Question

A person pulls a bucket of water from a well of depth h. If the mass of uniform rope is m and that of the bucket full of water is M, then work done by the person is:
(A) \[(M+\dfrac{m}{2})gh\]
(B) \[\dfrac{1}{2}(M+m)gh\]
 (C) \[(M+m)gh\]
(D) \[(\dfrac{M}{2}+m)gh\]

Answer 1

Hint:Here the person is applying force on the rope to pull it upwards. On the bucket, two forces are acting one is the gravitational pull acting downwards and other is tension force in the string which acts in the upward direction.

Complete step by step answer:

We need to find the work done by the person. Work is defined as the dot product of force and displacement and it is a scalar quantity. There are two forces acting and despite that, the bucket moves upward.
Force on the bucket acting downward= mg
Force on the bucket acting upwards is not given, so we have to use here the work-energy theorem.
Work energy theorem states that work done is equal to the change in the potential energy of the object. The rope has a uniform mass m. So, the centre of mass for the rope is at half of the length of the rope.
Work done= \[Mgh+mg\dfrac{h}{2}=(M+\dfrac{m}{2})gh\]

This matches option (A), so the correct option is (A).

Note: Work-energy theorem states that the net work done by the forces on an object equals the change in its kinetic energy. Here if the upward force and displacement of the bucket would have been given then we would have been able to solve this problem by using Newton’s second law.