The radius and acceleration due to gravity of moon are `1/4` and `1/5` that of earth. The ratio of the mass of earth to mass of moon is

Options

A. `1:80`
B. `80:1`
C. `1:20`
D. `20:1`

Answer

To solve the problem, we need to find the ratio of the mass of the Earth (M) to the mass of the Moon (m) given that the radius of the Moon is \( \frac{1}{4} \) that of the Earth and the acceleration due to gravity on the Moon is \( \frac{1}{5} \) that of the Earth.

### Step-by-step Solution:

1. **Identify the given values:**
- Let \( R_e \) be the radius of the Earth.
- The radius of the Moon \( R_m = \frac{1}{4} R_e \).
- Let \( g_e \) be the acceleration due to gravity on Earth.
- The acceleration due to gravity on the Moon \( g_m = \frac{1}{5} g_e \).

2. **Use the formula for acceleration due to gravity:**
The formula for the acceleration due to gravity is given by:
\[ g = \frac{GM}{R^2} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the celestial body, and \( R \) is its radius.

3. **Write the equations for Earth and Moon:**
For Earth:
\[ g_e = \frac{G M}{R_e^2} \]
For Moon:
\[ g_m = \frac{G m}{R_m^2} \]

4. **Substituting the values for the Moon:**
Substitute \( R_m = \frac{1}{4} R_e \) into the equation for \( g_m \):
\[ g_m = \frac{G m}{\left(\frac{1}{4} R_e\right)^2} = \frac{G m}{\frac{1}{16} R_e^2} = \frac{16 G m}{R_e^2} \]

5. **Set up the ratio of the accelerations:**
Now, we can set up the ratio of the gravitational accelerations:
\[ \frac{g_e}{g_m} = \frac{\frac{G M}{R_e^2}}{\frac{16 G m}{R_e^2}} = \frac{M}{16 m} \]

6. **Substituting the known values of \( g_m \) and \( g_e \):**
We know that \( g_m = \frac{1}{5} g_e \):
\[ \frac{g_e}{\frac{1}{5} g_e} = \frac{M}{16 m} \]
Simplifying gives:
\[ 5 = \frac{M}{16 m} \]

7. **Rearranging the equation:**
Rearranging gives:
\[ M = 80 m \]

8. **Finding the ratio of the masses:**
Therefore, the ratio of the mass of the Earth to the mass of the Moon is:
\[ \frac{M}{m} = 80 \]
This can be expressed as:
\[ \text{Ratio of mass of Earth to mass of Moon} = 80:1 \]

### Final Answer:
The ratio of the mass of Earth to the mass of Moon is \( 80:1 \).

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