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 $4 R$ from its centre is $E$ and $G$ is the gravitational constant, then the ratio of $\frac{E}{G R} is$

• 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 $d r$ and of radius $r$ is $d V=4 \pi r^{2} d r$

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