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} $
$\frac{r_{2}}{r_{1}} = \left(\frac{1}{2}\right)^{2/3} = \frac{1}{4^{1/3}} $
$r_{2} = \frac{r_{1}}{4^{1/3}}$