The gravitational force between two objects is described by Newton's law of gravitation, which states:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects, and
- \( r \) is the distance between the centers of the two objects.
If the distance \( r \) between the Earth and the Moon doubles (i.e., \( r \) becomes \( 2r \)), and if the masses \( m_1 \) and \( m_2 \) remain the same, the new gravitational force \( F' \) can be expressed as:
\[ F' = \frac{G \cdot m_1 \cdot m_2}{(2r)^2} \] \[ F' = \frac{G \cdot m_1 \cdot m_2}{4r^2} \] \[ F' = \frac{1}{4} \cdot \left( \frac{G \cdot m_1 \cdot m_2}{r^2} \right) \]
This shows that the gravitational force \( F' \) is \( \frac{1}{4} \) (or 25%) of the original force \( F \).
Thus, when the distance between the Earth and the Moon doubles, the gravitational force decreases by a factor of four (or goes down by 75%).
So, the correct conclusion is:
The gravitational force between Earth and the Moon would go down by a proportional amount.
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