Question

If $g$ is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass $m$ raised from the surface of the earth to a height equal to the radius $R$ of the earth is

• A. $\frac{1}{2}mgR$
• B. $2mgR$
• C. $mgR$
• D. $\frac{1}{4}mgR$

Answer 1

The correct option is A

$\frac{1}{2}mgR$

Step 1: Given data

$g\to$acceleration due to gravity

$m\to$mass of object

$R\to$radius of earth

We have to find the change in potential energy when an object is raised to a height $R$

Step 2: Formula used and solution

The potential energy at a point on the surface of the earth is given by,

$U_i=\frac{-GMm}{R}$

Potential energy at a height $R$ from the surface of the earth,

$U_f=\frac{-GMm}{2R}$

Change in potential energy,

$$\begin{gathered} ∆U=U_f-U_i \\ \Rightarrow ∆U=\frac{-GMm}{2R}-\frac{-GMm}{R} \\ \Rightarrow ∆U=\frac{-GMm}{R}\left(\frac{1}{2}-1\right) \\ \Rightarrow ∆U=\frac{GMm}{2R} \\ \Rightarrow ∆U=\frac{GM}{R^2}\frac{mR}{2}\left(\because g=\frac{GM}{R^2}\right) \\ \Rightarrow ∆U=\frac{mgR}{2} \end{gathered}$$

Therefore, gain in potential energy when raised to a height $R$ is equal to $\frac{1}{2}mgR$.

Hence, Option A is the correct answer.