Question

Consider a planet in some solar system which has a mass double the mass of the earth and density equal to the average density of the earth. An object weighing W on the earth will weight
(a) W
(b) 2 W
(c) W/2
(d) 2^1/3 W at the planet.

Answer 1

(d) 2^1/3 W at the planet

The weight of the object on the Earth is $W=m\frac{G{M}_e}{{R_e}^2}$.
Here, m is the actual mass of the object; M_e is the mass of the earth and R_e is the radius of the earth.
Let R_p be the radius of the planet.
Mass of the planet, $M_p=2M_e$
If $\rho$ is the average density of the planet then

$$\begin{gathered} \frac{4}{3}\pi {R_p}^3\times \rho =2\times \left(\frac{4}{3}\pi {R_e}^3\times \rho \right) \\ \Rightarrow R_p={\left(2\right)}^{\frac{1}{3}}{R}_e \end{gathered}$$

Now, weight of the body on the planet is given by
$$\begin{gathered} W_p=m\left(\frac{G{M}_p}{{R_p}^2}\right)=m\left(\frac{2G{M}_e}{2^{{\displaystyle \frac{2}{3}}}{R_e}^2}\right) \\ \Rightarrow W_p=2^{\frac{1}{3}}\times m\left(\frac{G{M}_e}{{R_e}^2}\right) \\ \Rightarrow W_p=2^{\frac{1}{3}}\times W \end{gathered}$$