of its depth to its breadth that is, it would be increased