Question

An object of mass m is raised from the surface of the earth to a height equal to the radius of the earth, that is, taken from a distance R to 2R from the centre of the earth. What is the gain in its potential energy?

Answer 1

Step 1: Draw the diagram as per given information.

Step 2 : Find the initial potential energy of the body.
Formula used: P.E = $\frac{-GMm}{R}$
Let mass of the earth = M
Radius of the earth = R
Mass of the body = m
The initial potential energy of body on the surface of earth
$(P.E{)}_i=\frac{-GMm}{R}$

Step 3 : Find the final potential energy of the body.
Formula used: $PE=\frac{-GMm}{R}$
As the object is raised from $R\to 2R$
So, the final potential energy of the object at a height equal to radius of the Earth,
$(PE{)}_f=\frac{-GMm}{2R}$

Step: 4 Find the gain in potential energy.
Gain in Potential Energy $=P{E}_f-P{E}_i$
$=\frac{-GMm}{2R}-\left(\frac{-GMm}{R}\right)$

$=\frac{GMm}{R}[-\frac{1}{2}+1]$

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

As, $GM=g{R}^2$

Gain in P.E. = $\frac{g{R}^{2}m}{2R}=\frac{1}{2}mgR$

Final Answer:
$\frac{1}{2}mgR$