Question

The density of a solid sphere of radius $R$ is $\rho (r)=20\frac{r^2}{R^2}$ where, $r$ is the distance from its centre. If the gravitational field due to this sphere at a distance $4R$ from its centre is $E$ and $G$ is the gravitational constant, then the ratio of $\frac{E}{GR}is$

• A. $\frac{\pi }{5}$
• B. $3\pi$
• C. $\frac{3\pi }{2}$
• D. $\pi$

Answer 1

Answer: (D)

Volume of a spherical shell of thickness $dr$ and of radius $r$ is $dV=4\pi r^{2}dr$

So, mass of this shell is
$dM=\rho dV=\frac{20r^2}{R^2}\cdot 4\pi r^2\cdot dr$
Total mass of complete solid sphere is
$M=\underset{0}{\overset{R}{\int }}dM=\underset{0}{\overset{R}{\int }}\frac{80\pi r^4}{R^2}dr$
$\Rightarrow M=\frac{80\pi R^5}{5R^2}=16\pi R^3$
Due to this mass $M$, gravitational field intensity at a distance $4R$ from its' centre is
$E=\frac{GM}{(4R{)}^2}$
$\Rightarrow E=\frac{G\cdot 16\pi R^3}{(4{)}^2{R}^2}$
$\Rightarrow E=G\pi R$
$\Rightarrow \frac{E}{GR}=\pi$