How does the gravitational force between Earth and the Moon change when the distance between the two objects doubles but mass doesn’t change?

The gravitational force between Earth and the Moon would go down by a proportional amount.
The gravitational force between Earth and the Moon would go down by a proportional amount.

The gravitational force between Earth and the Moon would stay the same.
The gravitational force between Earth and the Moon would stay the same.

The gravitational force between Earth and the Moon would goes up by a proportional amount.
The gravitational force between Earth and the Moon would goes up by a proportional amount.

The gravitational force between Earth and the Moon would completely disappear.
The gravitational force between Earth and the Moon would completely disappear.

1 answer

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.