Question

If we consider a solid sphere of mass $M$ and radius $R$, then the potential energy of gravitational interaction of matter forming this solid sphere is :

• A. $-\frac{G M^{2}}{R}$
• B. $-\frac{1}{2} \frac{G M^{2}}{R}$
• C. $-\frac{3}{5} \frac{G M^{2}}{R}$
• D. $-\frac{3}{2} \frac{G M^{2}}{R}$

Answer 1

Answer: (C)

Considering an elemental shell of radius $r$ and width $d r$ as shown

If inside mass is $m$ then interaction energy of shell with mass is
$d U =-\frac{G m d m}{r} $
$\Rightarrow d U=-\frac{G\left(\frac{4}{3} \pi r^{3} \rho\right)\left(4 \pi r^{2} d r \rho\right)}{r} $
$\Rightarrow d U=-\frac{16 \pi^{2} G \rho^{2}}{3} r^{4} d r$
net interaction energy is
$\Rightarrow U=-\frac{16}{3} \pi^{2} G\left(\frac{M}{\frac{4}{3} \pi R^{3}}\right)^{2} \int_{0}^{R} r^{4} d r$
$\Rightarrow U=-\left(\frac{16}{3} \pi^{2} G\right)\left(\frac{M^{2}}{\frac{16}{9} \pi^{2} R^{6}}\right)\left(\frac{R^{5}}{5}\right)$
$\Rightarrow U=-\frac{3}{5} \frac{G M^{2}}{R}$