EXPERIMENTS UPON THE WHEEL AND AXLE.
304. We shall commence a series of experiments upon the wheel a and axle b of [Fig. 46]. We shall first determine the velocity ratio, and then ascertain the mechanical efficiency by actual experiment. The wheel is of wood; it is about 30" in diameter. The string to which the power is attached is coiled round a series of hooks, placed near the margin of the wheel; the effective circumference is thus a little less than the real circumference. I measure a single coil of the string and find the length to be 88"·5. This length, therefore, we shall adopt for the effective circumference of the wheel. The axle is 0"·75 in diameter, but its effective circumference is larger than the circle of which this length is the diameter.
305. The proper mode of finding the effective circumference of the axle in a case where the rope bears a considerable proportion to the axle is as follows. Attach a weight to the extremity of the rope sufficient to stretch it thoroughly. Make the wheel and axle revolve suppose 20 times, and measure the height through which the weight is lifted; then the one-twentieth part of that height is the effective circumference of the axle. By this means I find the circumference of the axle we are using to be 2"·87.
306. We can now ascertain the velocity ratio in this machine. When the wheel and axle have made one complete revolution the power has been lowered through a distance of 88"·5, and the load has been raised through 2"·87. This is evident because the wheel and axle are attached together, and therefore each completes one revolution in the same time; hence the ratio of the distance which the power moves over to that through which the load is raised is 88"·5 ÷ 2"·87 = 31 very nearly. We shall therefore suppose the velocity ratio to be 31. Thus this wheel and axle has a far higher velocity ratio than any of the systems of pulleys which we have been considering.
307. Were friction absent the velocity ratio of 31 would necessarily express the mechanical efficiency of this wheel and axle; owing to the presence of friction the real efficiency is less than this—how much less, we must ascertain by experiment. I attach a load of 56 lbs. to the hook which is borne by the rope descending from the axle: this load is shown at d in [Fig. 46]. I find that a power of 2·6 lbs. applied at e is just sufficient to raise d. We infer from this result that the mechanical efficiency of this machine is 56 ÷ 2·6 = 21·5. I add a second 56 lb. weight to the load, and I find that a power of 5·0 lbs. raises the load of 112 lbs. The mechanical efficiency in this case is 112 ÷ 5·5 = 22·5. We adopt the mean value 22. Hence the mechanical efficiency is reduced by friction from 31 to 22.
308. We may compute from this result the number of units of energy which are utilized out of every 100 units applied. Let us suppose a load of 100 lbs. is to be raised one foot; a force of 100 ÷ 22 = 4·6 lbs. will suffice to raise this load. This force must be exerted through a space of 31', and consequently 31 × 4·6 = 143 units of energy must be expended; of this amount 100 units are usefully employed, and therefore the percentage of energy utilized is 100 ÷ 143 × 100 = 70. It follows that 30 per cent. of the applied energy is consumed in overcoming friction.
309. We can see the reason why the wheel and axle overhauls—that is, runs down of its own accord—when allowed to do so; it is because less than half the applied energy is expended upon friction.
310. A series of experiments which have been carefully made with this wheel and axle are recorded in Table XVIII.
Wheel of wood; axle of iron, in oiled brass bearings; weight of wheel and axle together, 16·5 lbs.; effective circumference of wheel, 88"·5; effective circumference of axle, 2"·87; velocity ratio, 31; mechanical efficiency, 22; useful effect, 70 per cent.; formula, P = 0·204 + 0·0426 R.
| Number of Experiment. | R. Load in lbs. | Observed power in lbs. | P. Calculated power in lbs. | Difference of the observed and calculated powers. |
|---|---|---|---|---|
| 1 | 28 | 1·4 | 1·4 | 0·0 |
| 2 | 42 | 2·0 | 2·0 | 0·0 |
| 3 | 56 | 2·6 | 2·6 | 0·0 |
| 4 | 70 | 3·2 | 3·2 | 0·0 |
| 5 | 84 | 3·7 | 3·8 | +0·1 |
| 6 | 98 | 4·4 | 4·4 | 0·0 |
| 7 | 112 | 5·0 | 5·0 | 0·0 |
By the method of the Appendix a relation connecting the power and the load has been determined; it is expressed in the form—
P = 0·204 + 0·0426 R.
311. Thus for example in experiment 5 a load of 84 lbs. was found to be raised by a power of 3·7 lbs. The value calculated by the formula is 0·204 + 0·0426 × 84 = 3·8. The calculated value only differs from the observed value by 0·1 lb., which is shown in the fifth column. It will be seen from this column that the values calculated from the formula represents the experiments with fidelity.
312. We have deduced the relation between the power and the load from the principle of energy, but we might have obtained it from the principle of the lever. The wheel and axle both revolve about the centre of the axle; we may therefore regard the centre as the fulcrum of a lever, and the points where the cords meet the wheel and axle as the points of application of the power and the load respectively.
313. By the principle of the lever of the first order ([Art. 237]), the power is to the load in the inverse proportion of the arms; in this case, therefore, the power is to the load in the inverse proportion of the radii of the wheel and the axle. But the circumferences of circles are in proportion to their radii, and therefore the power must be to the load as the circumference of the axle is to the circumference of the wheel.
314. This mode of arriving at the result is a little artificial; it is more natural to deduce the law directly from the principle of energy. In a mechanical power of any complexity it would be difficult to trace exactly the transmission of power from one part to the next, but the principle of energy evades this difficulty; no matter what be the mechanical arrangement, simple or complex, of few parts or of many, we have only to ascertain by trial how many feet the power must traverse in order to raise the load one foot; the number thus obtained is the theoretical efficiency of the machine.