Question

a satellite can be in a geostationary orbit around a planet at a distance r from the centre of the planet. If the angular velocity of the planet about its axis doubles, a satellite can now be in a geostationary orbit around the planet if its distance from the centre of the planet 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 $\frac{r}{(4{)}^{1/3}}$

Let,

Mass of satellite = $m$,

Mass of planet = $M$,

Angular velocity of planet = $\omega$

For a satellite to be a geostationary it has to have same angular velocity as that of planet, so angular velocity of planet will also be $\omega$

Using,

Centripetal force = gravitational force

$m{\omega }^{2}r=\frac{GmM}{r^2}$

${\omega }^2=\frac{GM}{r^3}$

$r\propto \frac{1}{{\omega }^{2/3}}$

So, If $\omega$ is doubled, i.e, ${\omega }^{\prime }=2\omega$, then

$r^{\prime }=\frac{r}{2^{2/3}}$

$r^{\prime }=\frac{r}{4^{1/3}}$