that is, the mechanical advantage of the inclined plane, or the ratio of the weight lifted to the force acting parallel to the plane, is the ratio of the length of the plane to the height of the plane. This is precisely the conclusion at which we arrived in another way in Chapter V, p. 63. FIG. 131. The jackscrew 138. The screw. The screw (Fig. 131) is a combination of the inclined plane and the lever. Its law is easily obtained from the principle of work. When the force which acts on the end of the lever has moved this point through one complete revolution, the weight R, which rests on top of the screw, has evidently been lifted through a vertical distance equal to the distance between two adjoining threads. This distance d is called the pitch of the screw. Hence, if we represent by 7 the length of the lever, the work principle gives that is, the mechanical advantage of the screw, or the ratio of the weight lifted to FIG. 132. The letter press the force applied, is equal to the ratio of the circumference of the circle moved over by the end of the lever to the distance between the threads of the screw. In actual practice the friction in such an arrangement is always very great, so that the effort exerted must always be considerably greater than that given by equation 7. The common jackscrew just described (and used chiefly for raising buildings), the letter press (Fig. 132), and the vise (Fig. 133) are all familiar forms of the screw. FIG. 133. The vise 139. A train of gear wheels. A form of machine capable of very high mechanical advantage is the train of gear wheels shown in Fig. 134. Let the student show from the principle of work, namely Es = Rs', that 140. The worm wheel. Another device of high mechanical advantage Show that if I is the length of the crank is the worm wheel (Fig. 135). FIG. 134. Train of gear gear decrease speed rather than to multiply force. It will be seen that the crank handle must make n turns while the cogwheel is making one. The worm-gear "drive" is generally used in the rear axles of auto trucks. (1) (2) 141. The differential pulley. In the differential pulley (Fig. 136) an endless chain passes first over the fixed pulley A, then down and around the movable pulley C, then up again over the fixed pulley B, which is rigidly attached to A, but differs slightly from it in diameter. On the circumference of all the pulleys are projections which fit between the links, and thus keep the chains from slipping. When the chain is pulled down at E, as in Fig. 136, (2), until the upper rigid system of pulleys has made one complete revolution, the chain between the upper and lower pulleys has been shortened by the difference between the circumferences of the E R G FIG. 136. The differential pulley pulleys A and B, for the chain has been pulled up a distance equal to the circumference of the larger pulley and let down a distance equal to the circumference of the smaller pulley. Hence the load R has been lifted by half the difference between the circumferences of A and B. The mechanical advantage is therefore equal to the circumference of A divided by one half the difference between the circumferences of A and B. QUESTIONS AND PROBLEMS 1. A 1500-pound safe must be raised 5 ft. The force which can be applied is 250 lb. What is the shortest inclined plane which can be used for the purpose? 2. A 300-pound barrel was rolled up a plank 12 ft. long into a doorway 3 ft. high. What force was applied parallel to the plank? 3. A force of 80 kg. on a wheel whose diameter is 3 m. balances a weight of 150 kg. on the axle. Find the diameter of the axle. 4. If the capstan of a ship is 12 in. in diameter and the levers are 6 ft. long, what force must be exerted by each of 4 men in order to raise an anchor weighing 2000 lb. ? D F G H R E FIG. 137. The compound lever 5. If, in the compound lever of Fig. 137, AC = 6 ft., BC = 1 ft., DF = 4 ft., FG = 8 in., HJ = 5 ft., and IJ = 2 ft., what force applied at E will support a weight of 2000 lb. at R? E ገ n FIG. 138. Hay scales FIG. 139. Windlass with gears 6. The hay scales shown in Fig. 138 consist of a compound lever with fulcrums at F, F', F", F"". If Fo and F'o' are lengths of 6 in., FE and F'E 5 ft., F"n 1 ft., F"m 6 ft., rF' 2 in., and FS 20 in., how many pounds at W will be required to balance a weight of a ton on the platform? 7. In the windlass of Fig. 139 the crank handle has a length of 2 ft., and the barrel a diameter of 8 in. There are 20 cogs in the small cogwheel and 60 in the large one. What is the mechanical advantage of the arrangement? 8. If in the crane of Fig. 140 the crank arm has a length of m., and the gear wheels A, B, C, and D have respectively 12, 48, 12, and 60 cogs, while the axle over which the chain runs has a radius of 10 cm., what is the mechanical advantage of the crane? 9. If a worm wheel (Fig. 135) has 30 teeth, and the crank is 25 cm. long, while the radius of the axle is 3 cm., what is the mechanical advantage of the arrangement? 10. A small jackscrew has 20 threads to the inch. Using a lever 3 in. long will give what mechanical advantage? (Use 3.1416.) 11. The screw of a letter press has 5 threads to the inch, and the diameter of the wheel is 12 in. If there were no friction, what pressure would result from a rotating force of 20 lb. applied to the wheel? 12. Eight jackscrews, each of which has a pitch of in. and a lever arm of 18 in., are being worked simultaneously to raise a building weighing 100,000 lb. What force would have to be exerted at the end of each lever if there were no friction? What if 75% were wasted in friction? 13. What is gained by using a machine whose mechanical advantage is? Name two or three household appliances whose mechanical advantage is less than 1. POWER AND ENERGY 142. Definition of power. When a given load has been raised a given distance a given amount of work has been done, whether the time consumed in doing it is small or great. Time is therefore not a factor which enters into the determination of work; but it is often as important to know the rate at which work is done as to know the amount of work accomplished. The rate of doing work is called power, or activity. Thus, if P represent power, I the work done, and t the time required to do it, W P (10) 143. Horse power. James Watt (1736-1819), the inventor of the steam engine, considered that an average horse could do 33,000 foot pounds of work per minute, or 550 foot pounds per second. The metric equivalent is 76.05 kilogram meters per second. This number is probably considerably too high, but it has been taken ever since, in English-speaking countries, as the unit of power, and named the horse power (H.P.). The power of steam engines has usually been rated in horse power. The horse power of an ordinary railroad locomotive is from 500 to 1000. Stationary engines and steamboat engines of the largest size often run from 5000 to 20,000 H.P. The power of an average horse is about H.P., and that of an ordinary man about H.P. 144. The kilowatt. In the metric system the erg has been taken as the absolute unit of work. The corresponding unit of power is an erg per second. This is, however, so small that it is customary to take as the practical unit 10,000,000 ergs per second; that is, one joule per second (see § 125, p. 106). This unit is called the watt, in honor of James Watt. The power of dynamos and electric motors is almost always expressed in kilowatts, a kilowatt representing 1000 watts; and in modern practice even steam engines are being increasingly rated in kilowatts rather than in horse power. A horse power is equivalent to 746 watts, or about of a kilowatt. A kilowatt is almost exactly equal to 102 kilogram meters per second. 145. Definition of energy. The energy of a body is defined as its capacity for doing work. In general, inanimate bodies possess energy only because of work which has been done upon them at some previous time. Thus, suppose a kilogram weight |