Q 11-07-135JEE MainJEE Main 2021 (27 Jul, Shift 2)Easy
The planet Mars has two moons, if one of them has a period $7$ hours, $30$ minutes and an orbital radius of $9.0\times10^3$ km. Find the mass of Mars. $\left\{\text{Given } \dfrac{4\pi^2}{G} = 6\times10^{11}\ \text{N}^{-1}\text{m}^{-2}\text{kg}^2\right\}$
Answer: (D) $6.00\times10^{23}$ kg
Kepler's third law: $T^2 = \dfrac{4\pi^2r^3}{GM} \Rightarrow M = \dfrac{4\pi^2}{G}\cdot\dfrac{r^3}{T^2}$.
$r = 9\times10^6$ m, $r^3 = 7.29\times10^{20}\ \text{m}^3$; $T = 7.5$ h $= 27000$ s, $T^2 = 7.29\times10^8\ \text{s}^2$.
$$M = 6\times10^{11}\times\frac{7.29\times10^{20}}{7.29\times10^8} = 6.00\times10^{23}\ \text{kg}$$
Solution by Sreeraj P, M.Sc Physics