It’s easy…if one of the two objects is the Earth… LOL!

OK more seriously, the importance of Cavendish figuring out what G was is that it let us determine masses of astronomical bodies from watching how satellites (or even just Clouseaued phones) behave. That’s especially helpful when the body in question is the Earth.

Newton’s law of gravity (shorn of vector notation and written for the limitations of the WordPus editor: F = GmM/r^2, where the two ms (lower case and capital) are the two masses, r is the distance between them and G is the fudge factor. Without G, the force between two 1kg masses one meter apart would work out to one newton (about the force an apple exerts on your hand when you hold it). Instead, it’s much, much, much less than that. When we’re fumbling phones, we have a phone of known mass (m), and a force of known amount (F), and we know the phone’s distance from the center of mass of Earth (r). [And we know we can treat the entire Earth as a point mass at that distance from the phone…but that’s calculus (which is a superpower) so I won’t go over it here and just do my Clark Kent impression.]

If you rearrange the known items on one side and the unknown items on the other, you end up with r^2F/m = GM. And so we can calculate GxM but not either of them by themselves. So without an actual value for G, we can’t figure out M (the mass of the Earth). And in practice, we generally ignore the mass of the phone and the force on it, and simply consider the acceleration of the phone, which is F/m = a. That makes the formula more generally useful since we can just watch an object of unknown mass at a known distance and still figure out what GM is for the body pulling on it; the acceleration at that distance, from that larger object will always be the same.

And this is what astrodynamicists do. In general, they just work with GM and call it the gravitational parameter and give it the symbol μ (Greek letter mu). And this number is very well known for the case of Earth. And the Sun, and Jupiter, etc. The astrondynamicist usually knows μ, and wants a, so he’ll use a = μ/r^2.

So the way to determine what G is, is to measure the force between objects of known mass, which is what this video is about. Cavendish had no idea what G was, so he had to make his apparatus as sensitive as possible; the man in the video did know and he could use that knowledge to debug his apparatus and realize it just wasn’t sensitive enough. But once Cavendish knew, from his experiment, what G is, he could then simply use μ for the Earth and calculate M.

So now we know the mass of the Earth. And for that matter, every other astronomical body where we could watch objects orbiting it. The Sun. Jupiter, Saturn, Uranus (at the time we had not yet discovered Mars’s tiny little moons). But not Venus and Mercuiry, they have no satellites, even today we know of none.

So what? Did I not just get through explaining that for astrodynamical purposes, knowing μ is good enough?

Yes, it’s good enough for astrodynamical purposes. But knowing the actual mass of an object helps us to know what it’s made of. We can take the mass of the Earth, its radius, and the fact that it’s a sphere, and determine its average density, which is considerably higher than the average density of the rocks we see around us. And thus we know that something much denser must be deep underground. And that puts us on the path to (eventually) knowing that we have an iron core, one that is compressed and therefore even denser than your engine block.

Or that Jupiter (and the other outer planets) and the Sun are basically big balls of gas. And so: the Sun is a big ball of gas, yet somehow generates all that energy it continuously generates. (It’s not some big pile of burning coal!) And we’re off to the races, able to ask (and eventually answer) intelligent questions about the universe.