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.