This consists of a heavy beam which terminates in a knob at one end; and the body to be weighed is placed at the other end, the fulcrum being moveable. Let AB be the beam; let P denote its weight, and G its centre of gravity. The body to be weighed is suspended from A,

and the fulcrum is moved about until there is equilibrium, when the beam is horizontal. The Danish Steel-yard might be graduated by the aid of theory; for Pat G must balance the body hung from A according to the Principle of the Lever. Or we may proceed by experiment as before. Let a body of one pound weight be suspended at A; move the fulcrum about until there is equilibrium with the beam horizontal, and mark the position of the fulcrum by the figure 1. Again, instead of the weight of one pound at A put a weight of two pounds; move the fulcrum about as before, and mark with the figure 2 the place which it has when the beam is horizontal and in equilibrium. Proceeding in this way the beam becomes graduated and the Steelyard is fit for use. It will be found in this case that the figures 1, 2, 3, 4, ... do not succeed at equal intervals on the beam. Thus in using the Danish Steel-yard to weigh a body if the fulcrum comes precisely under one of the figures marked on the beam we know the weight of the body; but if the fulcrum comes between two of the figures we cannot tell the weight exactly, but only two values between which it must lie.

218. There are some weighing machines which do not depend on the Principle of the Lever. They usually consist mainly of a strong spring which is drawn out to a greater extent the heavier the body is which is suspended from it; and a contrivance is furnished by which we can readily observe how far the spring has been drawn out. These machines may be graduated by experiment, that is by suspending known weights and recording the corresponding points to which the spring is drawn out.

XIV. THE WHEEL AND AXLE. THE TOOTHED WHEEL.

219. In this Chapter we shall consider two other Mechanical Powers, namely, the Wheel and Axle, and the Toothed Wheel.

220. The Wheel and Axle. This machine consists of

G

W

two cylinders which have a common axis; the larger cylinder is called the Wheel and the smaller the Axle. The two cylinders are rigidly connected with the common axis, which is supported in a horizontal position, so that the machine can turn round it. The Weight acts by a string which is fastened to the Axle and coiled round it; the Power acts by a string which is fastened to the Wheel and coiled round it. The Weight and the Power tend to turn the machine round the axis in opposite directions.

221. When there is equilibrium on the Wheel and Axle the Power must be to the Weight in the same proportion as the radius of the Axle is to the radius of the Wheel. For it is easy to see the close resemblance between this machine and a Lever of the first class. It will be obvious that the effect of the Weight must be the same whether it is placed

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

B

W

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

B

W

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