Question

A planet of mass $M$ has uniform density in a spherical volume of radius $R$. Calculate the work done by the external agent to deassemble the planet in eight identical spherical part against gravitational pull amongst its constitute particle.

• A. $\frac{9}{20}\frac{G{m}^2}{R}$.
• B. $\frac{19}{20}\frac{G{m}^2}{R}$.
• C. $\frac{20}{20}\frac{G{m}^2}{R}$.
• D. $\frac{21}{20}\frac{G{m}^2}{R}$.

Answer 1

The correct option is A $\frac{9}{20}\frac{G{m}^2}{R}$.

self energy of Big sphere $=\frac{-3}{5}\frac{G{M}^2}{R}$

$\frac{4}{3}\pi R^3=8\times \frac{4}{3}\pi r^3\Rightarrow r=\frac{R}{2}$

$\Rightarrow$ Self energy of small sphere $=\frac{-3}{5}\times \frac{4\times M^2}{64\times \frac{R}{2}}$

$=\frac{-3}{5}\times \frac{G{M}^2}{32R}$

$\Rightarrow U=\frac{-3}{5}\frac{G{M}^2}{R}+8\times \frac{3}{5}\times \frac{G{M}^2}{32R}=\frac{3}{4}\times \frac{3}{5}\frac{G{M}^2}{R}$

$\Rightarrow U=\frac{9}{20}\frac{G{M}^2}{R}$

Therefor option A is correct.