CHAPTER VII.

THE MECHANICAL POWERS (continued).

2 (a). THE WHEEL AND AXLE.

Introduction.-The wheel and axle is only another form of the lever. It is used when the distance through which the weight has to be raised is very great, as in the case of a bucket from a well.

When the distance through which the weight has to move is very small, the lever is admirably adapted to produce the effect. But if it were required to raise the weight through a considerable distance by means of an ordinary lever, either the arm would have to be inconveniently long, or the weight would have to be raised by successive efforts with a shorter lever, which would necessitate the raising of the fulcrum after each effort. In order to avoid this intermittent motion, the lever is modified into the machine called the wheel and axle. Description. This machine consists of two cylinders,

Fig. 40.

Fig. 41.

W

A B and CD (fig. 40), having a common axis, but differing in radius. The smaller cylinder is called the axle, and the larger the wheel.

The Power is applied at the circumference of the wheel, and the Weight acts on the circumference of the axle. Round the cylinder is wrapped a rope, to which the weight to be raised is attached.

Reference to fig. 41 will convince us that this machine is only a perpetual lever. In this figure F is

the centre of the axle, FB the radius of the axle, and FA the radius of the wheel. The power acts at A, the extremity of the long arm A F; and the weight acts at B, the extremity of the shorter arm B F. Consequently, P: W::AF: BF. In other words, the power is to the weight as the radius of the axle is to the radius of the wheel.

Virtual Velocity.-It is easy to perceive that the power descends in one revolution of the wheel through a space equal to the circumference of the wheel, and the weight ascends a space equal to the circumference of the axle. The virtual velocities of the power and weight are therefore equal-that is to say, the power multiplied by its velocity equals the weight multiplied by its velocity.1

The force is conveyed to the axle in a great variety of ways, of which a few of the principal will be now described.

BD

Windlass.-The windlass, is a modification of the wheel and axle used for raising heavy weights-such as a ship's anchor. Instead of a wheel, levers called hand-spikes are inserted in holes cut in the axle. In this case means must be used to prevent the reaction of the weight during the removal of the levers. Such a contrivance is represented in fig. 42, and is called a ratchet wheel.3

Fig. 42.

A is the ratchet-wheel having teeth curved in the direction on which the rope is coiled round the cylinder. B is a bolt called a paul, movable on its centre, D, and by its weight falling between the teeth of the wheel A.

1 In the wheel and axle we have two parallel forces, P and W, acting at the ends of the lever A B moving on its fulcrum F. The equation to produce the equilibrium is therefore PXAF=WxBF, if the power acts in the direction of a tangent, as in figure 40. Hence, if X be the point on the circumference at which the power acts, Xx F is the moment of X about F. But XF-A F, therefore, XxF=WxB F, Dutch, winden, to wind; as, an axis. Ital., rochetto, a spindle.

From the direction of the teeth, it is evident that the wheel can be moved round only in the direction of the arrow; any tendency to revolve in the opposite direction being checked by the paul B falling between the teeth, immediately the power is withdrawn.

Capstan. The axle is sometimes placed vertically; the machine is then called a capstan,1 and possesses

Fig. 43.

considerable advantages over the ordinary windlass, The levers are inserted into holes in the capstan head, and the men walk round, pushing the levers before them. Any number of men may thus exert their strength upon the axle; and the power is constant, which is not the case with the horizontal axis, as will be shown immediately.

The barrel of the capstan is always of a conical form. The rope is wound round the barrel till the end is reached. The part A is then slackened, or surged, and the coils slip up towards the narrow end of the barrel, and a second series is then wound on. The number of coils wound on to the capstan barrel to start with, depends upon the strain, three being generally sufficient to cause the rope to so bite the capstan that it may be held tight by a boy at A.

Winch. In another case the axle is terminated by a bent lever, called a winch, by means of which the axle can be turned by the workman.

In this case the power is not uniform. By referring to the figure it will be seen that when, in the revolu

1

1 L., capistrum, a halter; so called from being used with a rope.

tion of the handle, the operator is exerting his strength from A to B and C, his weight will assist him, but in

Fig. 44.

the opposite part of the revolution, from C through D to A, he has to raise his body as well as the handle. To equalize this motion as much as possible, a double winch is used, with handles fixed in opposite directions, so that one descends while the other ascends. The capstan has not this disadvantage, and is therefore superior to the winch, where a number of men are to act upon a single axis.

Steering Wheel.-In a ship's steering wheel, levers as spokes project from the rim of the wheel.

Mechanical Efficiency.-The mechanical efficiency of the wheel and axle, depending theoretically upon the ratio existing between the radii of the wheel and of the axle, may be increased or diminished by increasing or diminishing that ratio, that is by enlarging the wheel or by diminishing the diameter of the axle. These alterations, however, can only be carried to a certain extent. For it is evident that if the wheel be too large, the power will have to act through an inconveniently large space. Again, if the axle be very small, its strength will be reduced so that the machine can only be employed for the raising of small weights.

It is necessary also, in constructing the machine, to consider the direction of the axis. If it be horizontal, and the centre of gravity of the whole machine fall midway between the two pivots supporting it and the wheel, the weight of the machine will press equally upon both. If the axis be vertical, which is sometimes

a convenient position, the whole weight will rest upon the lowest pivot, which should therefore be correspondingly strengthened.

Omitted Resistance.-In order to obtain the real mechanical efficiency of the wheel and axle, deductions have to be made from its theoretical efficiency, on account of the stiffness, weight, and thickness of the rope. Its stiffness and weight are real additions to the weight to be raised; while its thickness, by increasing the radius of the axle (the effect of the rope being considered to act through its centre), diminishes the ratio between it and the radius of the wheel. In addition to these, the weight of the machine and friction have to be allowed for.

Differential Wheel and Axle, or Chinese Windlass. -In cases where great resistance has to be overcome, the difficulty of so proportioning the ratio of the power to the weight, as to make the machine efficient without reducing the strength of the axle or increasing the radius of the wheel too much, is overcome by a contrivance called the differential wheel and axle, or Chinese windlass.

Fig. 45 represents this machine. AB is the axle,

Fig. 45.

consisting of two parts of different diameters. D is a pulley to which the weight C is attached. The rope is rove through this pulley, and coiled on both the thick and thin parts of the axle in the same direction. By turning the handle of the winch E, the rope is coiled on the thicker part, and wound off the thinner one.

Every revolution of the axle will draw up a length of rope equal to the circumference of the thick axle, and at the same time a length equal to the circumference of the thin axle will be let down.