If the axle is made of two different thicknesses, as shown in the diagram, it is called the compound or differential wheel and axle. From the figure you will notice that the

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CAPSTAN.

cord coils in one direction on one section, and in the opposite direction on the other, so that when it is winding on to the larger part it is unwinding from the smaller. The load raised will, for every complete turn of the axle, be lifted

through a height equal to half the difference of the circumferences of the larger and smaller axles. The power developed in a machine of this character will be more fully illustrated in the lesson on the pulley.

The wheel and axle in its elementary form is not so frequently met with as modifications of it, for the greater part of wheel-work machinery comes under this principle. We still, however, find it employed in raising water and loads from the holds of ships in the form of a windlass. A much more powerful application of the wheel and axle is that of the capstan, where the block is vertical. By this means enormous anchors and chain cables are raised with ease. The steering wheel which works the ship's rudder is another application of the same principle. Whether in compound or simple machinery, no mechanical power, perhaps, admits of so many modifications and uses as the wheel and axle.

TOOTHED WHEELS.

For example, this power occurs in toothed wheels or cogged wheels, and in wheels connected by endless belts, in cranes, watches, steam engines, and machinery of all kinds. In all these applications of the principle, the relation of the power to the weight is the same as that given for the ordinary wheel and axle. You can easily understand that in ordinary practice, however, the friction between wheels armed with teeth is very great, so that much of the power applied is really lost in overcoming this.

GENERAL RULE: Multiply the power at the edge of the wheel by its radius, and divide the product by the radius of the axle. The quotient is the weight that the power will raise.

This rule is practically the same as that stated for the application of the lever.

EXERCISES

1. Find the power necessary to raise a weight of 400 lb. by an axle of 10 in. and a wheel of 50 in. in diameter.

2. If the radius of the wheel is 3 ft., the weight 18 lb., and the power 3 lb., what is the radius of the axle ?

3. If the radius of the wheel is 6 ft., the radius of the axle 2 ft., and the weight 36 lb., what must be the power to produce equilibrium?

4. If in a capstan the radius of the axle is 1 ft., and 6 men push, each with a force of 100 lb., on spokes 5 ft. long, how many pounds will they be able to support?

5. The radius of the wheel being three times that of the axle, and the cord on the wheel being only strong enough to support a tension equivalent to 36 lb., find the greatest weight which can be lifted.

Lesson No. 39. Simple Mechanics — Pulleys

The pulley is a wheel over which a cord, or chain, or cable, is passed, in order to transmit the force applied to the cord in another direction.

When the block in which the pulley turns is fixed, the pulley is said to be fixed. There is no mechanical advan

tage gained by the use of one or more fixed pulleys; but this contrivance is of the greatest use in enabling us to change the

A PULLEY WITH DOUBLE
BLOCKS.

direction of the force. Thus, it is much more convenient to raise a bucket from a well by drawing downward, as is the case where the rope passes over a fixed pulley above the head, than by drawing upward, leaning over the curbing.

From its portable form, its cheapness, and the facility with which it can be applied, the pulley is one of the most convenient and most useful of the mechanical powers.

When the pulley block moves up and down with the cord, it is called a movable pulley; and when movable and fixed pulleys are worked together, we get what is called a "system of pulleys."

In the foregoing illustration suppose that the weight is 8 lb. It is supported by two cords, A and B; that is, the two sections of the cord support 4 lb. each. Now the cord being continuous, the power must be 4 lb.

We leave out of consideration the weight of the pulley in this case, and also the friction of the various parts of the machine.

Notice how the law of virtual velocities comes in here. We have seen that the weight is sustained by two cords; if, therefore, it has been raised one foot, shortened one foot. To do this, the power must run down two feet. To get the full value of this machine the cords must be parallel.

each cord must be