In Newton's law of universal gravitation, the magnitude of the gravitational force between two point masses is $F=\frac{GmM}{r^2}$. Which two factors determine the value of $F$ for a given $G$?

• A. The distance between the objects ($r$) and the velocity of one object ($v$)
• B. The masses ($m$ and $M$) and the net electric charge on the objects ($q$)
• C. The masses ($m$ and $M$) and the acceleration due to gravity ($g$)
• D. The two masses ($m$ and $M$) and the distance between their centers ($r$)

Answer: (D)

Verified step by step guidance

1. Recall Newton's law of universal gravitation, which states that the gravitational force \(F\) between two point masses is given by the formula: \[F = \frac{G m M}{r^{2}}\] where \(G\) is the gravitational constant, \(m\) and \(M\) are the masses of the two objects, and \(r\) is the distance between their centers.
2. Identify the variables in the formula that directly affect the magnitude of the gravitational force \(F\). These are the masses \(m\) and \(M\), and the distance \(r\) between the two masses.
3. Understand that the gravitational force is proportional to the product of the two masses, meaning if either mass increases, the force increases proportionally.
4. Recognize that the gravitational force is inversely proportional to the square of the distance \(r\) between the masses, so as the distance increases, the force decreases rapidly.
5. Conclude that for a given gravitational constant \(G\), the two factors determining the value of \(F\) are the two masses (\(m\) and \(M\)) and the distance \(r\) between their centers.