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 is $\frac{r}{(x)^{1 / 3}} .$ Find the value of x.

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Detailed Solution

Angular velocity of geostationary satellite is same as that of the Earth

$ω = \sqrt{\frac{GM}{r^{3}}} \Rightarrow ω \propto \sqrt{\frac{1}{r^{3}}}$

$\frac{ω_{1}}{ω_{2}} = {(\frac{r_{2}}{r_{1}})}^{3 / 2} \Rightarrow \frac{r_{2}}{r_{1}} = {(\frac{ω_{1}}{ω_{2}})}^{2 / 3};$ $∴ r_{2} = \frac{r}{(4)^{1 / 3}}$

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