Question

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 $W$, the weight of the same object on that planet will be :

• A. $\sqrt{2}W$
• B. $W$
• C. $2^{\frac{1}{3}}W$
• D. $2W$

Answer 1

The correct option is C $2^{\frac{1}{3}}W$

Weight of an object is proportional to acceleration due to gravity

$W\propto g$

Acceleration due to gravity is given by,

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

The mass of the earth in terms of density is given by,

$M_E=\rho \left(\frac{4}{3}\pi R_E^3\right)$

As the Density of the planet is the same as that of Earth,

${\left(\frac{R_P}{R_E}\right)}^3=\frac{M_P}{M_E}$

Given that, $M_P=2M_E$

$\Rightarrow \frac{M_P}{M_E}=2\text{also}\frac{R_P}{R_E}=(2{)}^{\frac{1}{3}}$

$\frac{W_P}{W_E}=\frac{g_P}{g_E}=\frac{M_P}{M_E}{\left(\frac{R_E}{R_P}\right)}^2$

$\frac{W_P}{W_E}=2{\left(\frac{1}{2}\right)}^{\frac{2}{3}}$

$W_P=2^{\frac{1}{3}}W$

Hence, option $\left(C\right)$ is correct.