Question

The acceleration due to gravity at the surface of the moon is1.67m/s². The mass of the Earth is about 81 times more massive than the moon. What is the ratio of the radius of the Earth to that ofthe moon?

Answer 1

$g_e = 9.81\ m/s^2\\g_m = 1.67\ m/s^2$

Mass of earth is 81 times mass of moon. So
$M_e=81\ M_m$

We need to find $\frac{R_e}{R_m}$

$\frac{g_m}{g_e}= \frac{GM_m}{R_m^2} /\frac{GM_e}{R_e^2} \\ \\ \Rightarrow \frac{1.67}{9.81}= \frac{GM_m \times R_e^2}{GM_e \times R_m^2} \\ \\ \Rightarrow \frac{1.67}{9.81}= \frac{M_m}{M_e} \times (\frac{R_e}{R_m})^2 \\ \\ \Rightarrow \frac{R_e}{R_m}= \sqrt{\frac{1.67}{9.81} \times \frac{M_e}{M_m} } =\sqrt{\frac{1.67}{9.81} \times 81} \\ \\ \Rightarrow \frac{R_e}{R_m}= 3.71$

Ratio of radius of earth to moon is 3.71