Question
How would the gravitational force between the Earth and Moon change if the distance between them was halved?
2 times stronger
4 time weaker
4 times stronger
8 times stronger
Answers
GPT-4o mini
The gravitational force between two masses is described by Newton's law of universal gravitation, which states:
\[ F = G \frac{m_1 m_2}{r^2} \]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( r \) is the distance between the centers of the two masses.
If the distance \( r \) between the Earth and the Moon is halved, the new distance \( r' \) becomes \( \frac{r}{2} \). The new gravitational force \( F' \) can be written as:
\[ F' = G \frac{m_1 m_2}{\left(\frac{r}{2}\right)^2} \]
Simplifying this, we get:
\[ F' = G \frac{m_1 m_2}{\frac{r^2}{4}} \]
\[ F' = 4 \cdot \left(G \frac{m_1 m_2}{r^2}\right) \]
\[ F' = 4F \]
This means that the gravitational force would increase by a factor of 4 if the distance between the Earth and Moon was halved.
So, the answer is **4 times stronger**.
\[ F = G \frac{m_1 m_2}{r^2} \]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( r \) is the distance between the centers of the two masses.
If the distance \( r \) between the Earth and the Moon is halved, the new distance \( r' \) becomes \( \frac{r}{2} \). The new gravitational force \( F' \) can be written as:
\[ F' = G \frac{m_1 m_2}{\left(\frac{r}{2}\right)^2} \]
Simplifying this, we get:
\[ F' = G \frac{m_1 m_2}{\frac{r^2}{4}} \]
\[ F' = 4 \cdot \left(G \frac{m_1 m_2}{r^2}\right) \]
\[ F' = 4F \]
This means that the gravitational force would increase by a factor of 4 if the distance between the Earth and Moon was halved.
So, the answer is **4 times stronger**.