The planet Mars has two moons, if one of them has a period 7 hours, 30 minutes and an orbital radius of $9.0 \times 10^{3} \mathrm{~km}$. Find the mass of Mars. $\left\{\right.$ Given $\left.\frac{4 \pi^{2}}{\mathrm{G}}=6 \times 10^{11} \mathrm{~N}^{-2} \mathrm{~kg}^{2}\right\}$

(B) $6.00 \times 10^{23} \mathrm{~kg}$

(A) $3.25 \times 10^{21} \mathrm{~kg}$

(C) $5.96 \times 10^{19} \mathrm{~kg}$

(D) $7.02 \times 10^{25} \mathrm{~kg}$

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Gravitation

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 1 0^{3} km

. Find the mass of Mars.

{

Given

\frac{4 π ^{2}}{G} = 6 \times 1 0^{11} N^{- 2} kg^{2}}

Answer: (B)

Gravity supplies the centripetal force for the moon's circular orbit, which gives Kepler's third law

T^{2} = \frac{4 π ^{2} r ^{3}}{GM} \Rightarrow M = \frac{4 π ^{2}}{G} \cdot \frac{r ^{3}}{T ^{2}}

Here

T = 7 h 30 min = 2.7 \times 1 0^{4}

s and

r = 9.0 \times 1 0^{3} km = 9.0 \times 1 0^{6}

m, so

M = 6 \times 1 0^{11} \times \frac{( 9.0 \times 1 0 ^{6} ) ^{3}}{( 2.7 \times 1 0 ^{4} ) ^{2}} = 6 \times 1 0^{11} \times \frac{7.29 \times 1 0 ^{20}}{7.29 \times 1 0 ^{8}}

The mass of Mars is

6.00 \times 1 0^{23}

kg.

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Single correct option