300. When compared with the differential pulley as a means of raising a weight, this arrangement appears rather bulky and otherwise inconvenient, but, as we shall presently learn, it is a far more economical means of applying energy. In its practical application, moreover, the arrangement is simplified in various ways, two of which may be mentioned.
301. The capstan is essentially a wheel and axle; the power is not in this case applied by means of a rope, but by direct pressure on the part of the men working it; nor is there actually a wheel employed, for the pressure is applied to what would be the extremities of the spokes of the wheel if a wheel existed.
302. In the ordinary winch, the power of the labourer is directly applied to the handle which moves round in the circumference of a circle.
303. There are innumerable other applications of the principle which are constantly met with, and which can be easily understood with a little attention. These we shall not stop to describe, but we pass on at once to the important question of the relation between the power and the load.
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.