Mass of the moon is 1/81 of the earth but gravitational pull is 1/6 of the earth. It is due to the fact that

Options

A. Radius of the moon is 81/6 of the earth
B. Radius of the Earth is `sqrt(81/6)` of the earth
C. Radius of the moon is `sqrt(81/6)` of the earth
D. Moon is the satellite of the earth

Answer

To solve the problem, we need to analyze the relationship between the mass of the moon, the mass of the Earth, and their respective gravitational pulls.

### Step-by-Step Solution:

1. **Understanding the Given Data**:
- Mass of the moon (m) = (1/81) * Mass of the Earth (M)
- Gravitational pull of the moon (g_m) = (1/6) * Gravitational pull of the Earth (g_E)

2. **Using the Gravitational Force Formula**:
The gravitational force (F) between two masses is given by Newton's law of gravitation:
\[ F = \frac{G \cdot m \cdot M}{r^2} \]
where G is the gravitational constant, m is the mass of the smaller body, M is the mass of the larger body, and r is the distance between their centers.

3. **Setting Up the Equations**:
For the Earth:
\[ g_E = \frac{G \cdot M}{R^2} \]
For the Moon:
\[ g_m = \frac{G \cdot m}{r^2} \]

4. **Substituting the Mass of the Moon**:
Substitute \( m = \frac{M}{81} \) into the equation for gravitational pull of the moon:
\[ g_m = \frac{G \cdot \left(\frac{M}{81}\right)}{r^2} \]

5. **Relating Gravitational Pulls**:
We know that \( g_m = \frac{1}{6} g_E \). Therefore:
\[ \frac{G \cdot \left(\frac{M}{81}\right)}{r^2} = \frac{1}{6} \cdot \frac{G \cdot M}{R^2} \]

6. **Canceling G and M**:
Cancel G and M from both sides (assuming M ≠ 0):
\[ \frac{1}{81r^2} = \frac{1}{6R^2} \]

7. **Cross-Multiplying**:
Cross-multiply to solve for r:
\[ 6R^2 = 81r^2 \]

8. **Solving for r**:
Rearranging gives:
\[ r^2 = \frac{6}{81} R^2 \]
Taking the square root:
\[ r = R \cdot \sqrt{\frac{6}{81}} = R \cdot \frac{\sqrt{6}}{9} \]

9. **Final Relationship**:
Thus, the radius of the moon (r) can be expressed in terms of the radius of the Earth (R):
\[ r = \frac{\sqrt{6}}{9} R \]

### Conclusion:
The gravitational pull of the moon is less than that of the Earth despite its mass being smaller due to the significant difference in their radii. The gravitational pull is influenced not only by mass but also by the distance from the center of the mass.