If the distance between earth and sun is doubled, they by what factor will the gravitational force between them be changed ?

Explanation

To solve the problem of how the gravitational force between the Earth and the Sun 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 (in this case, the Earth and the Sun) is given by Newton's law of gravitation:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
where \( G \) is the gravitational constant, \( m_1 \) is the mass of the Earth, \( m_2 \) is the mass of the Sun, and \( r \) is the distance between them.

2. **Initial Force Calculation**: Let the initial distance between the Earth and the Sun be \( r \). The initial gravitational force \( F \) can be expressed as:
\[ F = \frac{G \cdot m_{\text{Earth}} \cdot m_{\text{Sun}}}{r^2} \]

3. **New Distance**: If the distance is doubled, the new distance becomes \( 2r \).

4. **New Force Calculation**: The new gravitational force \( F' \) when the distance is \( 2r \) can be calculated as:
\[ F' = \frac{G \cdot m_{\text{Earth}} \cdot m_{\text{Sun}}}{(2r)^2} \]
Simplifying this gives:
\[ F' = \frac{G \cdot m_{\text{Earth}} \cdot m_{\text{Sun}}}{4r^2} \]

5. **Finding the Factor of Change**: To find the factor by which the gravitational force has changed, we can take the ratio of the new force \( F' \) to the initial force \( F \):
\[ \frac{F'}{F} = \frac{\frac{G \cdot m_{\text{Earth}} \cdot m_{\text{Sun}}}{4r^2}}{\frac{G \cdot m_{\text{Earth}} \cdot m_{\text{Sun}}}{r^2}} \]
The \( G \), \( m_{\text{Earth}} \), and \( m_{\text{Sun}} \) terms cancel out:
\[ \frac{F'}{F} = \frac{1}{4} \]

6. **Conclusion**: This means that the new gravitational force \( F' \) is one-fourth of the initial gravitational force \( F \):
\[ F' = \frac{F}{4} \]

### Final Answer:
The gravitational force between the Earth and the Sun will be changed by a factor of \( \frac{1}{4} \) if the distance between them is doubled.