We return to a topic we have discussed only in the simplest of terms, but which has a great deal more depth. Of Newton’s many achievements, one of his greatest has to have been the amazing realization that the gravity force is not simply a terrestrial phenomenon. Until he came along, people thought that objects “naturally” fall when they are near the Earth, and that heavenly bodies “naturally” do their little dance. To make the connection that the motions of planets could be explained using the very same paradigm that explains why things fall to Earth is truly a great achievement in human thought. Newton (apocryphally after seeing an apple fall from a tree) called this his law of universal gravitation, with emphasis on “universal,” as it points out that the law applies both on Earth and in the heavens.
The key to Newton’s idea is that the gravitational force actually exists between two objects and depends upon the masses of each and their separation in space. The Earth is no more special than the apple – both attract each other with equal force (which we know from the third law already), and the magnitude of that force depends upon their masses and their separations.
This actually does not fit well with our current understanding of the gravity force. In particular, we have been saying that the force equals \(mg\), even as the height (distance from the Earth’s surface) changes, so how is this dependent upon separation? First of all, it turns out that it is not the separation of the outer surfaces of the objects that matters, but rather their centers. In fact, it is even more complicated than that, so to simplify it, let’s first just assume that the two gravitating objects are very small (point masses), so that their separation is well-defined: