Question
How does the gravitational force between Earth and the Moon change when the distance between the two objects doubles but mass doesn’t change?
Answers
Answered by
GPT-4o mini
The gravitational force between two objects is described by Newton's law of universal 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 (in this case, the Earth and the Moon), and
- \( r \) is the distance between the centers of the two masses.
If the distance \( r \) between the Earth and the Moon doubles (i.e., it becomes \( 2r \)), the new gravitational force \( F' \) can be calculated as follows:
\[
F' = \frac{G \cdot m_1 \cdot m_2}{(2r)^2} = \frac{G \cdot m_1 \cdot m_2}{4r^2}
\]
This shows that when the distance is doubled, the new gravitational force \( F' \) is one-fourth of the original gravitational force \( F \):
\[
F' = \frac{F}{4}
\]
Therefore, when the distance between the Earth and the Moon doubles while their masses remain unchanged, the gravitational force between them decreases to one-fourth of its original value.
\[
F = \frac{G \cdot m_1 \cdot m_2}{r^2}
\]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects (in this case, the Earth and the Moon), and
- \( r \) is the distance between the centers of the two masses.
If the distance \( r \) between the Earth and the Moon doubles (i.e., it becomes \( 2r \)), the new gravitational force \( F' \) can be calculated as follows:
\[
F' = \frac{G \cdot m_1 \cdot m_2}{(2r)^2} = \frac{G \cdot m_1 \cdot m_2}{4r^2}
\]
This shows that when the distance is doubled, the new gravitational force \( F' \) is one-fourth of the original gravitational force \( F \):
\[
F' = \frac{F}{4}
\]
Therefore, when the distance between the Earth and the Moon doubles while their masses remain unchanged, the gravitational force between them decreases to one-fourth of its original value.