Question

If a planet existed whose mass and radius were both half of that of the earth, what would be the acceleration due to gravity at the surface of the planet in terms of that on the surface of the earth?

Answer 1

Step 1: Given data

Let $M$ is the mass of the earth, $R$ is the radius of the earth and $G$ is the gravitational constant.

Mass of the planet, $M_p=\frac{M}{2}kg$

Radius of the planet, $R_p=\frac{R}{2}m$

Step 2: Finding the acceleration due to gravity $g_p$ for the planet

Acceleration due to gravity $g$ at surface of the earth can be given as below:

$g=\frac{GM}{R^2}$ [Equation-1]

Similarly, acceleration due to gravity $g_p$ at surface of the planet can be given as below:

$g_p=\frac{G{M}_p}{{R_p}^2}$

Substitute the values.

$g_p=\frac{G\left({\displaystyle \frac{M}{2}}\right)}{{\left({\displaystyle \frac{R}{2}}\right)}^2}$

$g_p=\frac{GM}{2\times {\displaystyle \frac{R^2}{4}}}$

$g_p=2\times \frac{GM}{R^2}$

$g_p=2g$ [From equation-1]

Hence, the acceleration due to gravity at the surface of the planet is $2$ times as that of the acceleration due to gravity on the surface of the earth.