Fig. 85.—A windmill.
119. Law of Machines.—When a body receives energy, work is done upon it. Therefore work is done upon a machine when it receives energy and the machine does work upon the body to which it gives the energy. In the operation of a machine, therefore, two quantities of work are to be considered and by the principle of the conservation of energy, these two must be equal. The work done by a machine equals the work done upon it, or the energy given out by a machine equals the energy received by it. These two quantities of work must each be composed of a force factor and a space factor. Therefore two forces and two spaces are to be considered in the operation of a machine. The force factor of the work done on the machine is called the force or effort. It is the force applied to the machine. The force factor of the work done by a machine is called the weight or resistance. It is the force exerted by the machine in overcoming the resistance and equals the resistance overcome.
If f represents the force or effort, and Df the space it acts through, and w represents the weight or resistance, and Dw the space it acts through, then the law of machines may be expressed by an equation, f × Df = w × Dw. That is, the effort times the distance the effort acts equals the resistance times the distance the resistance is moved or overcome. When the product of two numbers equals the product of two other numbers either pair may be made the means and the other the extremes of a proportion. The equation given above may therefore be expressed w: f = Df: Dw. Or the resistance is to the effort as the effort distance is to the resistance distance. The law of machines may therefore be expressed in several ways. One should keep in mind, however, that the same law of machines is expressed even though the form be different. What two ways of expressing the law are given?
120. The Simple Machines.—There are but six simple machines. All the varieties of machines known are simply modifications and combinations of the six simple machines. The six simple machines are more easily remembered if we separate them into two groups of three each. The first or lever group consists of those machines in which a part revolves about a fixed axis. It contains the lever, pulley and wheel and axle. The second or inclined plane group includes those having a sloping surface. It contains the inclined plane, the wedge, and the screw.
121. The Lever.—The lever is one of the simple machines most frequently used, being seen in scissors, broom, coal shovel, whip, wheelbarrow, tongs, etc. The lever consists of a rigid bar capable of turning about a fixed axis called the fulcrum. In studying a lever, one wishes to know what weight or resistance it can overcome when a certain force is applied to it. Diagrams of levers, therefore, contain the letters w and f. In addition to these, O stands for the fulcrum on which it turns. By referring to Fig. 86, a, b, c, one may notice that each of these may occupy the middle position between the other two. The two forces (other than the one exerted by the fulcrum) acting on a lever always oppose each other in the matter of changing rotation. They may be considered as a pair of parallel forces acting on a body, each tending to produce rotation.
Fig. 86.—The three classes of levers.
122. Moment of Force.—The effectiveness of each force may therefore be determined by computing its moment about the fixed axis (see Art. 84), that is, by multiplying each force by its distance to the fulcrum or axis of rotation. Let a meter stick have a small hole bored through it at the 50 cm. mark near one edge, and let it be mounted on a nail driven into a vertical support and balanced by sliding a bent wire along it. Suspend by a fine wire or thread a 100 g. weight, 15 cm. from the nail and a 50 g. weight 30 cm. from the nail, on the other side of the support. These two weights will be found to balance. When viewed from this side A (Fig. 87) tends to turn the lever in a clockwise direction (down at right), B in the counter-clockwise direction (down at left). Since the lever balances, the forces have equal and opposite effects in changing its rotation as may also be computed by determining the moment of each force by multiplying each by its distance from the fulcrum. Therefore the effectiveness of a force in changing rotation depends upon the distance from it to the axis as well as upon the magnitude of the force.
