Physics - Gravitation Question with Solution | TestHub

The acceleration due to gravity on the surface of a planet is given by $g=\frac{GM}{R^2}$.

Rearranging for radius squared, we get $R^2=\frac{GM}{g}$.

Taking the ratio of the square of the radius of the earth to that of the moon:

$\frac{R_e^2}{R_m^2}=\left(\frac{M_e}{M_m}\right)\times \left(\frac{g_m}{g_e}\right)$

Given that $M_m=\frac{1}{8}{M}_e\Rightarrow \frac{M_e}{M_m}=8$

And $g_m=\frac{1}{6}{g}_e\Rightarrow \frac{g_m}{g_e}=\frac{1}{6}$

Substituting these values into the ratio:

$\frac{R_e^2}{R_m^2}=8\times \frac{1}{6}=\frac{4}{3}$

Taking the square root on both sides:

$\frac{R_e}{R_m}=\sqrt{\frac{4}{3}}$

$R_e=\sqrt{\frac{4}{3}}{R}_m$

Thus, the radius of the earth is ${\left(\sqrt{\frac{4}{3}}\right)}^{\text{th}}$ of the moon's radius.

Answer: radius of the earth is $(\sqrt{4/3}{)}^{\text{th }}$ of the moon's radius.