A planet has double the mass of the earth. Its average density is equal to that of the earth. An object weighing $$W$$ on earth will weigh on that planet:

A planet has mass $$M_p = 2M_e$$ and average density $$\rho_p = \rho_e$$. We need the weight of an object on this planet.

Since density is the same: $$\rho = \dfrac{M}{\frac{4}{3}\pi R^3}$$.

$$\dfrac{M_p}{\frac{4}{3}\pi R_p^3} = \dfrac{M_e}{\frac{4}{3}\pi R_e^3}$$.

$$\dfrac{R_p^3}{R_e^3} = \dfrac{M_p}{M_e} = 2 \implies R_p = 2^{1/3}\,R_e$$.

$$\dfrac{g_p}{g_e} = \dfrac{M_p}{M_e} \cdot \dfrac{R_e^2}{R_p^2} = 2 \cdot \dfrac{R_e^2}{2^{2/3}\,R_e^2} = 2 \cdot 2^{-2/3} = 2^{1/3}$$.

$$W_p = \dfrac{g_p}{g_e} \cdot W = 2^{1/3}\,W$$.

The correct answer is Option A: $$2^{1/3}W$$.