Question

A satellite can be in a geostationary orbit around earth at a distance $r$ from the center. If the angular velocity of earth about its axis doubles, a satellite can now be in a geostationary orbit around earth, if its distance from the center is

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

Answer 1

The correct option is C $\frac{r}{(4{)}^{1/3}}$

Given, ${\omega }_2=2\omega$

For case-1:
$T_1=\frac{2\pi }{\omega }$

For case-2:
$T_2=\frac{\pi }{\omega }$ (As ${\omega }_2=2\omega$)

Using Kepler's 3rd law : $T^2\propto r^3$
or $\frac{T_1^2}{T_2^2}=\frac{r^3}{r_2^3}$
So, $\frac{T_1^2}{T_2^2}=\frac{4{\pi }^2}{{\omega }^2}\times \frac{4{\omega }^2}{4{\pi }^2}=4$
here, $\frac{r^3}{r_2^3}=4$
$\Rightarrow r_2=\frac{r}{(4{)}^{1/3}}$

Hence, option C is correct.