LECTURE V.. CONTENTS.-The Principle of Work-Work put in, Work lost, Useful Work-Efficiency of a Machine-Principle of Work applied to the Lever-Experiments I. II.-Wheel and Axle - The Principle of Moments applied to the Wheel and Axle-The Principle of Work applied to the Wheel and Axle-Experiment III.-The Winch Barrel —Example I.—Ship's Capstan—The Fusee-Questions. The Principle of Work.*-The principle of work is applicable to all machines, and may be stated as follows: The work put into a machine is equal to the work absorbed by the machine plus the work given out by the machine. Or, WORK PUT IN LOST WORK + USEFUL WORK. This is an axiom. But, nevertheless, many deluded would-be inventors have spent much time and money in devising "perpetual motion " appliances, or machines which should turn out as much work as, or even more than, was put into them! I. When a machine is employed to perform mechanical work, a certain force must be applied to one part of it in order to move the machine and to perform work at another part. The product of this applied force and the distance through which it acts constitute the whole work put into the machine. 2. Some of this work must be expended in merely keeping the different parts in motion, against natural resistances due to friction at the fulcra or journals, and friction between moving parts and the air or water in the case of an hydraulic apparatus. work so absorbed is termed lost work. The The mean value of the frictional resistances, multiplied by the mean distance through which they are overcome, constitute the work lost in the mechanism. One great object to be kept in view, in designing most machines, is to minimise this lost work by minimising the internal resistances to motion in the machine *The Principle of Work is usually stated as follows in books on Mechanics, but I find that engineering students much prefer the above definition. "If a system of bodies be at rest under the action of any forces, and be moved a very little, no work will be done." Conversely: If no work is one during this small movement, the forces are in equilibrium.”—Prof. eve's Manual of Mechanics," p. 73. 66 PRINCIPLE OF WORK APPLIED TO THE LEVER. 39 itself; but you must remember that these can never be entirely disposed of, as has only too often been conjectured by "perpetual motion" faddists. 3. The remainder goes to do the useful work for which the machine was designed, and therefore 4. The efficiency of a machine = the work got out. the work put in. To impress these facts on the mind of the student we present them in the following condensed form : = 1. Work put in force applied × the distance it acts. 2. Work lost =force absorbed in overcoming internal resistances x the distance it acts. 3. Useful workforce given out × the distance it acts. 4. Efficiency ratio of work got out to work put in. = 5. Work put in = lost work + useful work. A2 Principle of Work applied to the Lever.-In applying the above "principle of work" to the lever, we will take the liberty of neglecting the lost work. We shall therefore assume that the friction at the fulcrum is so small that it may be neglected for the purpose we have in view. EXPERIMENT I. Let AF be a straight lever without weight, having its fulcrum at F, a force, W, acting vertically downwards from the point B1, W PRINCIPLE OF WORK APPLIED TO A LEVER. and a force, P, acting vertically upwards at the end A,, keeping W in equilibrium. Now imagine the lever elevated to the position A,F. 1 The work put in at A1 =P × the vertical distance from A, to A,. The work got out at B1 = W × the vertical distance from B1 to B. Therefore, since we neglect all frictional resistances Or, i.e., The work put in But by Euclid the triangles A,FA, and B,FB, are similar in every respect. But this is the equation we proved in Lecture III. with respect to the lever as complying with the "principle of moments." Hence the "principle of work" and the "principle of moments" are in agreement. In the accompanying figure the force P has been shown as elevated through 12", and the force W as elevated through 6". Therefore, Or, P x 12" W x 6" P being half the magnitude of W, it has to be elevated through double the distance in order that the same amount of work may be done in the same time. EXPERIMENT II.-Consider the case of a simple lever, where a weight, W, at B is balanced by another weight, P, at A, around PRINCIPLE OF WORK APPLIED TO A LEVER. a fulcrum at F, without friction. Let the lever be turned through 90°, or a quarter of a revolution-i.e., from a horizontal position, AB, to a vertical position, A'B'. Then by the definition of work— The work put in at A = P × A'F, and The work got out at B= W × B'F. THE WHEEL AND AXLE. 41 It does not matter in the slightest degree how circuitous the paths P and W take in passing from their original to their new positions in this case, since all we require to know is the vertical distances through which P is depressed and W elevated. Consequently, by the "Principle of Work," .. Substituting AF for A'F, and BF for B'F, We get, Px AF = W x BF But this is the equation for the "principle of moments," which we have again deduced from the "principle of work" by another and simpler form of reasoning. We find that this latter method appeals more directly to the minds of young engineering students than the proofs usually found in books on Mechanics. The Wheel and Axle.-The wheel and axle has been used for centuries for drawing water by a bucket from a well. It is used by the navvy for lifting the material which he excavates from the earth, by the mason for raising stones, bricks and mortar, and by many other tradesmen for a variety of purposes; as well as by the quartermaster as a steeringgear, and the able seaman as a capstan. The accompanying illustration shows the form it takes when used for elevating goods in a store or mill.* It is simply a practical arrangement for continuing the action of the lever as long as required. So long as a sufficient pull is applied to the rope, which fits into the grooved wheel, to overcome the resistance of the load attached to the chain hook, the weight will be raised. The wheel and axle is therefore a form of lever by which a weight may be raised through any desired height. WHEEL AND AXLE. The Principle of Moments applied to the Wheel and Axle. In the diagram let the larger circle represent the circumference of a wheel of radius, R, to the periphery of which a force, *The above figure represents a wheel and axle as supplied by Messrs. P. & W. MacLellan, of Glasgow. -R B applied a resistance W. P, is applied. Let the smaller circle represent the circumference of the axle or barrel of radius, r, to the periphery of which is Let the forces P and W act in the same direction and vertically downwards. Join the points where the lines of action of the forces are tangents to the wheel and axle by a straight line, AB. Then, AB passes through the common centre of the circles-i.e., through their common centre of motion or fulcrum F, and AF is the effective arm for the force P, whilst BF is the effective arm for the force W. In fact, AFB is a straight lever in equilibrium, with the fulcrum at F. moments about F, we have W WHEEL AND AXLE. Therefore, taking The Principle of Work applied to the Wheel and Axle. EXPERIMENT III.-Take a model of the wheel and axle as illustrated by the accompanying figure. Let forces, P and W, act in equilibrium, as in the previous case, at radii R and r respectively. MODEL TO TEST THE PRINCIPLE OF WORK APPLIED TO THE WHEEL AND AXLE. Now mark carefully with a piece of coloured chalk or ink the exact positions where the tape supporting P is a tangent to the wheel, |