Moving along through our countdown to Newton’s birthday, we come to the next important physical quantity, angular momentum. For some obscure reason, this gets the symbol L, and the angular momentum for a single particle about some point A is given by:
This is probably the most deceptive equation we’ll see this season. Yesterday’s definition of work clearly showed its vector calculus roots, but to the untrained eye, this just looks like a simple multiplication: You take the momentum (p) and multiply by the distance (r) from point A, and you’re all set.
To those with a little mathematical training, though, that × symbol is a terrifying sight. In math and physics circles, ordinary multiplication is so basic that it doesn’t require a specific symbol. The × sign that ordinary people use to indicate multiplication is reserved for the most daunting of vector operations, the cross product. The linear momentum of the particle and the radius from point A are both vectors, having a direction as well as a magnitude. The cross product takes two vectors and produces a third vector– thus, the little arrows over all three quantities.
What’s more, the vector that results from the cross product is at right angles to the two vectors that were multiplied together to get it. So, if the momentum vector points north and the radius vector points west, the angular momentum vector points straight up. This is consistently the single most confusing thing in an introductory mechanics course, which makes it kind of a travesty that we only get to it in the last week or so of the term.
So, why do we bother with all this scary and confusing math?
Angular momentum is important because there is clearly something special about rotational motion. As anybody who has every played with a top or a gyroscope knows, spinning objects behave in slightly strange ways, to such a degree that every so often a crazy person will convince themselves that if they combine enough spinning objects in just the right way, they can defy the force of gravity. They can’t. Angular momentum is weird, but it’s not magic.
Angular momentum is the property we use to characterize the motion of things that move in closed loops, and it’s one of the most significant properties in all of physics. On the largest scales, angular momentum is a huge factor in determining the structure and behavior of planetary systems, galaxies, and black holes. On the smallest scales, angular momentum is what determines the structure and chemical behavior of atoms. Angular momentum is ultimately responsible for everything we see around us.
“But wait,” you object conveniently for my narrative, “You said that p is the linear momentum of a particle. How does something moving in a straight line have anything to do with rotation?”
It’s true, the connection is not obvious– objects moving in straight lines seem qualitatively different than objects that are spinning in place. But in the right circumstances, you can convert linear motion into rotational motion. If you throw a ball at a door, for example, and hit it in the right place, you can make that door swing open or closed. That interaction converts the linear motion of the ball, which flies through space in a straight line, into rotational motion of the door, which pivots about its hinges. This implies that even though it was moving in a linear fashion, the moving ball had some rotational character. That’s what you calculate with today’s equation.
“Yes, but that still doesn’t look anything like an object spinning about some axis,” you say. “The ball is moving through space, but a spinning wheel just sits there. They’re completely different.”
On a macroscopic level, they certainly look different, and it might not seem like this equation does you any good in describing a spinning wheel. But when you think about it, a spinning solid object is itself made up of mind-boggling numbers of individual atoms. At any given instant, each of those atoms is moving in a straight line through space, and thus has some (tiny) linear momentum. You can use that tiny linear momentum and the distance from the axis of rotation to calculate the individual atom’s angular momentum, and then repeat the calculation for each of the other atoms. Add up all 1026 of those individual atomic angular momenta, and that’s the angular momentum of the whole macroscopic object.
So, you can see, this is the most deceptive of all the equations we’ve looked at. Not only is it hiding scary vector business behind a pleasant facade, but it’s vastly more powerful than it lets on. So, take a moment to celebrate the existence of angular momentum, say, by doing a little spinning dance, and we’ll be back tomorrow with the next equation oft he season.