THE SINGLE MOVEABLE PULLEY.

187. We commence with the most simple case, that of the single moveable pulley ([Fig. 35]). The rope is firmly secured at one end a; it then passes down under the moveable pulley b, and upwards over a fixed pulley. To the free end, which depends from the fixed pulley, the power is applied while the load to be raised is suspended from the moveable pulley. We shall first study the relation between the power and the load in a simple way, and then we shall describe a few exact experiments.

188. When the load is raised the moveable pulley itself must of course be also raised, and a part of the power is expended for this purpose. But we can eliminate the weight of the moveable pulley, so far as our calculations are concerned, by first attaching to the power end of the rope a sufficient weight to lift up the moveable pulley when not carrying a load. The weight necessary for doing this is found by trial to be a little over 1·5 lbs. So that when a load is being raised we must reduce the apparent power by 1·5 lbs. to obtain the power really effective.

189. Let us suspend 14 lbs. from the load hook at b, and ascertain what power will raise the load. We leave the weight of the moveable pulley and 1·5 lbs. of the power at c out of consideration. I then find by experiment that 7 lbs. of effective power is not sufficient to raise the load, but if one pound more be added, the power descends, and the load is raised. Here, then, is a remarkable result; a weight of 8 lbs. has overcome 14 lbs. In this we have the first application of the mechanical powers to increase our available forces.

Fig. 35.

190. Let us examine the reason of this mechanical advantage. If the load be raised one foot, it is plain that the power must descend two feet: for in order to raise the load the two parts of the rope descending to the moveable pulley must each be shortened one foot, and this can only be done by the power descending two feet. Hence when the load of 14 lbs. is lifted by the machine, for every foot it is raised the power must descend two feet: this simple point leads to a conception of the greatest importance, on which depends the efficiency of the pulley. In the study of the mechanical powers it is essential to examine the number of feet through which the power must act in order to raise the load one foot: this number we shall always call the velocity ratio.

191. To raise 14 lbs. one foot requires 14 foot-pounds of energy. Hence, were there no such thing as friction, 7 lbs. on the power hook would be sufficient to raise the load; because 7 lbs. descending through two feet yields 14 foot-pounds. But there is a loss of energy on account of friction, and a power of 7 lbs. is not sufficient: 8 lbs. are necessary. Eight lbs. in descending two feet performs 16 foot-pounds; of these only 14 are utilised on the load, the remainder being the quantity of energy that has been diverted by friction. We learn, then, that in the moveable pulley the quantity of energy employed is really greater than that which would lift the weight directly, but that the actual force which has to be exerted is less.

192. Suppose that 28 lbs. be placed on the load hook, a few trials assure us that a power of 16 lbs. (but not less) will be sufficient for motion; that is to say, when the load is doubled, we find, as we might have expected, that the power must be doubled also. It is easily seen that the loss of energy by friction then amounts to 4 foot-pounds. We thus verify, in the case of the moveable pulley, the approximate law that the friction is proportional to the load.

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 2817·517·5 0·0
2 5733·533·3-0·2
3 8548·548·6+0·1
411364·063·8-0·2
514280·079·6-0·4
617094·594·9+0·4
7198110·5 110·2 -0·3
8226125·5 125·5  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.

In the fourth column the values of P calculated by means of this formula are given, and in the last we exhibit the discrepancies between the observed and the calculated values for the sake of comparison. It will be seen that the discrepancy in no case amounts to 0·5 lb., consequently the formula expresses the experiments very well. The mode of deducing it is given in the Appendix.

197. The quantity 2·21 is partly that portion of the power expended in overcoming the weight of the moveable pulley, and partly arises from friction.

198. We can readily calculate from the formula how much power will be required to raise a given weight; for example, suppose 200 lbs. be attached to the moveable pulley, we find that 111 lbs. must be applied as the power. But in order to raise 200 lbs. one foot, the power exerted must act over two feet; hence the number of foot-pounds required is 2 × 111 = 222. The quantity of energy that is lost is 22 foot-pounds. Out of every 222 foot-pounds applied, 200 are usefully employed; that is to say, about 90 per cent. of the applied energy is utilized, while the remaining 10 per cent. is lost by friction.