Question

The period of a satellite in a circular orbit around a planet is independent of

• A. the mass of the planet
• B. the radius of the circular orbit
• C. the mass of the satellite
• D. All the three parameters $\left(a\right),\left(b\right)$ and $\left(c\right)$

Answer 1

The correct option is C the mass of the satellite


Consider a planet of mass $M$ and a satellite of mass $m$ going in a circular orbit of radius $R$.

Here, gravitational force $F$ provides the centripetal force $F_c$. So,

$F=F_c$

$\Rightarrow \frac{GMm}{R^2}=\frac{m{v}^2}{R}$

where, $v$ is the orbital velocity

$\Rightarrow v=\sqrt{\frac{GM}{R}}$

Now, time period $T$ is given by

$T=\frac{2\pi R}{v}=\frac{2\pi R}{\sqrt{\frac{GM}{R}}}=\frac{2\pi R^{3/2}}{\sqrt{GM}}$

$\Rightarrow T\propto \frac{R^{3/2}}{\sqrt{M}}$

From above expression, $T$ is dependent on $R$ and $M$.

Hence, option $\left(c\right)$ is correct.

Why this question: To check student’s understanding of the derivation of time period for circular orbit. Tip: It is important to remember the derivation as the nature of force can be changed and time period asked.