Question

Two planets of radii in the ratio $2:3$ are made from the material of density in the ratio $3:2$ . Then, the ratio of acceleration due to gravity $\frac{g_1}{g_2}$ at the surface of the two planets will be

• A. $1$
• B. $2.25$
• C. $0.50$
• D. $0.12$

Answer 1

Answer: (A)

The acceleration due to gravity $(g)$ is given by
$g=\frac{GM}{R^2}$
where $M$ is mass, $G$ the gravitational constant and $R$ the radius.
Since, planets have a spherical shape
$V=\frac{4}{3}\pi r^3$
$Also,mass\left(M\right)=volume\left(V\right)\times density\left(\rho \right)$
$g=\frac{G\frac{4}{3}\pi R^3\rho }{R^2}$
$⟹g=\frac{4G\pi \rho R}{3}$
$Given,R_1:R_2=2:3$
${\rho }_1:{\rho }_2=\frac{3}{2}$
$\therefore \frac{g_1}{g_2}=\frac{{\rho }_1{R}_1}{{\rho }_2{R}_2}=\frac{3}{2}\times \frac{2}{3}=1$