219. To raise 280 lbs. one foot 280 foot-pounds of energy would be necessary, but in the differential pulley-block 46·09 lbs. must be exerted for a distance of 16 feet in order to accomplish this object. The product of 46·09 and 16 is 737·4. Hence the differential pulley-block requires 737·4 foot-pounds of energy to be applied in order to yield 280 useful foot-pounds; but 280 is only 38 per cent. of 737·4, and therefore with a load of 280 lbs. only 38 per cent. of the energy applied to a differential pulley-block is utilized. In general, we may state that not more than about 40 per cent. is profitably used, and that the remainder is expended in overcoming friction.
220. It is a remarkable and useful property of the differential pulley, that a weight which has been hoisted will remain suspended when the hand is removed, even though the chain be not secured in any manner. The pulleys we have previously considered do not possess this convenient property. The weight raised by the three-sheave pulley-block, for example, will run down unless the free end of the rope be properly secured. The difference in this respect between these two mechanical powers is not a consequence of any special mechanism; it is simply caused by the excessive friction in the differential pulley-block.
221. The reason why the load does not run down in the differential pulley may be thus explained. Let us suppose that a weight of 400 lbs. is to be raised one foot by the differential pulley-block; 400 units of work are necessary, and therefore 1,000 units of work must be applied to the power chain to produce the 400 units (since only 40 per cent. is utilized). The friction will thus have consumed 600 units of work when the load has been raised one foot. If the power-weight be removed, the pressure supported by the upper pulley-block is diminished. In fact, since the power-weight is about ¹/₆th of the load, the pressure on the axle when the power-weight has been removed is only ⁶/₇ths of its previous value. The friction is nearly proportional to that pressure: hence when the power has been removed the friction on the upper axle is ⁶/₇ths of its previous value, while the friction on the lower pulley remains unaltered.
We may therefore assume that the total friction is at least ⁶/₇ths of what it was before the power-weight was removed. Will friction allow the load to descend? 600 foot-pounds of work were required to overcome the friction in the ascent: at least ⁶/₇ × 600 = 514 foot-pounds would be necessary to overcome friction in the descent. But where is this energy to come from? The load in its descent could only yield 400 units, and thus descent by the mere weight of the load is impossible. To enable the load to descend we have actually to aid the movement by pulling the chain d ([Figs. 36] and [37]), which proceeds from the small groove in the upper pulley.
222. The principle which we have here established extends to other mechanical powers, and may be stated generally. Whenever more than half the applied energy is consumed by friction, the load will remain without running down when the machine is left free.
THE EPICYCLOIDAL PULLEY-BLOCK.
223. We shall conclude this lecture with some experiments upon a useful mechanical power introduced by Mr. Eade under the name of the epicycloidal pulley-block. It is shown in [Fig. 33], and also in [Fig. 49]. In this machine there are two chains: one a slight endless chain to which the power is applied; the other a stout chain which has a hook at each end, from either of which the load may be suspended. Each of these chains passes over a sheave in the block: these sheaves are connected by an ingenious piece of mechanism which we need not here describe. This mechanism is so contrived that, when the power causes the sheave to revolve over which the slight chain passes, the sheave which carries the large chain is also made to revolve, but very slowly.
224. By actual trial it is ascertained that the power must be exerted through twelve feet and a half in order to raise the load one foot; the velocity ratio of the machine is therefore 12·5.
225. If the machine were frictionless, its mechanical efficiency would be of course equal to its velocity ratio; owing to the presence of friction the mechanical efficiency is less than the velocity ratio, and it will be necessary to make experiments to determine the exact value. I attach to the load hook a weight of 280 lbs., and insert a few small hooks into the links of the power chain in order to receive weights: 56 lbs. is sufficient to produce motion, hence the mechanical efficiency is 5. Had there been no friction a power of 56 lbs. would have been capable of overcoming a load of 12·5 × 56 = 700 lbs. Thus 700 units of energy must be applied to the machine in order to perform 280 units of work. In other words, only 40 per cent. of the applied energy is utilized.
226. An extended series of experiments upon the epicycloidal pulley-block is recorded in Table XII.