Question

A weight hangs freely from the end of a spring. A boy then slowly pushes the weight upward until the spring becomes slack. The gain in gravitational potential energy of the weight during this process is equal to:
A. the work done by the boy against the force of gravity acting on the weight.
B. the loss of stored energy by the spring minus the work done by the tension in the spring
C. the work done on the weight by the boy plus the stored energy lost by the spring
D. the work done on the weight by the boy minus the work done by the tension in the spring plus the stored energy lost by the spring

Answer 1

Hint: Check the forces acting on the spring when stretched and when in equilibrium position. Using the law of conservation of energy determines the gain in kinetic energy of the weight during all the process.

Complete step by step answer:
The gain in the gravitational potential energy cannot be only equal to the work done against the gravitational force as there is also loss in the spring energy.

Hence, the option A is incorrect.

The loss of energy stored in the spring and the tension in the spring are the same for the present case. Hence, their subtraction is zero. So, the gain in gravitational potential energy comes out to be equal to zero which is not correct.

Hence, option B is incorrect.

The gain in the gravitational potential energy of the weight is equal to work done by the boy against the gravitational force minus the energy lost by the spring.

Hence, the option D is incorrect.

The gain in the gravitational potential energy of the weight is equal to the work done by the boy against the gravitational force.

The work done \[W\] by the boy on the weight is equal to the work done \[{U_G}\] by the boy against the gravitational force minus the energy lost \[{E_S}\] by the spring during the whole process.
\[W = {U_G} - {E_S}\]

This in result proves that the work done against the gravitational force is equal to the work done by the boy against the gravitational force and the energy lost by the spring.
\[{U_G} = W + {E_S}\]

Hence, the correct option is C.

So, the correct answer is “Option C”.

Note:
One may assume that the spring energy of the spring will become zero when it returns to its original position and gain in gravitational potential energy is only equal to work done against the gravitational force. But the spring energy in the equation is the loss in spring energy when it changes its position which is not zero.