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 | 56 | 10 | 12·3 | +2·3 |
| 2 | 112 | 20 | 20·8 | +0·8 |
| 3 | 168 | 31 | 29·2 | -1·8 |
| 4 | 224 | 38 | 37·7 | -0·3 |
| 5 | 280 | 48 | 46·1 | -1·9 |
| 6 | 336 | 54 | 54·6 | +0·6 |
| 7 | 392 | 64 | 63·1 | -0·9 |
| 8 | 448 | 72 | 71·5 | -0·5 |
| 9 | 504 | 80 | 80·0 | 0·0 |
| 10 | 560 | 86 | 88·4 | +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.