The acceleration due to gravity on the earth of radius `R_(e)` is `g_(e)` and that on moon of radius `R_(m)` is `g_(m)`. The ratio of the masses of the earth and moon is given by

Options

A. `(g_(e))/(g_(m))sqrt(((R_(e))/(R_(m))))`
B. `(g_(e))/(g_(m))*(R_(e))/(R_(m))`
C. `(g_(e)^(2))/(g_(m)^(2))xx(R_(e))/(R_(m))`
D. `(g_(e))/(g_(m))xx(R_(e)^(2))/(R_(m)^(2))`

Answer

To find the ratio of the masses of the Earth and the Moon, we can use the formula for acceleration due to gravity. The acceleration due to gravity \( g \) is given by the formula:

\[ g = \frac{GM}{R^2} \]

where \( G \) is the universal gravitational constant, \( M \) is the mass of the celestial body, and \( R \) is its radius.

### Step 1: Write the equations for gravity on Earth and Moon

For Earth:
\[ g_e = \frac{G M_e}{R_e^2} \]

For Moon:
\[ g_m = \frac{G M_m}{R_m^2} \]

### Step 2: Rearrange the equations to find the masses

From the equation for Earth, we can express the mass of the Earth \( M_e \):
\[ M_e = \frac{g_e R_e^2}{G} \]

From the equation for Moon, we can express the mass of the Moon \( M_m \):
\[ M_m = \frac{g_m R_m^2}{G} \]

### Step 3: Find the ratio of the masses

Now, we want to find the ratio of the masses of the Earth and the Moon:
\[ \frac{M_e}{M_m} = \frac{\frac{g_e R_e^2}{G}}{\frac{g_m R_m^2}{G}} \]

### Step 4: Simplify the ratio

Since \( G \) appears in both the numerator and denominator, it cancels out:
\[ \frac{M_e}{M_m} = \frac{g_e R_e^2}{g_m R_m^2} \]

### Final Answer

Thus, the ratio of the masses of the Earth and the Moon is given by:
\[ \frac{M_e}{M_m} = \frac{g_e R_e^2}{g_m R_m^2} \]