Get Ready for a Schooling in Angular Momentum

March 20, 2019 | Story | No Comments

It's almost always the last topic in the first semester of introductory physics—angular momentum. Best for last, or something? I've used this concept to describe everything from fidget spinners to standing double back flips to the movement of strange interstellar asteroids.

But really, what the heck is angular momentum?

Let me start with the following situation. Imagine that there are two balls in space connected by a spring. Why are there two balls in space? I don't know—just use your imagination.

Not only are these balls connected by a spring, but the red ball has a mass that is three times the mass of the yellow ball—just for fun. Now the two balls are pushed such that they move around each other—just like this.

Yes, this is a numerical calculation. If you want to take a look at the code and play with it yourself (and you should), here it is. If you want all the details about how to make something like this, take a look at this post on the three body problem.

When we see stuff like these rotating spring-balls, we think about what is conserved—what doesn't change. Momentum is a good example of a conserved quantity. We can define momentum as:

Let me just make a plot of the total momentum as a function of time for this spring-ball system. Since momentum is a vector, I will have to plot one component of the momentum—just for fun, I will choose the x-coordinate. Here's what I get.

In that plot, the red curve is the x-momentum of the red (heavier) ball and the blue curve is for the yellow ball (yellow doesn't show up in the graph very well). The black line is the total momentum. Notice that as one object increases in momentum, the other object decreases. Momentum is conserved. You could do the same thing in the y-direction or the z-direction, but I think you get the idea.

What about energy? I can calculate two types of energy for this system consisting of the balls and the spring. There is kinetic energy and there is a spring potential energy:

The kinetic energy depends on the mass (m) and velocity (v) of the objects where the potential energy is related to the stiffness of the spring (k) and the stretch (s). Now I can plot the total energy of this system. Note that energy is a scalar quantity, so I don't have to plot just one component of it.

The black curve is again the total energy. Notice that it is constant. Energy is also conserved.

But is there another conserved quantity that could be calculated? Is the angular velocity conserved? Clearly it is not. As the balls come closer together, they seem to spin faster. How about a quick check, using a plot of the angular velocity as a function of time.

Nope: Clearly, this is not conserved. I could plot the angular velocity of each ball—but they would just have the same value and not add up to a constant.

OK, but there is something else that can be calculated that will perhaps be conserved. You guessed it: It's called the angular momentum. The angular momentum of a single particle depends on both the momentum of that particle and its vector location from some point. The angular momentum can be calculated as:

Although this seems like a simple expression, there is much to go over. First, the L vector represents the angular momentum—yes, it's a vector. Second, the r vector is a distance vector from some point to the object and finally the p vector represents the momentum (product of mass and velocity). But what about that "X"? That is the cross product operator. The cross product is an operation between two vectors that produces a vector result (because you can't use scalar multiplication between two vectors).

I don't want to go into a bunch of maths regarding the cross product, so instead I will just show it to you. Here is a quick python program showing two vectors (A and B) as well as A x B (you would say that as A cross B).

You can click and drag the yellow A vector around and see what happens to the resultant of A x B. Also, don't forget that you can always look at the code by clicking the "pencil" icon and then click the "play" to run it. Notice that A X B is always perpendicular to both A and B—thus this is always a three-dimensional problem. Oh, you can also rotate the vectors by using the right-click or ctrl-click and drag.

But now I can calculate (and plot) the total angular momentum of this ball-spring system. Actually, I can't plot the angular momentum since that's a vector. Instead I will plot the z-component of the angular momentum. Also, I need to pick a point about which to calculate the angular momentum. I will use the center of mass for the ball-spring system.

There are some important things to notice in this plot. First, both the balls have constant z-component of angular momentum so of course the total angular momentum is also constant. Second, the z-component of angular momentum is negative. This means the angular momentum vector is pointing in a direction that would appear to be into the screen (from your view).

So it appears that this quantity called angular momentum is indeed conserved. If you want, you can check that the angular momentum is also conserved in the x and y-directions (but it is).

But wait! you say. Maybe angular momentum is only conserved because I am calculating it with respect to the center of mass for the ball-spring system. OK, fine. Let's move this point to somewhere else such that the momentum vectors will be the same, but now the r-vectors for the two balls will be something different. Here's what I get for the z-component of angular momentum.

Now you can see that the z-component for the two balls both individually change, but the total angular momentum is constant. So angular momentum is still conserved. In the end, angular momentum is something that is conserved for situations that have no external torque like these spring balls. But why do we even need angular momentum? In this case, we really don't need it. It is quite simple to model the motion of the objects just using the momentum principle and forces (which is how I made the python model you see).

But what about something else? Take a look at this quick experiment. There is a rotating platform with another disk attached to a motor.
What happens with the motor-disk starts to spin? Watch. (There's a YouTube version here.)

Again, angular momentum is conserved. As the motor disk starts to spin one way, the rest of the platform spins the other way such that the total angular momentum is constant (and zero in this case). For a situation like this, it would be pretty darn difficult to model this situation with just forces and momentum. Oh, you could indeed do it—but you would have to consider both the platform and the disk as many, many small masses each with different momentum vectors and position vectors. It would be pretty much impossible to explain with that method. However, by using angular momentum for these rigid objects, it's not such a bad physics problem.

In the end, angular momentum is yet another thing that we can calculate—and it turns out to be useful in quite a number of situations. If you can find some other quantity that is conserved in different situations, you will probably be famous. You can also name the quantity after yourself if that makes you happy.