10.6. Universal Gravitation

Satisfied that he’s on to something, he analyzed the motions of the moons of Jupiter and Saturn and eventually comets and showed that they obeyed Kepler’s Law, like Kepler’s planets and now, like the Moon. Suddenly, the whole solar system, including the moons of all planets hung together in a single mathematical system. One set of rules for the all of the celestial bodies in our visible universe and apparently fruit on our terrestrial home. But apples, planets, and moons…what’s special about them? Not much, but one quality is common to all of them: they have mass.

10.6.1. Breathtaking Leap

In Principia he made a breathtaking leap. He assumed that a gravitational attraction exists between any two objects with mass! Not just moons and planets. Right now you are being attracted by the Earth, but also by the sun, and the Moon, and Jupiter, but also…by the banana on your desk that you’re saving for lunch . All objects in the Universe attract one another according to his original planetary universal rule. For example:

In the top, two objects, 1 and 2, with masses $M_1$ and $m_2$ , are a distance $R_{12}$ from one another. In the bottom, they each exert a gravitational force on the other of equal magnitude, but opposite direction (according to Newton's First law)

The force \(F_{1,2}\) that some object #1 with a mass \(M_1\) attracts some other object #2, with a mass \(m_2\)…and visa versa. This attraction is along a line connecting their centers which are \(R_{1,2}\) apart. The relation that describes this is just an abstraction of his original Moon rule:

\[ F_{1,2} = G\cfrac{M_1m_2}{R_{12}^2},(\#eq:universal) \]

the Universal law of Gravitation. The constant of proportionality, \(G\) is called Newton’s Constant or the Gravitational Constant and is very tiny and not known well. The uncertainty on that number is 0.021 out of 6.67, or about 0.3%. For a fundamental constant of nature, that’s not very precise.