We kicked off our countdown to Newton’s birthday with his equations of motion, so it seems fitting to close out the section on classical mechanics with another of Newton’s equations, this time the Law of Universal Gravitation:
Like all the other equations to this point, I’m cribbing this from the formula sheet for my just-completed intro mechanics class, which means it’s in the notation used by Matter and Interactions. This is sub-optimal in some ways– I prefer to have subscripts on the r to remind you which way it points, but I don’t care enough to re-do the equation.
So, this is the famous inverse-square law of gravity, saying that every object with mass in the universe attracts every other object with mass in the universe with a force that is equal to some constant times the product of their masses divided by the square of the distance between them. The r-hat at the end is a unit vector telling you that the force is along the line between the two objects.
Why is this important?
This is important because it was this law, perhaps even more than the laws of motion, that made Newton’s name, and launched physics as a mathematical science. Newton hit upon this equation for the gravitational force in the 1660’s, and published it in the Principia Mathematica in 1686. There’s some argument as to whether he was the first to think of the inverse-square nature of the law (Robert Hooke even claimed that Newton stole the idea from him), but he unquestionably put it on a solid mathematical footing, showing that it was reasonable to use it not only for tiny particles, but also for planet-sized spherical masses.
This last bit was the killer app of the late seventeenth century, because this law of gravitation allowed Newton to explain and correctly predict the elliptical orbits of the planets. No other theory to that point had been able to do as much, and having a solid explanation of planetary motion was a huge milestone in the path to modern science.
(Popular legend, of course, has Newton coming up with this law after being struck by a falling apple in his family’s orchard. This is almost certainly apocryphal, but Newton did do a lot of productive thinking in the 1660’s while staying with family in the country, an outbreak of plague having closed the universities. In some sense, then, the birth of physics can be credited to the Black Death, which is either ironic or appropriate, depending on your feelings about the subject. )
We now know that this law of gravity is a good description of reality over a huge range of sizes– masses ranging from a few kilograms to billions of times the mass of the Sun, and separations of less than the thickness of a human hair up to millions of light-years. This mathematical form also provides the paradigm for thinking about forces and interactions in physics: it depends on intrinsic properties of the two interacting objects, and the distance between them, and that’s it. It’s a universal law, that does not depend on the composition or history of the objects, and that’s a big part of what makes it powerful.
This is also the first equation I’ve put on this advent calendar that is not fully correct. In fact, we now have a more general theory of gravity, Einstein’s general relativity, which gives a more accurate description of reality in certain circumstances, and also provides an explanation of why gravity his this form, something Newton was unable to do. (Pressed to explain the mechanism by which gravity acted, he huffily replied (in Latin) “hypotheses non fingo,” meaning “I feign no hypotheses.”) The math involved in general relativity is fiendishly complex, though, and for the vast majority of situations in which you might want to know the gravitational force between two objects, Newton’s law is more than sufficient.
We’ll return to this question again later, though. For now, just take a moment to appreciate Sir Isaac’s greatest achievement, and come back later for the next equation of the season.
(I intended this to be Sunday’s advent calendar post, but getting and decorating (with SteelyKid’s help) our Christmas tree was more important, so instead you get two equations today…))