193. By means of a moveable pulley a man is able to raise a weight nearly double as great as he could lift directly. From a series of careful experiments it has been found that when a man is employed in the particular exertion necessary for raising weights over a pulley, he is able to work most efficiently when the pull he is required to make is about 40 lbs. A man could, of course, exert greater force than this, but in an ordinary day’s work he is able to perform more foot-pounds when the pull is 40 lbs. than when it is larger or smaller. If therefore the weights to be lifted amount to about 80 lbs., energy may really be economized by the use of the single moveable pulley, although by so doing a greater quantity of energy would be actually expended than would have been necessary to raise the weights directly.
194. Some experiments on larger loads have been tried with the moveable pulley we have just described; the results are recorded in Table IX.
Table IX.—Single Moveable Pulley.
Moveable pulley of cast iron 3"·25 diameter, groove 0"·6 wide, wrought iron axle 0"·6 diameter; fixed pulley of cast iron 5" diameter, groove 0"·4 wide, wrought iron axle 0"·6 diameter, axles oiled; flexible plaited rope 0"·25 diameter; velocity ratio 2, mechanical efficiency 1·8, useful effect 90 per cent.; formula P = 2·21 + 0·5453 R.
| Number of Experiment. | R. Load in lbs. | Observed power in lbs. | P. Calculated power in lbs.. | Discrepancies between observed and calculated powers. |
|---|---|---|---|---|
| 1 | 28 | 17·5 | 17·5 | 0·0 |
| 2 | 57 | 33·5 | 33·3 | -0·2 |
| 3 | 85 | 48·5 | 48·6 | +0·1 |
| 4 | 113 | 64·0 | 63·8 | -0·2 |
| 5 | 142 | 80·0 | 79·6 | -0·4 |
| 6 | 170 | 94·5 | 94·9 | +0·4 |
| 7 | 198 | 110·5 | 110·2 | -0·3 |
| 8 | 226 | 125·5 | 125·5 | 0·0 |
The dimensions of the pulleys are precisely stated because, for pulleys of different construction, the numerical coefficients would not necessarily be the same. An attentive study of this table will, however, show the general character of the relation between the power and the load in all arrangements of this class.
The table consists of five columns. The first contains merely the numbers of the experiments for convenience of reference. In the second column, headed R, the loads, expressed in pounds, which are raised in each experiment, are given; that is, the weight attached to the hook, not including the weight of the lower pulley. The weight of this pulley is not included in the stated loads. In the third column the powers are recorded, which were found to be sufficient to raise the corresponding loads in the second column. Thus, in experiment 7, it is found that a power of 110·5 lbs. will be sufficient to raise a load of 198 lbs. The third column has thus been determined by gradually increasing the power until motion begins.
195. From an examination of the columns showing the power and the load, we see that the power always amounts to more than half the load. The excess is partly due to a small portion of the power (about 1·5 lbs.) being employed in raising the lower block, and partly to friction. For example, in experiment 7, if there had been no friction and if the lower block were without weight, a power of 99 lbs. would have been sufficient; but, owing to the presence of these disturbing causes, 110·5 lbs. are necessary: of this amount 1·5 lbs. is due to the weight of the pulley, 10 lbs. is the force of friction, and the remaining 99 lbs. raises the load.
196. By a calculation based on this table we have ascertained a certain relation between the power and the load; they are connected by the formula which may be enunciated as follows:
The power is found by multiplying the weight of the load into 0·5453, and adding 2·21 to the product. Calling P the power and R the load, we may express the relation thus: P = 2·21 + 0·5453 R. For example, in experiment 5, the product of 142 and 0·5453 is 77·43, to which, when 2·21 is added, we find for P 79·64, very nearly the same as 80 lbs., the observed value of the power.