Question

The acceleration due to gravity on the moon is only one sixth that of earth, If the earth moon are assumed to have same density. The ratio of the radii of moon and earth will be

• A. $\frac{1}{6}$
• B. $\frac{1}{6^{(}1/3)}$
• C. $\frac{1}{36}$
• D. $\frac{1}{6^{(}2/3)}$

Answer 1

The correct option is A $\frac{1}{6}$

Given that,

The acceleration due to gravity on the moon is only one sixth that of earth

$g_m=\frac{g_E}{6}$

Now, The acceleration due to gravity on moon

$g_m=\frac{4}{3}\pi R_{m}G\rho ....\left(I\right)$

We know that,

The acceleration due to gravity on earth

$g_E=\frac{Gm}{R^2}$

$g_E=\frac{G\left(\frac{4}{3}\pi R^3\rho \right)}{R^2}$

$g_E=\frac{4}{3}\pi GR\rho .....\left(II\right)$

Now, from equation (II) and (I)

The ratio of the acceleration due to gravity on earth and moon

$\frac{g_m}{g_E}=\frac{R_m}{R_E}$

$\frac{g_E}{6\times g_E}=\frac{R_m}{R_E}$

$\frac{R_m}{R_E}=\frac{1}{6}$

Hence, the ratio of the radii of moon and earth will be $\frac{1}{6}$