as in the diagram, or whether it is placed at that part of the Axle which is close to the Wheel; and the effect of the Power must be the same whether it is placed as in the diagram, or whether it is placed at that part of the Wheel which is close to the Axle. Then if we imagine these changes to be made in the position of the Weight and the Power we obtain the following diagram:

Here CA is the radius of the Wheel, and CB is the radius of the Axle. We may consider ACB as a Lever of which C is the fulcrum. The Weight W, and the Power P, act in the manner shewn in the diagram; and in order that there may be equilibrium P must be to W in the same proportion as CB is to CA.

222. We have hitherto supposed that the Power

acts by means of a string, but it may act by the direct application of a man's hand, as in the familiar example of the machine used to draw up a bucket of water from a well.

223. The important principle of Art. 208 holds with respect to this machine. Suppose for instance that the radius of the Wheel is four times the radius of the Axle; then a weight of four pounds hanging round the Axle can be supported by a weight of one pound hanging round the Wheel. Thus a Power only a very little greater than one` pound will be sufficient to move the Weight of four pounds; but still to raise the Weight through any space the Power must descend through four times that space. Thus if the machine turns round just once, so as to raise the Weight through a space equal to the circumference of the Axle, then the Power descends through a space equal to the circumference of the Wheel; and these circumferences are in the same proportion as the radii, so that the circumference of the Wheel is four times that of the Axle.

224.

A Windlass and a Capstan may be considered as cases of the Wheel and Axle. The Windlass scarcely differs from the machine used to draw up water from a well; it has however more than one fixed handle for the convenience of working it, or there may be a moveable handle which can be shifted from one place to another. In the Capstan the fixed axis round which the machine turns is vertical; the hand which supplies the Power describes a circle in a horizontal plane, and the Weight is some heavy body which is attached to the Axle by a rope passing in a horizontal direction.

225. Toothed Wheels. Let two wheels of wood or

metal have their circumferences cut into equal teeth at equal distances. Let the Wheels be moveable about their centres, and in their own planes, and let them be placed in the same plane so that their edges touch, one tooth of one circumference lying between two teeth of the other circumference. If one of the Wheels of this pair be turned round its centre by any means the other Wheel will also be made to turn round its centre. Or a force which tends to turn one Wheel round may be balanced by a suitable force which tends to turn the other Wheel round in the contrary direction. The two forces may be supposed to act by means of strings on Axles belonging to the Toothed Wheels. Thus the Power P may be supposed to act at A, and the Weight W to act at B; also M is the common centre of one

Toothed Wheel and Axle, and N the common centre of the other Toothed Wheel and Axle.

226. The condition of equilibrium is somewhat complex; the reader may take it as verified by experiment: when there is equilibrium on a pair of Toothed Wheels the moment of the Power round the centre of its Axle must be to the moment of the Weight round the centre of its Axle in the same proportion as the radius of the Power Wheel is to the radius of the Weight Wheel. The principle of Art. 208 may be shewn to hold with respect to this machine.

227. In practice this machine is used to transmit motion; and then it is necessary to pay great attention to the form of the teeth, in order to secure uniform action in the machine, and to prevent the grinding away of the surfaces. On this subject, however, the student must consult works on mechanism. Toothed wheels are extensively applied in all machinery, as in cranes and steamengines, and especially in watch-work and clock-work.

228. Wheels are sometimes turned by means of straps passing over their circumferences: in such cases the minute protuberances of the surfaces prevent the sliding of the straps. The strap passing partly round a Wheel exerts a force on the Wheel at both points where it leaves the Wheel: the effect at each point would be measured by the moment of the tension of the strap at that point round the centre of the Wheel. If it were not for the friction, and the weight and stiffness of the strap, the tension would be the same throughout, and so the action at one point of the Wheel would balance the action at the other point.

XV. THE PULLY.

229. The Pully consists of a small circular plate or wheel which can turn round an axis passing through the centres of its faces, and having its ends supported by a framework which is called the block. The circular plate

has a groove cut in its edge to prevent a string from slipping off when it is put round the Pully.

230. Let A denote a Pully, the block of which is fixed; and suppose a Weight attached to the end of a string passing round the Pully. If the string be pulled at the other end by a Power equal to the Weight, there will be equilibrium; if the string be pulled by a Power somewhat greater, the Weight will be raised. Thus a fixed Pully is a machine by the aid of which we can change the direction of a force without changing its magnitude. For example, we might have a Power which could most conveniently act in

W

a direction inclined at a certain angle to the horizon, and we might wish to use it in supporting a Weight, that is in balancing a vertical force; then by transmitting the Power by means of a string, and passing the string round a fixed Pully, as in the diagram, we can support a Weight equal to the Power. A fixed Pully is often used when weights are to be raised, as for example the sails of ships. Thus a fixed Pully, though it may be very convenient, does not afford us any mechanical advantage: see Art. 207. We shall presently see that by the aid of a moveable Pully, or of a system of moveable Pullies, we do obtain mechanical advantage.

231. It might at first sight appear that nothing is gained by making the circular plate of the Pully capable of turning round its axis; but practically this is very important. When the circular plate can thus turn round, it is found by trial that in the state of equilibrium the tension of the string is almost exactly the same on both sides of the Pully, so that a Weight can be moved by a Power which is very slightly greater. But when the circular plate cannot turn round it is found that there may be a considerable difference between the tension of the string on the two sides of the Pully; and so a Weight could not be moved unless the Power were considerably greater.

This is owing to Friction, which we shall explain hereafter.

232. Now consider the case of a single moveable Pully. Let a string pass round the Pully A, have one end fixed as at K and be pulled vertically upwards by a Power P at the other end. Let a Weight W be attached to the block of the Pully. Thus it is found on trial that there is equilibrium if the Power is equal to half the Weight. In fact we may consider the block to be acted on by three parallel forces; namely the Weight downwards, and the two forces upwards arising from the tension of the two

parts of the string. Thus the Weight must be equal to the sum of the two upward forces. But the tension of the string is throughout equal to the Power. Therefore twice P is equal to W. The reader will see that there is nothing strange in this result. The Weight W has to be supported in some manner, and on examining we find that the result is the same as if half the Weight were supported by a fixed beam at K, and half by the Power.

233. If the Power be only a little greater than half the Weight, the Weight will be raised. According to the principle of Art. 208 if the Weight is raised through any space the end of the string at which the Power acts must be raised through twice that space. This may be easily shewn.

For suppose the Weight to be raised through any space, say one inch; then the part KC of the string between the fixed end and the Pully must be shortened by one inch, and to keep the string stretched the end at which Pacts must be raised through two inches. Thus the Power end of the string moves through twice the space through which the Weight moves.

K

B