A mass m is moved from the bottom of a deep mine of depth R 2 to a height R above the surface of the earth. The total increase in its gravitational potential energy is: ( g = acceleration due to gravity on the surface of the earth, R = radius of the earth)

Step-by-step solution

The gravitational potential energy of a mass m at a distance r from the centre of the earth ( r U_ inside = - GMm 2R^3 (3R^2 - r^2) Initially, the mass is at a depth d = R 2 , which means its distance from the centre is r = R - R 2 = R 2 . U_i = - GMm 2R^3 [3R^2 - ( R 2 )^2 ] = - GMm 2R^3 (3R^2 - R^2 4 ) = - 11GMm 8R Finally, the mass is at a height h = R above the surface, so its distance from the centre is r = R + R = 2R . The potential energy outside the earth is: U_f = - GMm r = - GMm 2R The increase in potenti