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Simple Machines

Efficiency of Machines. — Every machine wastes energy because of friction. Consequently, the work put into the machine is always more than that obtained from it. This loss decreases the efficiency of the machine. The efficiency of a machine is defined to be the ratio of the work done by the machine to the work done on the machine. It is therefore the output of the machine divided by the input.

work done output energy

Efficiency = –––––––––– = ––––––––––

energy suppliedinput energy

Levers. — A lever is a very simple form of machine.

The simplest kind of lever is one in which the arms are of equal length. The scale beam on a pair of ordinary balances is such a lever. In this case equal forces, or weights, at the ends of the lever just balance each other. Usually the distances of the forces from the point at which the lever is supported are not equal, and for equilibrium in such cases the forces must also be unequal. The larger force, at a smaller distance from the fulcrum, then has the same tendency to tip the lever as does the smaller force at a greater distance.

Law of the Lever. — Any lever is balanced when the sum of the moments of force tending to produce rotation clockwise is equal to the sum of the moments of force tending to produce rotation coun­terclockwise. If only one force is applied and one force overcome, this law may be stated as follows: A lever is balanced when

Weight x weight arm = force x fоrсе arm.

This law states that when a lever is balanced under the action of two forces, the forces applied to the lever are inversely proportional to their distances from the fulcrum.

Mechanical Advantage.— In all simple machines like levers a certain advantage is obtained by the use of the machine. This advan­tage does not consist in an increase or in a decrease of the work per­formed by the machine. Neglecting friction, the work done on the machine must always be the same as the work done by the machine. This law follows from the law of the conservation of energy. However, by means of a suitable lever or other machine it is possible to exert a large force by the application of a small force. The large force will net through a small distance, and the small force through a large dis­tance, so that the work done in the two cases will be the same.

The ratio of the resistance, or the force, overcome to the applied force is called the actual mechanical advantage. Because of friction the actual mechanical advantage is always less than the ideal, or theoretical, mechanical advantage.

The Pulley. — A pulley consists of a wheel with a grooved rim, called a sheave, which is free to move about an axle which is mounted in a frame called a block. A flexible rope or cord passes over the groove in the rim of the wheel. To the ends of this rope are applied the weight and the force that overcomes this weight. In the case of a simple fixed pulley, equal forces or weights applied to the ends of the rope just balance each other. Neglecting friction, the tension in the rope is everywhere the same and the mechanical advantage of the pulley is unity. Therefore there is no advantage in such a pulley except that it is sometimes more convenient to pull down on the rope than it is to lift the weight directly.

When the pulley is movable and the weight is attached to it, it is evident that the weight is supported by two parts of the cord. Therefore it is necessary for each part to exert a pull equal to only one-half of the weight. If the weight is lifted, it moves only one-half as far as the free end of the cord to which F is ap­plied. By applying the principle of work to this simple machine, it is seen that if the weight W is lifted α ft and the force F moves b ft, then

W x α =F x b,

W b

–– = –– = 2 = the mechanical advantage

F α

The Jackscrew. — When large forces must be exerted, a jackscrew is often used (Figure 21). The pitch of such a screw is the distance between successive threads. Let p be the pitch of the screw, W the weight to be lifted, F the force applied to the lever arm, and r the length of the lever arm. In one complete turn of the screw, the output is the weight lifted times the distance through which it is lifted.

W x p

Weight

Lever arm

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