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:

Gravitation PYQ — JEE Main 2023 · 6 Apr · Shift 1 · Q54 with Solution

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Correct answer: D

For a spherical planet,

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

Given that the planet has the same average density as Earth,

\frac{M_p}{R_p^3} = \frac{M_e}{R_e^3}

The planet has double the mass of Earth:

M_p = 2M_e

So,

\frac{2M_e}{R_p^3} = \frac{M_e}{R_e^3}

Cancelling $M_e$ ,

\frac{2}{R_p^3} = \frac{1}{R_e^3}

R_p^3 = 2R_e^3

R_p = 2^{1/3}R_e

Surface gravity is

g = \frac{GM}{R^2}

Thus,

\frac{g_p}{g_e} = \frac{M_p/M_e}{(R_p/R_e)^2}

Substitute the values:

\frac{g_p}{g_e} = \frac{2}{\left(2^{1/3}\right)^2} = \frac{2}{2^{2/3}} = 2^{1/3}

So,

g_p = 2^{1/3} g_e

Weight is

W = mg

If the object weighs $W$ on Earth, then on the planet its weight will be

W_p = m g_p = m(2^{1/3}g_e) = 2^{1/3}W

The object will weigh

\boxed{2^{1/3}W}

So the correct option is D.