The mean radius of earth is R, its angular speed on its own axis is ω and the acceleration due to gravity at earth’s surface is g. What will be the radius of the orbit of a geostationary satellite ?

A
${(R^{2} g / ω^{2})}^{1 / 3}$
B
${(Rg / ω^{2})}^{1 / 3}$
C
${(R^{2} ω^{2} / g)}^{1 / 3}$
D
${(R^{2} g / ω)}^{1 / 3}$

Solution:

Gravitational force = Centripetal force
$\begin{array}{l} \frac{GMm}{r^{2}} = {mrω}^{2} \\ \frac{GM}{r^{3}} = ω^{2} but, g = \frac{GM}{R^{2}} \Rightarrow GM = {gR}^{2} \\ ∴ r^{3} = \frac{GM}{ω^{2}} = \frac{{gR}^{2}}{ω^{2}} \Rightarrow r = {(\frac{{gR}^{2}}{ω^{2}})}^{\frac{1}{3}} \end{array}$

The mean radius of earth is R, its angular speed on its own axis is $ω$ and the acceleration due to gravity at earth’s surface is g. What will be the radius of the orbit of a geostationary satellite ?

Solution:

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