Question

If mass density of earth varies with distance $r$ from centre of earth as $\rho =kr$ and ${}^{\prime }{R}^{\prime }$ is radius of earth, then find the orbital velocity of an object revolving around earth at a distance ${}^{\prime }{R}^{\prime }$ from its centre.

• A. $\sqrt{\frac{\pi k{R}^{3}G}{4}}$
• B. $\sqrt{\frac{\pi k{R}^{3}G}{2}}$
• C. $\sqrt{\frac{\pi k{R}^{3}G}{8}}$
• D. $\sqrt{\pi k{R}^{3}G}$

Answer 1

Answer: (B)

Let ${}^{\prime }{M}^{\prime }$ be total mass of earth. Consider a shell of thickness ${}^{\prime }d{r}^{\prime }$ and mass ${}^{\prime }d{m}^{\prime }$ at a distance ${}^{\prime }{r}^{\prime }$ from centre inside earth,
$\Rightarrow dm=\rho 4\pi r^{2}dr$
$M=\int dm$
$={\int }_0^{R}4\pi k{r}^{3}dr$
$=\frac{4\pi k{R}^4}{4}=\pi k{R}^4$
Let field due to earth’s gravity at a distance $‘2R^{\prime }$ from centre be ${}^{\prime }{I}^{\prime },I\times A=4\pi G{m}_{inside}$

$\Rightarrow I\times 4\pi {\left(2R\right)}^2=4\pi G\left(\pi k{R}^4\right)$
$\Rightarrow I=\frac{\pi k{R}^{4}G}{4R^2}$
$\Rightarrow I=\frac{\pi k{R}^{4}G}{4R^2}$
For a satellite of mass $‘m^{\prime }$ moving in orbit of $‘2R^{\prime }$ radius.
$mI=\frac{m{v}^2}{\left(2R\right)}$
$\Rightarrow I=\frac{V^2}{2R}$
$\Rightarrow \frac{\pi k{R}^{2}G}{4}=\frac{V^2}{2R}$
$V=\sqrt{\frac{\pi k{R}^{3}G}{2}}$