Question

Planets A and B have the same average density. Radius of A is twice that of B. The ratio of acceleration due to gravity on the surface of A and B is
A. $2:1$
B. $1:2$
C. $1:4$
D. $4:1$

Answer 1

For solving this problem, we need to use the formula for the average density of the planet. This formula includes the radius of the planet and acceleration due to gravity for that planet along with the gravitational constant. We will equate the average density of both the planets and use the relation given between their radii to find the required ratio of their acceleration due to gravity.

Formula used:
$\rho = \dfrac{{3g}}{{4\pi RG}}$,
where. $\rho$ is the average density of a planet, $g$ is the acceleration due to gravity, $R$ is the radius of the planet and $G$ is the gravitational constant.

Complete step by step answer:
First, let us consider the average densities of both the planets.
${\rho _A} = \dfrac{{3{g_A}}}{{4\pi {R_A}G}}$,
where, ${\rho _A}$ is the average density of the planet A, ${g_A}$ is the acceleration due to gravity on the surface of planet A, ${R_A}$ is the radius of the planet A and $G$ is the gravitational constant.
${\rho _B} = \dfrac{{3{g_B}}}{{4\pi {R_B}G}}$,
where, ${\rho _B}$ is the average density of the planet B, ${g_B}$ is the acceleration due to gravity on the surface of planet B, ${R_B}$ is the radius of the planet B and $G$ is the gravitational constant.
We are given that both these planets have the same average densities.
\Rightarrow {\rho _A} = {\rho _B} \\\ \Rightarrow \dfrac{{3{g_A}}}{{4\pi {R_A}G}} = \dfrac{{3{g_B}}}{{4\pi {R_B}G}} \\\ \Rightarrow \dfrac{{{g_A}}}{{{g_B}}} = \dfrac{{{R_A}}}{{{R_B}}} \\\
It is given that the Radius of A is twice that of B.
{R_A} = 2{R_B} \\\ \Rightarrow \dfrac{{{R_A}}}{{{R_B}}} = \dfrac{2}{1} \\\
But, we have determined that

$$\dfrac{{{g_A}}}{{{g_B}}} = \dfrac{{{R_A}}}{{{R_B}}} \\\ \therefore\dfrac{{{g_A}}}{{{g_B}}} = \dfrac{2}{1} \\\$$

Thus, the ratio of acceleration due to gravity on the surface of A and B is $2:1$.

Hence, option A is the correct answer.

Note: Here, we have seen that the ratio of the acceleration of the gravity of both the planets is equal to the ratio of their radii. This means that the gravitational acceleration on the surface of any planet is directly proportional to its radius. This is because as the radius increases, the distance of the surface from the center of the planet increases which ultimately increases the gravitational force and hence the gravitational acceleration.