320. We next place the barrel upon the axis already experimented upon and shown in [Fig. 46] at b. The circumference of the wheel is 88"·5; the circumference of the barrel is 14"·9. The proper mode of finding the circumference of the barrel is to suspend a weight from the rope, then raise this weight by making one revolution of the wheel, and the distance through which the weight is raised is the effective circumference of the barrel. The velocity ratio of the wheel and barrel is then found, by dividing 14·9 into 88·5, to be 5·94.
321. The mechanical efficiency of this machine is determined by experiment. I suspend a weight of 56 lbs. from the hook, and apply power to the wheel. I find that 10·1 lbs. is just sufficient to raise the load.
322. The mechanical efficiency is to be found by dividing 10·1 into 56; the quotient thus obtained is 5·54. The mechanical efficiency does not differ much from 5·94, the velocity ratio; and consequently in this machine but little power is expended upon friction.
323. We can ascertain the loss by computing the percentage of applied energy which is utilized. Let us suppose a weight of 100 lbs. has to be raised one foot: for this purpose a force of 100 ÷ 5·54 = 18·1 lbs. must be applied. This is evident from the definition of the mechanical efficiency; but since the load has to be raised one foot, it is clear from the meaning of the velocity ratio that the power must move over 5'·94: hence the number of units of work to be applied is to be measured by the product of 5·94 and 18·1, that is, by 107·5; in order therefore to accomplish 100 units of work 107·5 units of work must be applied. The percentage of energy usefully employed is 100 ÷ 107·5 × 100 = 93. This is far more than 70, which is the percentage utilized when the axle was used without the barrel ([Art. 309]).
324. A series of experiments made with care upon the wheel and barrel are recorded in Table XIX.
Table XIX.—The Wheel and Barrel.
Wheel of wood, 88"·5 in circumference, on the same axle as a cast iron barrel of 14"·9 circumference; axle is of wrought iron, 0"·75 in diameter, mounted in oiled brass bearings; power is applied to the circumference of the wheel, load raised by rope round barrel; velocity ratio, 5·94; mechanical efficiency, 5·54; useful effect, 93 per cent.; formula, P = 0·5 + 0·169 R.
| Number of Experiment. | R. Load in lbs. | Observed power in lbs. | P. Calculated power in lbs. | Difference of the observed and calculated values. |
|---|---|---|---|---|
| 1 | 14 | 2·7 | 2·9 | +0·2 |
| 2 | 28 | 5·3 | 5·2 | -0·1 |
| 3 | 42 | 7·7 | 7·6 | -0·1 |
| 4 | 56 | 10·1 | 10·0 | -0·1 |
| 5 | 70 | 12·4 | 12·4 | 0·0 |
| 6 | 84 | 14·7 | 14·7 | 0·0 |
| 7 | 98 | 17·1 | 17·1 | 0·0 |
| 8 | 112 | 19·4 | 19·5 | +0·1 |
The formula which represents the experiments with the greatest amount of accuracy is P = 0·5 + 0·169 R. This formula is compared with the experiments, and the column of differences shows that the calculated and the observed values agree very closely. The constant part 0·5 is partly due to the constant friction of the heavy barrel and wheel, and partly, it may be, to small irregularities which have prevented the centre of gravity of the whole mass from being strictly in the axle.
325. Though this machine is more economical of power than the wheel and axle of Art 305, yet it is less powerful; in fact, the mechanical efficiency, 5·54, is only about one-fourth of that of the wheel and axle. It is therefore necessary to inquire whether we cannot devise some method by which to secure the advantages of but little friction, and at the same time have a large mechanical efficiency: this we shall proceed to investigate.