What will be the gain in potential energy of a body of mass m at a height equal to three times the radius R of the earth?

1. mgR 2. 2mgR

3. (mgR)/3 4. (3mgR)/4

The (gravitational) potential energy is defined as the work done against the force of gravity to transfer an object from one point to another. For an object stationary on the Earth's surface the gravitational potential energy would be

U _s = -GMm/R

here G is the universal gravitational constant, M is the mass of the Earth, m is the mass of the object and R is the radius of the earth (or the distance of the object from the centre of the Earth).

If the same object were to be lifted to the height h = R then the gravitational potential energy would be

U _h = -GMm/(R+3R)

So, U _h > U _s

The gain in potential would be

dU = U _h - U _s

=> dU = -GMm/(R+3R) - [-GMm/(R) ]

=> dU = GMm[(1/R )– (1/4R)]

=> dU = 3GMm/4R

which is the gain in potential energy
so, correct answer is option (4).