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

Question

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:

Accepted Answer

1. **Concept:** The work done to lift a body against gravity is equal to the change in its gravitational potential energy.
   $$ W = \Delta U = U_{	ext{final}} - U_{	ext{initial}} $$
2. **Formulas:**
   - Gravitational potential energy at a distance $r$ from the center of the Earth: $U = -\frac{GMm}{r}$ [Eq. 7.23, 7.24].
   - Relation between $G$ and $g$: $g = \frac{GM}{R^2} \implies GM = gR^2$.
3. **Initial State:** At the surface of the Earth, $r_1 = R$.
   $$ U_i = -\frac{GMm}{R} $$
4. **Final State:** At a height $h = R$ above the surface, the distance from the center is $r_2 = R + h = R + R = 2R$.
   $$ U_f = -\frac{GMm}{2R} $$
5. **Calculation:**
   $$ W = -\frac{GMm}{2R} - \left( -\frac{GMm}{R} ight) $$
   $$ W = \frac{GMm}{R} - \frac{GMm}{2R} = \frac{GMm}{2R} $$
   Substitute $GM = gR^2$:
   $$ W = \frac{(gR^2)m}{2R} = \frac{1}{2}mgR $$

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