Question

The acceleration due to gravity on the surface of the earth is $10m{s^{ - 2}}$ . The mass of the planet Mars as compared to earth is $1/10$ and radius is $1/2$ . Determine the gravitational acceleration of a body on the surface of Mars.

Answer 1

Hint : n this solution, we will use the formula of gravitational acceleration on an object on the surface of the Earth. We will use the relation of gravitational acceleration with the mass and radius of the planet to determine the acceleration on the surface of Mars.

Formula used: In this solution, we will use the following formula:
$g = \dfrac{{GM}}{{{R^2}}}$ where $g$ is the gravitational acceleration on a planet with mass $M$ and radius $R$ .

Complete step by step answer:
We’ve been asked to find the gravitational acceleration on the surface of mars whose mass is $1/10$ of the Earth’s mass and whose radius is $1/2$ of that of Earth's. We know that the gravitational acceleration at the surface of a planet can be calculated as
$g = \dfrac{{GM}}{{{R^2}}}$
Now for Earth, we can write that
${g_E} = \dfrac{{G{M_E}}}{{{R_E}^2}}$
Similarly, for mars, the gravitational acceleration will be
${g_m} = \dfrac{{G{M_m}}}{{{R_m}^2}}$
Taking the ratio for the gravitational acceleration of these two planets, we get
$\dfrac{{{g_E}}}{{{g_M}}} = \dfrac{{{M_E}}}{{{M_m}}} \times \dfrac{{R_m^2}}{{R_E^2}}$
Now, for Mars, its mass is $1/10$ of the Earth’s mass and whose radius is $1/2$ of that of Earth's so we can write
$\dfrac{{{M_E}}}{{{M_m}}} = 10$ and $\dfrac{{{R_E}}}{{{R_m}}} = 2$ .So substituting these values in the equation of the ratio of gravitational acceleration, we get
$\dfrac{{{g_E}}}{{{g_m}}} = 10 \times \dfrac{1}{4}$
Which gives us
$\dfrac{{{g_E}}}{{{g_m}}} = 10 \times \dfrac{1}{4}$
${g_m} = {g_E} \times \dfrac{4}{{10}}$
Since ${g_E} = 10\,m{s^{ - 2}}$ , the gravitational acceleration on the surface of mars will be
${g_m} = 4\,m{s^{ - 2}}$
This value of gravitational acceleration is less than the gravitational acceleration on the surface of the earth.

Note:
To answer such questions, we should be aware of the formula of gravitational acceleration. In reality, the mass of mars and the radius of mars are different from the relations mentioned in the questions so we should only focus on the relations mentioned in the question.