Question

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 $'R'$ 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 $'M'$ be total mass of earth. Consider a shell of thickness $'dr'$ and mass $'dm'$ at a distance $'r'$ from centre inside earth,
$\Rightarrow dm=\rho 4\pi r^{2}dr$
$M=∫dm$
$=\displaystyle \int _{0}^{R}4\pi kr^{3}dr$
$=\frac{4 \pi k R^{4}}{4}=\pi kR^{4}$
Let field due to earth’s gravity at a distance $`2R'$ from centre be $'I',I\times A=4\pi Gm_{inside}$

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