In Newton's law of universal gravitation, how does the gravitational force depend on the sizes (masses) of two objects separated by a distance $r$?

• A. It depends only on distance and not on mass: $F=G\frac{1}{r^2}$.
• B. It is inversely proportional to the product of their masses: $F\propto \frac{1}{m_1\times m_2}$.
• C. It is directly proportional to the product of their masses: $F=G\frac{m{m}_1\times m_2}{r^2}$.
• D. It is directly proportional to the sum of their masses: $F\propto m_1+m_2$.

Answer: (C)

Verified step by step guidance

1. Recall Newton's law of universal gravitation, which states that the gravitational force \(F\) between two masses is proportional to the product of their masses and inversely proportional to the square of the distance between them.
2. Express this relationship mathematically as: \(F = G \frac{m_1 \times m_2}{r^2}\), where \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses of the two objects, and \(r\) is the distance between their centers.
3. Understand that the force increases if either mass increases, showing direct proportionality to both \(m_1\) and \(m_2\).
4. Recognize that the force decreases as the distance \(r\) increases, specifically with the square of the distance, indicating an inverse square law dependence.
5. Conclude that the gravitational force depends directly on the product of the two masses and inversely on the square of the distance separating them.