1. Use density relation to find the planet's radius

Given:

• Mass of planet: $M_p = 2M_e$
• Density of planet equals density of earth: $\rho_p = \rho_e$

We know $\rho = \frac{M}{\frac{4}{3}\pi R^3}$

Since densities are equal, $\frac{M_p}{R_p^3} = \frac{M_e}{R_e^3}$

Substitute $M_p = 2M_e$: $\frac{2M_e}{R_p^3} = \frac{M_e}{R_e^3}$

Cancel $M_e$: $\frac{2}{R_p^3} = \frac{1}{R_e^3}$ $R_p^3 = 2R_e^3$ $R_p = 2^{1/3} R_e$

2. Use surface gravity formula

Surface gravity is $g = \frac{GM}{R^2}$

So for the planet, $\frac{g_p}{g_e} = \frac{M_p/M_e}{(R_p/R_e)^2}$

Substitute values: $\frac{g_p}{g_e} = \frac{2}{(2^{1/3})^2} = \frac{2}{2^{2/3}} = 2^{1/3}$

Hence, $g_p = 2^{1/3} g_e$

3. Relate weight to gravity

Weight is $W = mg$.

If the object's weight on earth is $W$, then on the planet: $W_p = m g_p = m\left(2^{1/3} g_e\right) = 2^{1/3}(mg_e) = 2^{1/3}W$

4. Match with options

Thus the correct option is: $\boxed{\text{C: } 2^{1/3}W}$