9. What force is required with a single fixed pulley to raise a weight of 200 lbs.? How far will the effort move in raising the weight 10 ft.? What is the mechanical advantage?
10. In the above problem substitute a single movable pulley for the fixed pulley and answer the same questions.
11. What is the smallest number of pulleys required to lift a weight of 600 lbs. with a force of 120 lbs.? How should they be arranged?
12. A derrick in lifting a safe weighing 2 tons uses a system of pulleys employing 3 sections of rope. What is the force required?
13. Name three instances where pulleys are used to do work that otherwise would be difficult to do.
14. Draw a diagram for a set of pulleys by means of which 100 lbs. can lift 400 lbs.
(5) The Inclined Plane. Efficiency
126. Efficiency.—The general law of machines which states that the work done by a machine equals the work put into it requires a modification, when we apply the law in a practical way, for the reason that in using any machine there is developed more or less friction due to parts of the machine rubbing on each other and to the resistance of the air as the parts move through it. Hence the statement of the law that accords with actual working conditions runs somewhat as follows: The work put into a machine equals the useful work done by the machine plus the wasted work done by it. The efficiency of a machine is the ratio of the useful work done by it to the total work done on the machine. If there were no friction or wasted work, the efficiency would be perfect, or, as it is usually expressed, would be 100 per cent. Consider a system of pulleys into which are put 600 ft.-lbs. of work. With 450 ft.-lbs. of useful work resulting, the efficiency would be 450 ÷ 600 = {3/4}, or 75 per cent. In this case 25 per cent. of the work done on the machine is wasted. In a simple lever the friction is slight so that nearly 100 per cent. efficiency is often secured.
Some forms of the wheel and axle have high efficiencies as in bicycles with gear wheels. Other forms in which ropes are employed have more friction. Pulleys have sometimes efficiencies as low as 40 per cent. when heavy ropes are used.
127. Inclined Plane.—We now come to a type of simple machine of lower efficiency than those previously mentioned. These belong to the inclined plane group, which includes the inclined plane (see Fig. 101), the wedge and the screw. They are extensively used, however, notwithstanding their low efficiency, on account of often giving a high mechanical advantage. The relation between these machines may be easily shown, as the wedge is obviously a double inclined plane. In Art. 82 it is shown that the effort required to hold a weight upon an inclined plane is to the weight supported as the height of the plane is to its length.