If mass density of earth varies with distance ‘r’ from centre of earth as $\rho =kr$ and ^'$R$^' is radius of earth, then find the orbital velocity of an object revolving around earth at a distance ^'$2R$^' from its centre.

Let $\text{'}M\text{'}$ be total mass of earth.
Consider a shell of thickness $\text{'}dr\text{'}$ and mass $\text{'}dm\text{'}$ at a distance $\text{'}r\text{'}$ from centre inside earth,

$\Rightarrow dm=\rho 4\pi r^{2}dr$

$M=\int dm=\underset{0}{\overset{R}{\int }}4\pi k{r}^{3}dr=\frac{4\pi k{R}^4}{4}=\pi k{R}^4$

Let field due to earth’s gravity at a distance $\text{'}2R\text{'}$ from centre be $T,I\times A=4\pi G{m}_{inside}$

$\Rightarrow I\times 4\pi {\left(2R\right)}^2=4\pi G\left(\pi k{R}^4\right)\Rightarrow I=\frac{\pi k{R}^{4}G}{4R^2}$

For a satellite of mass $\text{'}m\text{'}$ moving in orbit or $\text{'}2R\text{'}$ radius.

$mI=\frac{m{v}^2}{\left(2R\right)}\Rightarrow I=\frac{V^2}{2R}\Rightarrow \frac{\pi k{R}^{2}G}{4}=\frac{V^2}{2R}$

$V=\sqrt{\frac{\pi k{R}^{3}G}{2}}$