than itself as the radius of the wheel is times greater than the radius of the axle. The radii in Formula 31 can be replaced by either the circumferences or the diameters if it is more convenient.

The mechanical advantage of the wheel and axle is the ratio of R to r. If the power and weight are disposed as

in Fig. 76, the mechanical advantage is one of force; if the points of application of P and W are interchanged, it is one of speed.

The wheel and axle is used to raise water from a well, to hoist ore from a mine, as with the windlass, to move buildings, and to raise anchors, as with the capstan. In the capstan no wheel is used, but instead straight bars, called hand-spikes, are

FIG. 77.-The Capstan

put into holes in the head of the capstan, and the power is applied to these.

110. Combinations of the Wheel and Axle, with the axle of one system working upon the FIG. 78.-Automobile Transmission Gearing wheel of another, are used, not only where great weights are to be lifted, but also where it is desired to make a great difference in speed between the movement of the power and of the resistance.

These results are usually secured by the use of a train of cog wheels such as is shown in Fig. 78, which represents a set of automobile transmission gearing.

111. The Pulley. The fixed pulley, The fixed pulley, in which the axis of the pulley is held in a fixed position is a modified lever of the first class; but in this machine the power arm is always equal to the weight arm, so that there is no gain in using it, except change in direction. This may be seen readily by reference to Fig. 79. The power is applied at one end of a rope that passes around the pulley in a groove cut in its edge, and is tangent at the points A and B. Apply the law of the lever, and the proportion will stand

P

W

FIG. 80

P: Wr: R;
.. P = W.

or

but r = R,

(32)

112. The Movable Pulley, in which the axis of the pulley can move with it, is a modified lever, but it is of the second class, the fulcrum being at B (Fig. 80), the weight (including the weight of the pulley) being applied at C with a lever arm CB=r, and the power at A with a lever arm AB = D. The formula for the single movable pulley is P: Wr: D, and since D is the diameter and r is the radius of the pulley, this becomes

FIG. 79

(33)

113. Combinations of Fixed Pulleys. — Figure 81 shows how, by a combination of fixed pulleys, the horizontal pull of a horse can be used to raise a heavy weight. The mechanical advantage secured by the movable pulley would frequently be useless if it were not for combining with it one or more fixed pulleys by which the direction of the pull can be changed.

114. Systems of Fixed and Movable Pulleys. Where great weights are to be raised, systems of pulleys are used. Usually a number of "sheaves " or pulleys are arranged side by side in the same block, and a single rope is passed alternately around the sheaves in two of these blocks, called the "block and tackle" (Fig. 82). Another arrangement is shown in Fig. 83. The weight is attached to the movable block A (Figs. 82 and 83), and since the rope is continuous there must be the same pull on each branch between the blocks. If we let n represent the number of branches extending to the movable block (n=6 in Figs. 82 and 83), then by § 62 each branch must A

A

P

FIG. 81

(34)

P

W

FIG. 83

FIG. 84. Fixed and Movable Pulleys in Use.

115. The Inclined Plane. Any plane surface that makes an angle with a horizontal surface forms an inclined plane. A ball placed upon a horizontal plane will retain its position and will press upon the plane with its entire weight. As soon, however, as one end of the plane is raised, the entire weight of the ball will not rest upon the plane, and it will begin to roll toward the lower end. The only way in which an inclined plane can be used efficiently is to have the moving

force act in a direction that is parallel to the inclined surface of the plane. Inclined planes are used for the purpose of lifting a weight to a certain height by the use of a small power. The power which moves a body from the bottom of the plane to the top lifts it through the height of the plane against gravity and hence the general law of machines will apply. This may be modified to read

=

(35)

in which H is the vertical height of the plane and L is the length along the slope,