Question
If the mass of a planet is cut in half. How would the force of gravity between that planet and the Sun change?
Answers
GPT-4o mini
The force of gravity between two objects is described by Newton's Law of Universal Gravitation, which states that the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) is given by the formula:
\[
F = G \frac{m_1 m_2}{r^2}
\]
where:
- \( G \) is the gravitational constant,
- \( m_1 \) is the mass of the first object (in this case, the Sun),
- \( m_2 \) is the mass of the second object (the planet),
- \( r \) is the distance between the centers of the two objects.
If the mass of the planet is cut in half, its new mass \( m'_2 \) will be:
\[
m'_2 = \frac{1}{2} m_2
\]
Substituting this into the gravitational force formula gives:
\[
F' = G \frac{m_1 m'_2}{r^2} = G \frac{m_1 \left(\frac{1}{2} m_2\right)}{r^2} = \frac{1}{2} G \frac{m_1 m_2}{r^2} = \frac{1}{2} F
\]
Thus, the new gravitational force \( F' \) will be half of the original gravitational force \( F \).
In conclusion, if the mass of the planet is cut in half, the gravitational force between that planet and the Sun will also be halved.
\[
F = G \frac{m_1 m_2}{r^2}
\]
where:
- \( G \) is the gravitational constant,
- \( m_1 \) is the mass of the first object (in this case, the Sun),
- \( m_2 \) is the mass of the second object (the planet),
- \( r \) is the distance between the centers of the two objects.
If the mass of the planet is cut in half, its new mass \( m'_2 \) will be:
\[
m'_2 = \frac{1}{2} m_2
\]
Substituting this into the gravitational force formula gives:
\[
F' = G \frac{m_1 m'_2}{r^2} = G \frac{m_1 \left(\frac{1}{2} m_2\right)}{r^2} = \frac{1}{2} G \frac{m_1 m_2}{r^2} = \frac{1}{2} F
\]
Thus, the new gravitational force \( F' \) will be half of the original gravitational force \( F \).
In conclusion, if the mass of the planet is cut in half, the gravitational force between that planet and the Sun will also be halved.