Question

Derive the formula for increase in gravitational potential energy of a Earth – mass system when the mass is lifted to a height h provided h << R.

Answer 1

1. If the object is on the surface of Earth, r = R

U₁ = \(-\frac{GMm}{R}\)

If the object is lifted to height h above the surface of Earth, the potential energy becomes -GMm 12 ~~ R+h

U₂ = \(-\frac{GMm}{R+h}\)

2. Increase in the potential energy is given by

3. If g is acceleration due to gravity on the surface of Earth. GM = gR²

∴ ∆U = mgh \((\frac{R}{R+h})\) … (1)

4. Equation (1) gives the work to be done to raise an object of mass rn to a height h, above the surface of the Earth.

5. If h << R, we can use R + h ≈ R.

∴ ∆U = mgh

Thus, mgh is increase in the gravitational potential energy of the Earth – mass system if an object of mass m is lifted to a height h, provided h << R.