Question

The density and acceleration due to gravity of the moon $3/5^{th}$ and $1/6^{th}$ those of the earth respectively. If the radius of the earth is $5.4\times {10}^{6}m$ , how much is it far form the moon :

• A. $1.5\times {10}^{6}m$
• B. $3.6\times {10}^{6}m$
• C. $4.5\times {10}^{6}m$
• D. $5.4\times {10}^{6}m$

Answer 1

The correct option is A $1.5\times {10}^{6}m$

Relationship of density between moon and earth is,

${\rho }_m=\frac{3}{5}{\rho }_e$

$\frac{m_m}{\frac{4}{3}\pi r_m^3}=\frac{3}{5}\times \frac{m_e}{\frac{4}{3}\pi r_e^3}$

$\frac{5}{3}\times \frac{r_e^3}{r_m^3}=\frac{m_e}{m_m}$

Relationship of acceleration of gravity between the moon and earth is,

$\frac{G{m}_m}{r_m^2}=\frac{1}{6}\frac{G{m}_e}{r_e^2}$

$\frac{6}{1}\times \frac{r_e^2}{r_m^2}=\frac{m_e}{m_m}$

Comparing the two equations we get,

$\frac{6}{1}\times \frac{r_e^2}{r_m^2}=\frac{5}{3}\times \frac{r_e^3}{r_m^3}$

$\frac{18}{5}=\frac{r_e}{r_m}$

$r_m=1.5\times \times {10}^{6}m$