A mass m is moved from the bottom of a mine of depth d = R 2 to a height h = R 2 above the surface of the Earth. If R is the radius of the Earth and g is the acceleration due to gravity at its surface, the total work done in this process is:

Step-by-step solution

The gravitational potential energy of a mass m at a distance r from the centre of the Earth (where r U = - GMm 2R^3 (3R^2 - r^2) At a depth d = R 2 , the distance from the centre is r = R - R 2 = R 2 . Initial potential energy, U_i = - GMm 2R^3 (3R^2 - ( R 2 )^2 ) = - GMm 2R^3 (3R^2 - R^2 4 ) = - 11GMm 8R At a height h = R 2 , the distance from the centre is r = R + R 2 = 3R 2 . Final potential energy, U_f = - GMm r = - GMm 3R 2 = - 2GMm 3R The total work done is the change in potential energy: W = U_f - U_i = - 2G