In Newton's law of universal gravitation, $F=G\frac{m_1{m}_2}{r^2}$, how does increasing the masses affect the gravitational force between two objects if the distance $r$ remains constant?

• A. The gravitational force increases in direct proportion to the product $m_1{m}_2$.
• B. The gravitational force increases in proportion to ${m_1{m}_2}^2$.
• C. The gravitational force increases in direct proportion to the sum $m_1+m_2$.
• D. The gravitational force is independent of mass and depends only on $r$.

Answer: (A)

Verified step by step guidance

1. Recall Newton's law of universal gravitation, which states that the gravitational force \(F\) between two masses \(m_1\) and \(m_2\) separated by a distance \(r\) is given by the formula: \[F = G \frac{m_1 m_2}{r^2}\] where \(G\) is the gravitational constant.
2. Note that in this formula, the force \(F\) is directly proportional to the product of the two masses \(m_1\) and \(m_2\). This means if either mass increases, the force increases proportionally.
3. Since the distance \(r\) remains constant, the denominator \(r^2\) does not change, so the only variables affecting the force are the masses.
4. Therefore, increasing the masses will increase the gravitational force in direct proportion to the product \(m_1 m_2\).
5. This means if you double one mass, the force doubles; if you double both masses, the force increases by a factor of four, illustrating the direct proportionality to the product of the masses.