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(D) Given that the density of the planet is equal to the density of Earth,$\rho_p = \rho_e$.Since density $\rho = \frac{M}{\frac{4}{3}\pi R^3}$,we hav

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$2^{1/3} W$

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(D) Given that the density of the planet is equal to the density of Earth,$ho_p = ho_e$.Since density $ho = \frac{M}{\frac{4}{3}\pi R^3}$,we have $\frac{M_p}{R_p^3} = \frac{M_e}{R_e^3}$.This implies $\frac{R_p}{R_e} = \left(\frac{M_p}{M_e}ight)^{1/3}$.Given $M_p = 2 M_e$,we get $\frac{R_p}{R_e} = 2^{1/3}$.The weight of an object is $W = mg = m \frac{GM}{R^2}$.Therefore,$\frac{W_p}{W_e} = \frac{M_p}{M_e} \left(\frac{R_e}{R_p}ight)^2$.Substituting the values,$\frac{W_p}{W_e} = 2 	imes \left(\frac{1}{2^{1/3}}ight)^2 = 2 	imes 2^{-2/3} = 2^{1 - 2/3} = 2^{1/3}$.Thus,$W_p = 2^{1/3} W$.

Consider a planet in some solar system which has a mass double the mass of Earth and density equal to the average density of Earth. If the weight of an object on Earth is $W$,then the weight of the same object on that planet will be

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