Question

A satellite can be in a geostationary orbit around earth at a distance $r$ from the centre. 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 centre 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

Answer: (C)

$\frac{r}{(4{)}^{1/3}}$

for a geostationary satellite, angular velocity of satellite is same as that of earth i.e. ${\omega }_s={\omega }_e$
initial ${\omega }_s=\sqrt{\frac{GM}{r^3}}$. if earth's angular velocity is doubled, lets change the radius to $R$ to make it geostationary.
So, final ${\omega }_s=\sqrt{\frac{GM}{R^3}}=2\sqrt{\frac{GM}{r^3}}$
solving this we get, $R=\frac{r}{4^{\left(\frac{1}{3}\right)}}$