Now that we’ve defined angular momentum, the next equation on our countdown to Newton’s birthday tells us what to do with it:
This is the Angular Momentum Principle, and as with energy and momentum before it, this relates the time derivative of the angular momentum (that is, how quickly it’s changing its value) to a quantity related to the interactions with other objects, in this case the torque.
So, why is this important?
As with energy, with the proper choice of system, we can often ensure that there is no net external torque on the system, in which case the right-hand side of this equation is zero. In such a system, the angular momentum is conserved, and does not change in time, so we know that the total angular momentum at the end of the problem must be the same as the total angular momentum at the start of the problem, though it can be redistributed among the components of the system.
This is a very powerful principle, which comes into all sorts of exotic scenarios. On the very small scale, it’s what lets us measure the angular momentum of a single photon— since the total angular momentum of the photon plus the optical elements must be a constant, we know that any optical element that changes the angular momentum of the photon must itself acquire some angular momentum, and start rotating. On the very large scale, it’s why we can detect black holes– as gas spirals into a black hole, the total angular momentum of the system must remain constant, which means the speed of the gas has to increase. As the gas moves faster, it heats up, eventually becoming so hot that it emits x-rays that we can detect here on Earth.
The black hole example also demonstrates a neat feature of angular momentum, which is that it depends not just on how much stuff you have and how fast it’s moving, but also on how that stuff is arranged. If you change the radius at which the mass of a moving object is located, that would tend to change its angular momentum, so in order to keep the angular momentum constant, the speed will change as the object moves. This is why ice skaters, gymnasts, and high-divers spin very rapidly when their limbs are drawn in tight, but slow down when they stretch out. And it’s why the merry-go-round spins faster when SteelyKid moves from the outer edge to the center, and slows down when she returns to the outside.
As with energy and momentum, the fact that angular momentum is conserved reflects a deep symmetry in nature. In this case, it’s a symmetry under rotation: the laws of physics work exactly the same way whether your facing north or east. Because of this, there must be some quantity that remains unchanged as you rotate from one orientation to another, and that quantity is the angular momentum.
Angular momentum is thus a very fundamental property in physics, and comes into play even for things that are not literally spinning. Fundamental particles– quarks, leptons, photons, all have intrinsic angular momentum called “spin,” which exists even though there is no sensible way to associate it with the rotation of a physical object. This spin angular momentum combines with angular momentum associated with the physical movement of objects to produce the total angular momentum for a composite system, which is then a conserved quantity. Angular momentum conservation also works in quantum physics, where an electron orbiting an atom does not have a definite position or radius, but exists as a diffuse ball of probability (unless you’re a Bohmian). That spread-out wavefunction still has angular momentum associated with it, which is conserved in cases where there are no external interactions to change it.
So, take a moment today to appreciate angular momentum and its conservation, and come back tomorrow for the next equation of the season.