The work done to raise a mass m from the surface of the earth to a height h, which is equal to the radius of the earth, is:

CONCEPT:

The work done of the mass is equal to the difference in the final gravitational potential energy to the initial gravitational potential energy of the earth or we can say that the work done is equal to the change in the potential energy and it is written as;

W = Uf – U_i ----(1)

Here, U_f is the final gravitational potential energy, and Ui is the initial gravitational potential energy.

The potential energy at the earth's surface is directly proportional to the mass of the earth and the mass from the earth's surface.

\(U=\frac{-GMm}{R}\)

Here, G is the gravitational constant, M is the mass of the earth, and m is the mass from the earth's surface, R is the distance.

CALCULATION:

In the figure below we are having the mass of the earth as 'M' and m as the mass from the surface of the earth which is raised by a height of h.

Now, the initial gravitational potential energy at the earth's surface is

\(U_i=\frac{-GMm}{R}\)

and the final gravitational potential energy at height h = R

\(U_f=\frac{-GMm}{2R}\)

Now, by using equation (1) we have;

 W = U_f – U_i

⇒ \( W=- \frac{{GMm}}{{2R}}- (-\frac{{GMm}}{{R}}) \)

⇒ \(W= \frac{GMm}{2R}\) ----(2)

Now, as we know that GM = gR², putting this value in equation 2), we have;

\(W= \frac{gR^2m}{2R}\)

⇒ W = \(\frac{1}{2}\)mgR

Hence, option 3) is the correct answer.