If the distance between the earth and moon were doubled, then the gravitational force between them will be

Options

A. Halved
B. Doubled
C. Quadrupled
D. Reduced to 1/4th

Answer

To solve the problem of how the gravitational force between the Earth and the Moon changes when the distance between them is doubled, we can follow these steps:

### Step-by-Step Solution:

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

2. **Identify the Initial Distance**:
Let the initial distance between the Earth and the Moon be \( r \).

3. **Determine the New Distance**:
If the distance is doubled, the new distance \( r' \) becomes:
\[ r' = 2r \]

4. **Calculate the New Gravitational Force**:
Substitute the new distance into the gravitational force formula:
\[ F' = \frac{G m_1 m_2}{(2r)^2} \]
Simplifying this gives:
\[ F' = \frac{G m_1 m_2}{4r^2} \]

5. **Relate the New Force to the Initial Force**:
The initial gravitational force \( F \) was:
\[ F = \frac{G m_1 m_2}{r^2} \]
Now, we can express \( F' \) in terms of \( F \):
\[ F' = \frac{1}{4} \cdot \frac{G m_1 m_2}{r^2} = \frac{1}{4} F \]

6. **Conclusion**:
Therefore, when the distance between the Earth and the Moon is doubled, the gravitational force between them decreases to one-fourth of its original value:
\[ F' = \frac{F}{4} \]

### Final Answer:
The gravitational force between the Earth and the Moon will be reduced to one-fourth of its original value.

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