Newton’s Law of Universal Gravitation states that the magnitude $F$ of the force exerted by a point with mass $M$ on a point with mass $m$ is
$$F= \frac{GmM}{r^2}$$
where $G$ is a constant and $r$ is the distance between the bodies. Assuming that the points are moving, find a formula for
the instantaneous rate of change of $F$ with respect to $r$.
I found the question in Calculus by Howard Anton book. I was trying to solve the problem following way
$$\frac{dF}{dr}=\frac{d}{dr}\left((GMm)(r^{-2})\right)=-2(GMm)(r^{-3})=-2\frac{GMm}{r^3}$$
The answer is looking like wrong. I am not sure. If it is wrong than how to proceed it?
Best Answer
It's correct. You can interpret it like this: since $\frac{dF}{dr} < 0$, $F$ is a decreasing function of $r$, i.e., moving the masses farther from each other decreases the gravitational force between them.