XIII. The Wheel and Axle. The Toothed Wheel.

180. The present Chapter will be devoted to the Wheel and Axle, and the Toothed Wheel. It will be seen that these two Mechanical Powers are only modifications of the Lever.

W

181. The Wheel and Axle. This machine consists of 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.

182. When there is equilibrium on the Wheel and Axle, the Power is to the Weight as the radius of the Axle is to the radius of the Wheel.

Let two circles having the common centre Crepresent sections of the wheel and axle respectively, made by planes perpendicular to the axis of the cylinder.

It may be assumed, that the effects of the Power and the Weight will not be altered if we suppose them both to act in the same plane perpendicular to the axis. Let the string by which the power, P, acts leave the wheel at A, and the string by which the weight, W, acts leave the axle at B. Then CA and CB will be perpendicular to the line of action of P and W. We may regard

ACB as a lever of which C is the fulcrum, and hence, by Art. 165, the necessary and sufficient condition for equilibrium is

183. If we wish to take into account the thickness of the strings by which P and W act, we may consider that the line of action of each of these forces coincides with the middle of the respective strings. Thus, in the condition of equilibrium, CA will denote the radius of the wheel increased by half the thickness of the string by which P acts, and CB will denote the radius of the axle increased by half the thickness of the string by which W acts.

184. We have supposed that the Power in the Wheel and Axle acts by means of a string; but the Power may act by means of the hand, as in the familiar example of the machine used to draw up a bucket of water from a well.

A windlass and a capstan may also be considered as cases of the Wheel and Axle.

The windlass scarcely differs from the machine used to draw up water from a well: the windlass however has more than one fixed handle for the convenience of working

it; or it may have a moveable handle which can be shifted from one place to another.

In the capstan the fixed axis of the machine is vertical; the hand which supplies the Power describes a circle in a horizontal plane, and the rope to which the Weight is attached leaves the axle in a horizontal direction.

185. In the Wheel and Axle, as described in Art. 182, the whole pressure on the fixed supports is equal to the sum of the Weight and the Power; for the machine resembles a Lever with parallel and like forces. If the Power be directed vertically upwards, the Power and the Weight being then on the same side of the axis of the machine, the whole pressure on the fixed support is equal to the difference of the Weight and the Power.

186. The Toothed Wheel. Let two circles of wood or metal have their circumferences cut into equal teeth at equal distances. Let the circles be moveable about axes perpendicular to their planes, and let them be placed with their axes parallel, 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 axis by any means, the other wheel will also be made to turn round its axis. 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 opposite direction.

187. When there is equilibrium on a pair of Toothed Wheels, the moments of the Power and the Weight about the centres of their respective wheels are as the perpendiculars from the centres of the wheels on the direction of the pressure between the teeth in contact.

Let M and N be the fixed centres of the wheels. Suppose the Power, P, and the Weight, W, to act by strings which are attached to axles concentric with the wheels. Let these strings leave the axles at A and B respectively. Then MA and NB will be perpendicular to the lines of action of P and W.

Let Q denote the mutual pressure at the point of contact of the teeth; so that a force Q acts at the point of contact in opposite directions on the two wheels. Draw perpendiculars from M and N on the line of action of Q, meeting it at m and n respectively.

Then, since the wheel which can turn round M is in equilibrium, the moments round M must be equal; that is, P× AM=Q× Mm.

Similarly, since the wheel which can turn round N is in equilibrium,

this establishes the proposition.

Draw the straight line MN meeting mn at 0: then, by similar triangles,

Mm

MO

If the teeth are very small compared with the radii of the wheels, O will nearly coincide with the point of contact

of the teeth, and MO and NO will be nearly the radii of the wheels. Thus we have very nearly

moment of P round M radius of Power-wheel

=

moment of W round N radius of Weight-wheel'

188. In practice the 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 which treat specially of mechanism.

189. Toothed wheels are extensively applied in all machinery, as in cranes and steam engines, and especially in watch-work and clock-work.

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; and the condition of equilibrium coincides with that given at the end of Art. 187.

EXAMPLES. XIII.

1. Find the radius of the Wheel to enable a Power of 1 lbs. to support a weight of 28 lbs., the diameter of the Axle being 6 inches.

2. Find what Weight suspended from the Axle can be supported by 3 lbs. suspended from the Wheel, if the radius of the Axle is 1 feet, and the radius of the Wheel is 31 feet.

HAS

3. A man whose weight is 12 stone has to balance by his weight 15 cwt.: shew how to construct a Wheel and Axle which will enable him to do this.

4. A Weight of 14 ounces is supported by a certain Power on a Wheel and Axle, the radii being 28 inches and