First slide

Motion of planets and satellites

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, & satellite can now be in a geostationary orbit around earth of radius

Moderate

A

$\frac{r}{2}$
B

$\frac{r}{2 \sqrt{2}}$
C

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

$\frac{r}{(2)^{1 / 3}}$

Solution

Angular speed of earth = angular speed of geostationary satellite
$\begin{array}{l} T \propto \frac{1}{ω} \\ \Rightarrow \frac{T_{2}}{T_{1}} = \frac{ω_{1}}{ω_{2}} = \frac{1}{2} \Rightarrow T_{2} = \frac{T_{1}}{2} \end{array}$
$Also, T \propto r^{3 / 2}$
$∴ {(\frac{r_{2}}{r_{1}})}^{3 / 2} = \frac{T_{2}}{T_{1}} = \frac{1}{2} \Rightarrow r_{2} = \frac{r_{1}}{4^{1 / 3}} = \frac{r_{1}}{4^{1 / 3}}$

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