JEE Main 2023 — Gravitation Question with Solution

The weight of an object on a planet is given by the equation $W=mg$, where $m$ is the mass of the object and $g$ is the acceleration due to gravity.

The acceleration due to gravity on a planet is given by the equation $g=\frac{GM}{R^2}$, where $G$ is the gravitational constant, $M$ is the mass of the planet, and $R$ is the radius of the planet.

In this case, the mass of the planet is double that of Earth ($M=2M_E$), but the density is the same. Density is defined as mass divided by volume, so if the mass is doubled and the density stays the same, the volume must also double.

Since the volume of a sphere (like a planet) is given by the equation $V=\frac{4}{3}\pi R^3$, a doubling of the volume implies that the radius of the planet is $R=2^{1/3}{R}_E$.

Substituting these values back into the equation for $g$, we get:

$g_{\text{planet}}=G\ast \frac{2M_E}{(2^{1/3}{R}_E{)}^2}=2^{1/3}\ast g_E$

So the weight of the object on the planet is $W_{\text{planet}}=m\ast g_{\text{planet}}=m{2}^{1/3}\ast g_E=2^{1/3}{W}_E$