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 | 57 | 15·5 | 15·9 | +0·4 |
| 2 | 114 | 29·5 | 29·5 | 0·0 |
| 3 | 171 | 43·5 | 43·1 | -0·4 |
| 4 | 228 | 56·0 | 56·6 | +0·6 |
| 5 | 281 | 70·0 | 69·2 | -0·8 |
| 6 | 338 | 83·0 | 82·8 | -0·2 |
| 7 | 395 | 97·0 | 96·4 | -0·6 |
| 8 | 452 | 109·0 | 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.
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.