Question

What is the mass of the planet that has a satellite whose time period is t and orbital radius as r?
$\begin{aligned} & a)\dfrac{4{{\pi }^{2}}{{r}^{3}}}{G{{T}^{2}}} \\\ & b)\dfrac{4{{\pi }^{2}}{{r}^{3}}}{G{{T}^{1}}} \\\ & c)\dfrac{4{{\pi }^{2}}{{r}^{3}}}{G{{T}^{3}}} \\\ & d)\dfrac{4{{\pi }^{2}}T}{G{{T}^{2}}} \\\ \end{aligned}$

Answer 1

The time period of satellite orbiting any planet is independent of the mass of the satellite. It depends on the mass of the planet it is orbiting, radius of the orbit, height of that satellite from the planet. As we have information about the satellite, we can calculate the mass of the planet using the time period of the satellite.
Formulas used:
$T=2\pi \sqrt{\dfrac{{{r}^{3}}}{GM}}$

Complete answer:
The time period of the satellite depends on the mass of the planet, distance between the planet and the satellite that is the orbital distance. As the time period of the satellite and the orbital distance is given, we can derive the mass of the planet from the formula of time period of the satellite.
$T=2\pi \sqrt{\dfrac{{{r}^{3}}}{GM}}$
Square both sides,
$\begin{aligned} & {{T}^{2}}=\dfrac{4{{\pi }^{2}}{{r}^{3}}}{GM} \\\ & M=\dfrac{4{{\pi }^{2}}{{r}^{3}}}{G{{T}^{2}}} \\\ \end{aligned}$
Therefore, the above mass of the planet matches with option a.

So Correct option is option a.

Additional information:
The period of the satellite is the time taken by the satellite to make one complete orbit around the planet. The period of earth as it travels one orbit around the sun is approximately one year. If so, we know the speed of the satellite and the orbital radius or the radius at which it orbits, we can easily find out the time period of the satellite. They are also satellites called stationary satellites which orbit in the same spot relative to the earth. The time period of such stationary satellites will be twenty-four hours, i.e., equal to the time period of the earth.

Note:
In the time period formula of the satellite, the radius taken is the orbital radius which is equal to the distance between the centre of earth and centre of the satellite. Most of us take it as the radius of the planet. But, it’s the radius of earth plus the height at which it is orbiting. Also, the mass in the formula is the mass of the planet but not the mass of the satellite.