Question

The ratio between masses of two planets is $2:3$ and the ratio between their radii is $3:2$. The ratio between acceleration due to gravity on these two planets is:
a) $4:9$
b) $8:27$
c) $9:4$
d) $27:8$

Answer 1

The acceleration due to gravity of a planet is given by a unique formula. We don’t have the absolute value of the parameters but we are given their ratios. Using these ratios along with the basic formulae, we are going to find the answer.

Formula used:
$g=\dfrac{GM}{{{r}^{2}}}$

Complete step by step answer:
Acceleration due to gravity has a unique formula for any given planet. It does not change with the different places situated at different points on the planet. It is directly proportional to the mass of the planet and inversely proportional to the square of the radius of the planet.
We know, that the acceleration due to gravity on a planet having a mass of $M$ and a radius of $r$ is given as:
$g=\dfrac{GM}{{{r}^{2}}}$ ----(i)
Here, we need to find the ratio of the acceleration due to gravity on two planets.
Here, let us consider the masses of two given planets as ${{M}_{1}}$ and ${{M}_{2}}$. Let the radii of the given planets be ${{r}_{1}}$ and ${{r}_{2}}$ respectively.
According to question, it is given that:
$\dfrac{{{M}_{1}}}{{{M}_{2}}}=\dfrac{2}{3}$ ----(ii)
Also, $\dfrac{{{r}_{1}}}{{{r}_{2}}}=\dfrac{3}{2}$ ----(iii)
We know that the acceleration due to gravity at planet $1$ is given by:
${{g}_{1}}=\dfrac{G{{M}_{1}}}{{{r}_{1}}^{2}}$
Also, the acceleration due to gravity at planet $2$ is given by:
${{g}_{2}}=\dfrac{G{{M}_{2}}}{{{r}_{2}}^{2}}$
Now, the ratio of acceleration due to gravity of the two given planets is:
$\dfrac{{{g}_{1}}}{{{g}_{2}}}=\dfrac{\dfrac{G{{M}_{1}}}{{{r}_{1}}^{2}}}{\dfrac{G{{M}_{2}}}{{{r}_{2}}^{2}}}$
$\Rightarrow \dfrac{{{g}_{1}}}{{{g}_{2}}}=\dfrac{{{M}_{1}}}{{{M}_{2}}}\times {{\left( \dfrac{{{r}_{2}}}{{{r}_{1}}} \right)}^{2}}$
$\Rightarrow \dfrac{{{g}_{1}}}{{{g}_{2}}}=\dfrac{2}{3}\times {{\left( \dfrac{2}{3} \right)}^{2}}$
$\Rightarrow \dfrac{{{g}_{1}}}{{{g}_{2}}}=\dfrac{8}{27}$
Hence, clearly, the ratio of acceleration due to gravity of the planets having masses in the ratio of $2:3$ and radii in the ratio of $3:2$, is $8:27$. Thus, the correct option is option (b).

Note:
The acceleration due to gravity is independent of the location of any point on the planet. It is constant throughout the planet. It is given by a unique formula valid throughout the planet. It is not affected by any other parameters such as weather, terrain etc.