Question

The amount of work done to raise a mass ‘m’ from the surface of the Earth to a height equal to the radius of the Earth ‘R’, will be: ____.

• A. mgR
• B. 2mgR
• C. mgR/4
• D. mgR/2

Hint:

A useful shortcut for work done to lift a mass to height $h$ is $W = \frac{mgh}{1 + h/R}$. Here $h=R$, so $W = \frac{mgR}{1 + R/R} = \frac{mgR}{2}$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When an object is moved through a significant distance relative to Earth's radius, we cannot use the simplified formula $W=mgh$. We must use the change in gravitational potential energy ($U = -GMm/r$).

Step 2: Key Formula or Approach:
1. $W = \Delta U = U_f - U_i$ 2. $U = -\frac{GMm}{r}$ 3. Relationship: $g = \frac{GM}{R^2} \implies GM = gR^2$

Step 3: Detailed Explanation:
1. Initial distance from center: $r_i = R$ (Surface) 2. Final distance from center: $r_f = R + R = 2R$ (at height $R$) 3. Work Done ($W$): \[ W = \left( -\frac{GMm}{2R} \right) - \left( -\frac{GMm}{R} \right) \] \[ W = \frac{GMm}{R} - \frac{GMm}{2R} = \frac{GMm}{2R} \] 4. Substitute $GM = gR^2$: \[ W = \frac{(gR^2)m}{2R} = \frac{mgR}{2} \]

Step 4: Final Answer:
The work done is mgR/2.