538. Our study of the subject will be facilitated by some
considerations founded on the principles of energy. In the experiment
of Fig. 71 let A be 14 lbs., and B, on the ground, be 56 lbs. Since the
rope is 15' long, A is 3' from the ground, and therefore 6' from the
pulley. I raise A to the pulley, and, in doing so, expend 6 × 14 = 84
units of energy. Energy is never lost, and therefore I shall expect to
recover this amount. I allow A to fall; when it has fallen 6', it is
then precisely in the same condition as it was before being raised,
except that it has a considerable velocity of descent. In fact, the 84
units of energy have been expended in giving velocity to A. B is then
lifted to a maximum height X, in which 56 × X units of energy have
been consumed. At the instant when B is at the summit X, A must be at
a distance of 6 + X feet from the pulley; hence the quantity of work
performed by A is 14 × (6 + X). But the work done by A must be equal to
that done upon B, and therefore