Question

A point mass m is situated inside a thick spherical shell of mass M, inner radius R and outer radius 2R. Work required to take this mass m from inside the shell to infinity, slowly is

• A. $\frac{GMm}{2R}$
• B. $\frac{7GMm}{15R}$
• C. $\frac{9GMm}{14R}$
• D. $\frac{3GMm}{7R}$

Answer 1

The correct option is C $\frac{9GMm}{14R}$


$dV=\frac{-Gdm}{r}=\frac{-G}{r}\rho 4\pi r^{2}drV=\int dV=-G\rho 4\pi {\int }_R^{2R}{r}^{}dr=\frac{-G\rho 4\pi }{2}3R^{2}where\rho =\frac{M}{\frac{4}{3}\pi (8R^3-R^3)}=\frac{3M}{4\pi \left(7R^3\right)}\Rightarrow V=\frac{-9\mathrm{GM}}{14R}Workdone=-V\times m=\frac{9\mathrm{GMm}}{14R}$