Question

A planet of mass $4$ times that of the earth spins about itself and completes one rotation is $96$ hours. The radius of a stationary satellite about this planet in comparisons to the radius of the geostationary orbit around the earth is

• A. $4$ times
• B. $\left(\frac{1}{4}\right)$ times
• C. $2$ times
• D. $\left(\frac{1}{2}\right)$ times

Answer 1

Answer: (A)

$T=\frac{2 \pi r^{\frac{3}{2}}}{\sqrt{G M}}$
$\therefore Forearth,24=\frac{2 \pi r^{\frac{3}{2}}}{\sqrt{G M}} \, \, \, ..........\left(\right. 1 \left.\right) \\ Forplanet,96=\frac{2 \pi r_{1}^{\frac{3}{2}}}{\sqrt{G \times 4 M}}..........\left(\right. 2 \left.\right) \, \\ divide \, \left(\right. 2 \left.\right)\text{ by }\left(\right. 1 \left.\right) \, \\ \frac{96}{24}=\frac{1}{2}\left(\right. \frac{r_{1}}{r} \left.\right)^{\frac{3}{2}}$
$8=\left(\frac{r_{1}}{r}\right)^{\frac{3}{2}}\Rightarrow \frac{r_{1}}{r}=4$
$r_{1}=4r$