Question

The planet Mars has two moons, if one of them has a period $7\text{hours},30\text{minutes}$ and an obial radius of $9.0\times {10}^3\text{km}$. Find the mass of Mars.
[Given, $\frac{4{\pi }^2}{G}=6\times {10}^{11}{\text{N}}^{-1}{\text{m}}^{-2}{\text{kg}}^2$]

• A. $5.96\times {10}^{19}\text{kg}$
  
• B. $3.25\times {10}^{21}\text{kg}$
• C. $7.02\times {10}^{25}\text{kg}$
• D. $6.00\times {10}^{23}\text{kg}$

Answer 1

The correct option is D $6.00\times {10}^{23}\text{kg}$

Given, orbital radius, $r=9.0\times {10}^3\text{km}=9\times {10}^6\text{m}$
Time period of revolution, $T=7\text{hours},30\text{minutes}=2.7\times {10}^4\text{sec}$
Also, $\frac{4{\pi }^2}{G}=6\times {10}^{11}{\text{N}}^{-1}{\text{m}}^{-2}{\text{kg}}^2$
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
$M=(6\times {10}^{11})\times \frac{(9\times {10}^6{)}^3}{(2.7\times {10}^4{)}^2}$
$\Rightarrow M=6\times {10}^{23}\text{kg}$
Hence, option (d) is correct.