Source: High school physics (Chinese)
Problem Sets:
Problem
As shown in Figure below, an object of mass $m$ is pulled by a fixed pulley, causing the object to accelerate upwards from rest at a constant acceleration $a$. The weight of the rope and friction on the pulley axle are negligible.
P0484-problem-1
Applying Newton's second law to the mass $m$:
$$F - mg = ma$$The required pulling force is $F = m(g+a)$.
The velocity of the object at time $t$ starting from rest is given by kinematics:
$$v = at$$Since it is a fixed pulley, the velocity of the rope where the force is applied is equal to the velocity of the object, $v$.
The instantaneous power $P$ is the product of the force and the velocity:
$$P = Fv$$Substituting the expressions for $F$ and $v$:
$$P = m(g+a)(at)$$