THE DIFFERENTIAL PULLEY-BLOCK.

208. By increasing the number of sheaves in a pair of pulley-blocks the power may be increased; but the length of rope (or chain) requisite for several sheaves becomes a practical inconvenience. There are also other reasons which make the differential pulley-block, which we shall now consider, more convenient for many purposes than the common pulley blocks when a considerable augmentation of power is required.

209. The principle of the differential pulley is very ancient, and in modern times it has been embodied in a machine of practical utility. The object is to secure, that while the power moves over a considerable distance, the load shall only be raised a short distance. When this has been attained, we then know by the principle of energy that we have gained a mechanical advantage.

210. Let us consider the means by which this is effected in that ingenious contrivance, Weston’s differential pulley-block. The principle of this machine will be understood from [Fig. 36] and [Fig. 37].

Fig. 36.

It consists of three parts,—an upper pulley-block, a moveable pulley, and an endless chain. We shall briefly describe them. The upper block p is furnished with a hook for attachment to a support. The sheave it contains resembles two sheaves, one a little smaller than the other, fastened together: they are in fact one piece. The grooves are provided with ridges, adapted to prevent the chain from slipping. The lower pulley q consists of one sheave, which is also furnished with a groove; it carries a hook, to which the load is attached. The endless chain performs a part that will be understood from the sketch of the principle in [Fig. 36]. The chain passes from the hand at a up to l over the larger groove in the upper pulley, then downwards at b, under the lower pulley, up again at c, over the smaller groove in the upper pulley at a, and then back again by d to the hand at a. When the hand pulls the chain downwards, the two grooves of the upper pulley begin to turn together in the direction shown by the arrows on the chain. The large groove is therefore winding up the chain, while the smaller groove is lowering.

211. In the pulley which has been employed in the experiments to be described, the effective circumference of the large groove is found to be 11"·84, while that of the small groove is 10"·36. When the upper pulley has made one revolution, the large groove must have drawn up 11"·84 of chain, since the chain cannot slip on account of the ridges; but in the same time the small groove has lowered 10"·36 of chain: hence when the upper pulley has revolved once, the chain between the two must have been shortened by the difference between 11"·84 and 10"·36, that is by 1"·48; but this can only have taken place by raising the moveable pulley through half 1"·48, that is, through a space 0"·74. The power has then acted through 11"·84, and has raised the resistance 0"·74. The power has therefore moved through a space 16 times greater than that through which the load moves. In fact, it is easy to verify by actual trial that the power must be moved through 16 feet in order that the load may be raised 1 foot. We express this by saying that the velocity ratio is 16.

Fig. 37.

212. By applying power to the chain at d proceeding from the smaller groove, the chain is lowered by the large groove faster than it is raised by the small one, and the lower pulley descends. The load is thus raised or lowered by simply pulling one chain a or the other d.

213. We shall next consider the mechanical efficiency of the differential pulley-block. The block ([Fig. 37]) which we shall use is intended to be worked by one man, and will raise any weight not exceeding a quarter of a ton.

We have already learned that with this block the power must act through sixteen feet for the load to be raised one foot. Hence, were it not for friction, the power need only be the sixteenth part of the load. A few trials will show us that the real efficiency is not so large, and that in fact more than half the work exerted is merely expended upon overcoming friction. This will lead afterwards to a result of considerable practical importance.

214. Placing upon the load hook a weight of 200 lbs., I find that 38 lbs. attached to a hook fastened on the power chain is sufficient to raise the load; that is to say, the power is about one-sixth of the load. If I make the load 400 lbs. I find the requisite power to be 64 lbs., which is only about 3 lbs. less than one-sixth of 400 lbs. We may safely adopt the practical rule, that with this differential pulley-block a man would be able to raise a weight six times as great as he could raise without such assistance.

215. A series of experiments carefully tried with different loads have given the results shown in Table XI.

Table XI.—The Differential Pulley-block.

Circumference of large groove 11"·84, of small groove 10"·36; velocity ratio 16; mechanical efficiency 6·07; useful effect 38 per cent.; formula P = 3·87 + 0·1508 R.

Number of
Experiment. 		R.
Load in lbs. 		Observed
power
in lbs.		P.
Calculated
power
in lbs..		Differences of the
observed and
calculated powers.
1 561012·3+2·3
21122020·8+0·8
31683129·2-1·8
42243837·7-0·3
52804846·1-1·9
63365454·6+0·6
73926463·1-0·9
84487271·5-0·5
95048080·0 0·0
10 5608688·4+2·4

The first column contains the numbers of the experiments, the second the weights raised, the third the observed values of the corresponding powers. From these the following rule for finding the power has been obtained:—

216. To find the power, multiply the load by 0·1508, and add 3·87 lbs. to the product; this rule may be expressed by the formula P = 3·87 + 0·1508 R. ([See Appendix].)

217. The calculated values of the powers are given in the fourth column, and the differences between the observed and calculated values in the last column. The differences do not in any case amount to 2·5 lbs., and considering that the loads raised are up to a quarter of a ton, the formula represents the experiments with satisfactory precision.

218. Suppose for example 280 lbs. is to be raised; the product of 280 and 0·1508 is 42·22, to which, when 3·87 is added, we find 46·09 to be the requisite power. The mechanical efficiency found by dividing 46·09 into 280 is 6·07.

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.