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
$\Rightarrow M_{2}=8 M_{1}$ ... (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.
$\Rightarrow R_{2}^{3}=8 R_{1}^{3}$
or $R_{2}=2 R_{1}$ ...(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} $
$=\frac{8 M_{1}}{M_{1}} \times\left(\frac{R_{1}}{2 R_{1}}\right)^{2}=\frac{2}{1}$
So, $g_{2} =2 g_{1}$