Q.Show that period of a satellite revolving around the Earth depends upon mass of the Earth.

Equating gravitational force to the centripetal force needed for a satellite's circular orbit gives $T^2=\dfrac{4\pi^2 r^3}{GM}$, which shows $T$ depends on the Earth's mass $M$.

Consider a satellite of mass $m$ revolving in a circular orbit of radius $r$ around the Earth of mass $M$. The gravitational force of attraction provides the necessary centripetal force:

$\frac{GMm}{r^2} = \frac{mv^2}{r} = m\omega^2 r$

Since $\omega = \dfrac{2\pi}{T}$,

$\frac{GM}{r^2} = \left(\frac{2\pi}{T}\right)^2 r$

$\frac{GM}{r^2} = \frac{4\pi^2 r}{T^2}$

Solving for $T^2$:

$T^2 = \frac{4\pi^2 r^3}{GM}$

$T = 2\pi\sqrt{\frac{r^3}{GM}}$

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