Question

12. The planet Mars has two moons, if one of them has a period 7 hours 30 minutes and an orbital radius of \( 9.0 \times 10^{3} km \). Find the mass of Mars. [2021] \( \left\{\right. \) Given \( \left.\frac{4 \pi^{2}}{G}=6 \times 10^{11} N^{-1} m^{-2} kg ^{2}\right\} \) (1) \( 5.96 \times 10^{19} kg \) (2) \( 3.25 \times 10^{21} kg \) (3) \( 7.02 \times 10^{25} kg \) (4) \( 6.00 \times 10^{23} kg \)

Answer 1

Correct option is (D) 6.00 × 10²³ kg

Given, orbital radius, r = 9.0 × 10³ km = 9 × 10⁶ m

Time period of revolution, T = 7 hours, 30 minutes = 2.7 × 10⁴ sec

Also, \(\frac{4\pi^2}{G} = 6 \times 10^{11}N^{-1}m^{-2} kg^2\)

Let mass of Mars be M

We have from the formula of time period of revolution,

\(T^2 = \frac{4\pi^2}{GM}.r^3\)

or, \(M = \frac{4\pi^2}{G}. \frac{r^3}{T^2}\)

By putting values, we have

\(M = (6 \times 10^{11}) \times \frac{(9\times 10^6)^3}{(2.7 \times 10^4)^2}\)

⇒ \(M = 6 \times 10^{23} kg\)