# 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) r/2 (B) r/2√2 (C) r/(4)^1/3 (D) r/(2)^1/3

Let angular velocity ω ⇒ mrω2 = GMm/r2

⇒ ω2 = GM/r3

So,

$$\omega _{1}^{2}r_{1}^{3}=\omega _{2}^{2}r_{2}^{3}\\r_{2}^{3} = r_{1}^{3}\times (\frac{\omega _{1}}{\omega _{2}})^{2}$$

Given that ω2 = 2ω1

$$r_{2}^{3} = r_{1}^{3}\times \left ( \frac{\omega _{1}}{2\omega _{1}} \right )^{2} = \frac{r_{1}^{3}}{4}\\r_{2}=\frac{r_{1}}{4^{\frac{1}{3}}}$$

Therefore, the correct option is (C)