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 $\left(\right.g\left.\right)$ is given by
$ \, \, g=\frac{G M}{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}}$
$\Longrightarrow \, g=\frac{4 G \pi \rho R}{3}$
$Given, \, \, \, R_{1}:R_{2}=2:3$
$ \, \, \, \rho _{1}:\rho _{2}=\frac{3}{2}$
$\therefore \, \frac{g_{1}}{g_{2}}=\frac{\rho _{1} R_{1}}{\rho _{2} R_{2}}=\frac{3}{2}\times \frac{2}{3}=1$