Question
Assuming the radius of the earth as $$R,$$ the change in gravitational potential energy of a body of mass $$m,$$ when it is taken from the earth's surface to a height $$3R$$ above its surface, is
A.
$$3\,mg\,R$$
B.
$$\frac{3}{4}\,mg\,R$$
C.
$$1\,mg\,R$$
D.
$$\frac{3}{2}\,mg\,R$$
Answer :
$$\frac{3}{4}\,mg\,R$$
Solution :
Gravitational potential energy $$\left( {GPE} \right)$$ on the surface of earth,
$$\eqalign{
& {E_1} = - \frac{{GMm}}{R} \cr
& GPE\,{\text{at}}\,3R,{E_2} = - \frac{{GMm}}{{\left( {R + 3R} \right)}} = - \frac{{GMm}}{{4R}} \cr
& \therefore {\text{Change}}\,{\text{in}}\,GPE = {E_2} - {E_1} = - \frac{{GMm}}{{4R}} + \frac{{GMm}}{R} = \frac{{3GMm}}{{4R}} \cr
& = \frac{{3g{R^2}m}}{{4R}}\left( {\because g = \frac{{GM}}{{{R^2}}}} \right) = \frac{3}{4}mg\,R \cr} $$