Question

The diameters of two planets are in ratio $4:1$. Their mean densities have ratio $1:2$. The ratio of $‘g^{\prime }$ on the planets will be

• A. $1:2$
• B. $1:4$
• C. $2:1$
• D. $4:1$

Answer 1

The correct option is C $2:1$

Given: ratio of diameter
$\frac{D_1}{D_2}=\frac{4}{1}$
$\frac{R_1}{R_2}=\frac{4}{1}$
and ratio of densities
$\frac{{\rho }_1}{{\rho }_2}=\frac{1}{2}$
$\frac{g^{\prime }}{g}=\frac{G{M}_1}{G{M}_2}{\left(\frac{R_2}{R_1}\right)}^2$
$=\frac{G\frac{4}{3}\pi R_1^3}{G\frac{4}{3}\pi R_2^3}{\left(\frac{R_2}{R_1}\right)}^2\left(\frac{{\rho }_1}{{\rho }_2}\right)$
$\frac{g^{\prime }}{g}=\left(\frac{{\rho }_1}{{\rho }_2}\right)\left(\frac{R_1}{R_2}\right)$
$\frac{g^{\prime }}{g}=\frac{1}{2}\times \frac{4}{1}=\frac{2}{1}$