If magnitude of the gravitational force between the earth and the moon is denoted by A and between the earth and the sun by B then

Options

A. A = B
B. A < B
C. B < A
D. A = 2B

Answer

To solve the problem, we need to compare the gravitational forces between the Earth and the Moon (denoted as A) and between the Earth and the Sun (denoted as B). We will use the formula for gravitational force and the given relationships between the masses and distances.

### Step-by-Step Solution:

1. **Identify the Gravitational Force Formula:**
The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[ F = \frac{G m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant.

2. **Calculate the Gravitational Force between Earth and Moon (A):**
Let:
- \( m_e \) = mass of the Earth
- \( m_m \) = mass of the Moon
- \( r_{em} \) = distance between Earth and Moon

The gravitational force between the Earth and the Moon is:
\[ A = \frac{G m_e m_m}{r_{em}^2} \]

3. **Calculate the Gravitational Force between Earth and Sun (B):**
Let:
- \( m_s \) = mass of the Sun
- \( r_{es} \) = distance between Earth and Sun

The gravitational force between the Earth and the Sun is:
\[ B = \frac{G m_e m_s}{r_{es}^2} \]

4. **Use Given Relationships:**
We know:
- The mass of the Sun \( m_s \) is \( 27 \times m_m \).
- The distance between Earth and Sun \( r_{es} \) is \( 390 \times r_{em} \).

5. **Substitute the Relationships into B:**
Substitute \( m_s \) and \( r_{es} \) into the equation for B:
\[ B = \frac{G m_e (27 m_m)}{(390 r_{em})^2} \]
Simplifying this gives:
\[ B = \frac{G m_e \cdot 27 m_m}{390^2 \cdot r_{em}^2} \]

6. **Relate B to A:**
Now, we can express B in terms of A:
\[ B = A \cdot \frac{27}{390^2} \]

7. **Compare A and B:**
Since \( 390^2 \) is a large number, we can conclude that:
\[ B = A \cdot \frac{27}{390^2} \quad \text{(where } \frac{27}{390^2} \text{ is a small fraction)} \]
Therefore, \( B \) is greater than \( A \) because the numerator (27) is much larger than the denominator (390^2).

### Conclusion:
Thus, we can conclude that \( B > A \).