Question

If the ratio of radii of $2$ planets is $1:4$ and the ratio of their density is $2:1$. What will be the ratio of their acceleration due to gravity?

Answer 1

Hint : First, we write the given data. We will use the formula for acceleration due to gravity and modify it according to the given data. We will write mass as volume multiplied by the density. Then, we will find the acceleration due to gravity is directly proportional to the radius and density.

Complete step-by-step solution:
Let $r_{1}$ and $r_{2}$ are radii of the two planets.
$\rho_{1}$ and $\rho_{2}$ are densities of the two planets.
We have given
$\dfrac{r_{1}}{r_{2}} = \dfrac{1}{4}$
$\dfrac{\rho_{1}}{\rho_{2}} = \dfrac{2}{1}$
We know the formula for acceleration due to gravity:
$g = \dfrac{GM}{r^{2}}$
We can write mass is equal to density multiplied by the volume.
$M = \dfrac{4}{3} \pi r^{3} \rho$
Now put the value of mass in acceleration due to gravity.
$g = \dfrac{G \dfrac{4}{3} \pi r^{3} \rho }{r^{2}}$
$\implies g =\dfrac{G4 \pi r \rho}{3}$
So, by the above formula,
$g \propto r \rho$
We write the ratio of acceleration due to gravity of two planets.
$\dfrac{g_{1}}{g_{2}} = \dfrac{r_{1} \rho_{1}}{ r_{2} \rho_{2}}$
Now put the given values.
$\dfrac{g_{1}}{g_{2}} = \dfrac{1 \times 2}{4 \times 1}$
$\implies \dfrac{g_{1}}{g_{2}} = \dfrac{1 }{2}$
The ratio of the acceleration due to gravity is $1:2$.

Note: The acceleration of an object varies with height. The variation in gravitational acceleration with length from the Earth's center obeys an inverse-square law. This indicates that gravitational acceleration is inversely proportional to the square of the distance from the Earth's center. As the distance is doubled, the acceleration due to gravity decreases by a factor of four.