THE THREE-SHEAVE PULLEY-BLOCK.

199. The next arrangement we shall employ is a pair of pulley-blocks s t, [Fig. 35], each containing three sheaves, as the small pulleys are termed. A rope is fastened to the upper block, s; it then passes down to the lower block t under one sheave, up again to the upper block and over a sheave, and so on, as shown in the figure. To the end of the rope from the last of the upper sheaves the power h is applied, and the load g is suspended from the hook attached to the lower block. When the rope is pulled, it gradually raises the lower block; and to raise the load one foot, each of the six parts of the rope from the upper block to the lower block must be shortened one foot, and therefore the power must have pulled out six feet of rope. Hence, for every foot that the load is raised the power must have acted through six feet; that is to say, the velocity ratio is 6.

200. If there were no friction, the power would only be one-sixth of the load. This follows at once from the principles already explained. Suppose the load be 60 lbs., then to raise it one foot would require 60 foot-pounds; and the power must therefore exert 60 foot-pounds; but the power moves over six feet, therefore a power of 10 lbs. would be sufficient. Owing, however, to friction, some energy is lost, and we must have recourse to experiment in order to test the real efficiency of the machine. The single moveable pulley nearly doubled our power; we shall prove that the three-sheave pulley-block will quadruple it. In this case we deal with larger weights, with reference to which we may leave the weight of the lower block out of consideration.

201. Let us first attach 1 cwt. to the load hook; we find that 29 lbs. on the power hook is the smallest weight that can produce motion: this is only 1 lb. more than one-quarter of the load raised. If 2 cwt. be the load, we find that 56 lbs. will just raise it: this time the power is exactly one-quarter of the load. The experiment has been tried of placing 4 cwt. on the hook; it is then found that 109 lbs. will raise it, which is only 3 lbs. short of 1 cwt. These experiments demonstrate that for a three-sheave pulley-block of this construction we may safely apply the rule, that the power is one-quarter of the load.

202. We are thus enabled to see how much of our exertion in raising weights must be expended in merely overcoming friction, and how much may be utilized. Suppose for example that we have to raise a weight of 100 lbs. one foot by means of the pulley-block; the power we must apply is 25 lbs., and six feet of rope must be drawn out from between the pulleys: therefore the power exerts 150 foot-pounds of energy. Of these only 100 foot-pounds are usefully employed, and thus 50 foot-pounds, one-third of the whole, have been expended on friction. Here we see that notwithstanding a small force overcomes a large one, there is an actual loss of energy in the machine. The real advantage of course is that by the pulley-block I can raise a greater weight than I could move without assistance, but I do not create energy; I merely modify it, and lose by the process.

203. The result of another series of experiments made with this pair of pulley-blocks is given in Table X.

Table X.—Three-Sheave Pulley-blocks.

Sheaves cast iron 2"·5 diameter; plaited rope 0"·25 diameter; velocity ratio 6; mechanical advantage 4; useful effect 67 per cent.; formula P = 2·36 + 0·238 R.

Number of
Experiment. 		R.
Load in lbs. 		Observed
power
in lbs.		P.
Calculated
power
in lbs..		Discrepancies
between
observed and
calculated powers.
1 5715·515·9+0·4
211429·529·5 0·0
317143·543·1-0·4
422856·056·6+0·6
528170·069·2-0·8
633883·082·8-0·2
739597·096·4-0·6
8452109·0 109·9 +0·9

204. This table contains five columns; the weights raised (shown in the second column) range up to somewhat over 4 cwt. The observed values of the power are given in the third column; each of these is generally about one-quarter of the corresponding value of the load. There is, however, a more accurate rule for finding the power; it is as follows.

205. To find the power necessary to raise a given load, multiply the loads in lbs. by 0·238, and add 2·36 lbs. to the product. We may express the rule by the formula P = 2·36 + 0·238 R.

206. To find the power which would raise 228 lbs.; the product of 228 and 0·238 is 54·26; adding 2·36, we find 56·6 lbs. for the power required; the actual observed power is 56 lbs., so that the rule is accurate to within about half a pound. In the fourth column will be found the values of P calculated by means of this rule. In the fifth column, the discrepancies between the observed and the calculated values of the powers are given, and it will be seen that the difference in no case reaches 1 lb. Of course it will be understood that this formula is only reliable for loads which lie between those employed in the first and last of the experiments. We can calculate the power for any load between 57 lbs. and 452 lbs., but for loads much larger than 452 or less than 57 it would probably be better to use the simple fourth of the load rather than the power computed by the formula.

207. I will next perform an experiment with the three-sheave pulley-block, which will give an insight into the exact amount of friction without calculation by the help of the velocity ratio. We first counterpoise the weight of the lower block by attaching weights to the power. It is found that about 1·6 lbs. is sufficient for this purpose. I attach a 56 lb. weight as a load, and find that 13·1 lbs. is sufficient power for motion. This amount is partly composed of the force necessary to raise the load if there were no friction, and the rest is due to the friction. I next gradually remove the power weights: when I have taken off a pound, you see the power and the load balance each other; but when I have reduced the power so low as 5·5 lbs. (not including the counterpoise for the lower block), the load is just able to overhaul the power, and run down. We have therefore proved that a power of 13·1 lbs. or greater raises 56 lbs., that any power between 13·1 lbs. and 5·5 lbs. balances 56 lbs., and that any power less than 5·5 lbs. is raised by 56 lbs.

When the power is raised, the force of friction, together with the power, must be overcome by the load. Let us call X the real power that would be necessary to balance 56 lbs. in a perfectly frictionless machine, and Y the force of friction. We shall be able to determine X and Y by the experiments just performed. When the load is raised a power equal to X + Y must be applied, and therefore X + Y = 13·1. On the other hand, when the power is raised, the force X is just sufficient to overcome both the friction Y and the weight 5·5; therefore X = Y + 5·5.

Solving this pair of equations, we find that X = 9·3 and Y = 3·8. Hence we infer that the power in the frictionless machine would be 9·3; but this is exactly what would have been deduced from the velocity ratio, for 56 ÷ 6 = 9·3 lbs. In this result we find a perfect accordance between theory and experiment.