Question

A satellite can be in a geostationary orbit around earth in an orbit of radius $r$. If the angular velocity of earth about its axis doubles, a satellite can now be in a geostationary orbit around earth of radius

• A. $\frac{r}{2}$
• B. $\frac{r}{2 \sqrt{2}}$
• C. $\frac{r}{(4)^{1 / 3}}$
• D. $\frac{r}{(2)^{1 / 3}}$

Answer 1

Answer: (C)

Angular speed of earth = angular speed of geostationary satellite
$T \propto \frac{1}{\omega}$
$\Rightarrow \frac{T_{2}}{T_{1}}=\frac{\omega_{1}}{\omega_{2}}=\frac{1}{2}$
$\Rightarrow T_{2}=\frac{T_{1}}{2}$
Also, $T \propto r^{3 / 2}$
$\therefore \left(\frac{r_{2}}{r_{1}}\right)^{3 / 2}=\frac{T_{2}}{T_{1}}=\frac{1}{2}$
$\Rightarrow r_{2}=\frac{r_{1}}{4^{1 / 3}}=\frac{r_{1}}{4^{1 / 3}}$