Question

A stationary light, smooth pulley can rotate without friction about a fixed horizontal axis. A light rope passes over the pulley. One end of the rope supports a ladder with man and the other end supports a counterweight of mass M. Mass of the man is m. Initially, the centre of mass of the counterweight is at a height h from that of man as shown in figure.
If the man starts to climb up the ladder slowly, calculate work done by him to reach his centre of mass in level with that of the counterweight

Answer 1

Let the centre of mass is a height y from the man.

$Y_{cen}=\frac{m\times 0+Mh}{M+m}$

$=\frac{M}{(m+M)}h$

Force on man is gravitational Force = mg work done by man to reach centre of mass at y distance above

W = mgy

$=\frac{mMgh}{m+M}$