Question

Consider two spherical planets of same average density. Second planet is $8$ times as massive as first planet. The ratio of the acceleration due to gravity of the second planet to that of the first planet is

• A. 1
• B. 2
• C. 4
• D. 8

Answer 1

Answer: (B)

Given, mass of second planet
$=8\times$ mass of first planet
$\Rightarrow M_2=8M_1$ ... (i)
$\Rightarrow \frac{4}{3}\pi R_2^3\times \rho =8\times \frac{4}{3}\pi R_1^3\times \rho$
$\therefore$ Density of both planets is same.
$\Rightarrow R_2^3=8R_1^3$
or $R_2=2R_1$ ...(ii)
So, ratio of acceleration due to gravity of the second planet to that of the first planet is
$\frac{g_2}{g_1}=\frac{\left(\frac{G{M}_2}{R_2^2}\right)}{\left(\frac{G{M}_1}{R_1^2}\right)}=\left(\frac{M_2}{M_1}\right)\times {\left(\frac{R_1}{R_2}\right)}^2$
$=\frac{8M_1}{M_1}\times {\left(\frac{R_1}{2R_1}\right)}^2=\frac{2}{1}$
So, $g_2=2g_1$