Question

What is the mass of the planet that has a satellite whose time period is $T$ and orbit radius is $r$?

[Assume, satellite revolves around the planet in a circular orbit]

• A. $\frac{4{\pi }^2{r}^2}{G{T}^2}$
• B. $\frac{4{\pi }^2{r}^0}{G{T}^2}$
• C. $\frac{4{\pi }^2{r}^3}{G{T}^2}$
• D. $\frac{4{\pi }^2{r}^{-3}}{G{T}^2}$

Answer 1

The correct option is C $\frac{4{\pi }^2{r}^3}{G{T}^2}$

Suppose a satellite of mass $m$ describes a circular orbit around a planet of mass $M$.
The gravitational force provides the necessary centripetal force, hence
$$\frac{GMm}{r^2}=m{\omega }^{2}r$$ Where, $r$ is the radius of circular path and $G$ is gravitional constant for the planet.

$\Rightarrow \frac{GM}{r^2}={\left(\frac{2\pi }{T}\right)}^{2}r[\because \omega =\frac{2\pi }{T}]$

$\Rightarrow M=\frac{4{\pi }^2{r}^3}{G{T}^2}$

Hence, option (c) is the correct answer.