Question

Given that the acceleration of gravity at the surface of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 km, determine the mass of Mars.

Short answer

The mass of Mars is $6.5\times {10}^{23}\mathrm{kg}$.

Step-by-step solution

Step 1. Given Data

The acceleration due to gravity of Mars is $g_m=0.38g$.

The radius of Mars is $r_m=3400\mathrm{km}$.

Step 2. Understanding the relation of acceleration due to gravity

In order to estimate the mass of the planet Mars, the formula of acceleration due to gravity will be used along with the radius of the Mars.

Step 3. Estimating the mass of Mars

The relation of gravitational acceleration is given by,

$\begin{array}{c}{g}_m=0.38g\\ \frac{G{m}_m}{{r_m}^2}=0.38\frac{Gm}{r^2}\\ m_m=0.38{r_m}^2\frac{m}{r^2}\end{array}$

Here, g is the Earth’s gravitational acceleration, $m_m$is the mass of Mars, r is the radius of the Earth and m is the mass of the Earth.

On plugging the values in the above relation.

$\begin{array}{l}{m}_m=0.38{\left(3400\mathrm{km}\right)}^2\left(\frac{\left(5.98\times {10}^{24}\mathrm{kg}\right)}{{\left(6380\mathrm{km}\right)}^2}\right)\\ m_m=6.5\times {10}^{23}\mathrm{kg}\end{array}$

Thus, $m_m=6.5\times {10}^{23}\mathrm{kg}$is the required mass.