## What is angular momentum?

I’ve spent some time today preparing for my theoretical mechanics class. We’re slated to go over the angular versions of Newton’s laws, with a heavy emphasis on conservation of angular momentum. And . . . I’m stuck.

This actually happens to me every time I come around to teaching angular momentum. I try really hard to come up with a way that makes it seem reasonable or “from the gut” and I fail. This post is really just getting my thoughts down so I don’t argue with myself so much the next time around.

## Quantify spinning

When I talk to students about linear momentum, they don’t seem to have a hard time internalizing the concept that mass and speed should be involved in describing/quantifying motion. But when I shift to angular momentum, I struggle. I asked this on twitter today and got some great suggestions:

One was “strength of spinnyness” from my friend Fran Poodry, and I like it, but I’m not sure if students would go from that to $\vec{r}\times\vec{p}$. The same goes with the phrase I used to use: “it’s a measure of how much you’re moving around me (or the origin).”

In fact, I’ve stopped using that latter phrase because I think that it leads to students wanting to put r (the distance from the origin) in the denominator. For example, consider a car traveling down a nearby road. If you’re close to the road, that car is “going around you” much faster than if you’re far from the road (the portion of the circle it’s traveling is larger). But, from the perspective of angular momentum, the reverse is true.

I’ve also tried Kepler’s 2nd law approach, but that’s a little unsatisfying as well. The argument goes that measuring the speed at which the particle is sweeping out area is like measuring the “strength of spinnyness.” I’ve had a hard time selling that, though it certainly does lead to $\vec{r}\times\vec{p}$ rather nicely (as long as you’re willing to live with a factor of 2 times the mass).

## Rigid body approach

My twitter friends are encouraging me that going with defining the moment of inertia first can be really helpful. If students can get I into their gut, then $L=I\omega$ isn’t that hard. I agree, but I think there’s some lurking difficulty in the definition of I. Specifically, why does having mass away from the axis (the distance is squared in the definition of I) matter? You can certainly have students interact with things to show them that makes sense, and you can also show that $\vec{r}\times\vec{F}$ is a useful concept, but, really, shouldn’t angular momentum for a simple particle be important, before jumping into rigid bodies?

## Noether’s theorem

If you consider a system whose mechanical properties are unaffected by the rotation of the system. Noether’s theorem shows how that leads to the conservation of $\vec{r}\times\vec{p}$. That’s cool, but not something I’m prepared to do at the beginning of this class.

## Ideas I want to get across

I want to show that if you take a time derivative of angular momentum, you find torque. So, no torque leads to conservation of angular momentum. But, if I can’t get a good picture of angular momentum into my students’ guts, how is this helpful? I also want a tie between “strength of spinnyness” and $\vec{r}\times\vec{p}$.

I don’t want to hand it down on a silver platter. I want them to be as comfortable with it as they are with linear momentum. I’m finding that’s hard because I’m not as comfortable.

Ok, this is a pretty shoddy post, but most of my ideas are down now. Please feel free to join the conversation. We don’t spend time on this until next Wednesday, so there’s plenty of time to call me an idiot help me out.

Associate professor of physics at Hamline.
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### 5 Responses to What is angular momentum?

1. bretbenesh says:

I have no ideas. But, as a person who has some experience with physics but is not an expert, I am happy to hear that I am not the only person who cannot completely put the meaning of angular momentum into words.

Thanks for making me feel better about myself!

2. cgoedde says:

This is a great question. After thinking about it for a while, here are my thoughts.

First, I wouldn’t try to shoehorn it into a single description. I think it’s better to use two. After all, we talk about both spin and orbital angular momentum. For spin, something like Fran’s strength of spinning is fine, but I might use “momentum of spinning” instead. For orbital, the best I can come up with right now is “momentum past a point, scaled by distance”.

For the latter, I would introduce it by something like the following:

“You are standing on skates on ice. Someone throws a baseball to you. Compare your rate of spin after you catch the ball in the following three cases:

(a) The ball is thrown straight at you.
(b) The ball is thrown slightly to your right or left, so that you can easily catch it.
(c) The ball is thrown well to your right or left, so that you really have to stretch out your arm to catch it.”

The goal here is to tie orbital to spin angular momentum and to lead up to r x p. I think the answers should be reasonably intuitive, and lead the students to the idea that both r and p are important in considering the “spininess” of a moving object. The r = 0 case is especially important, for this, I think.

• Andy "SuperFly" Rundquist says:

Wow, I really like that notion of the skates. How much would this make you spin around if you caught it? that’s a great questions and has all the hallmarks you need for both angular momentum and torque.

3. Steve Maier says:

Ditto on the example of a person on ice skates: that’s about the best concrete way to get at it as I’ve heard. I like that it ties linear momentum with the new context of circular motion.

In colloquial language usage, “momentum” means something very close to it’s operational definition. Just ask your students to give you examples of objects with “little” or “a lot” of momentum before it’s ever discussed in class. They’re usually spot on. The problem is, “momentum” is used pretty interchangeably with other terms like energy, force, strength, velocity, mass, etc. Listening to sportscasters during a football game will bring this to light.

For angular momentum, it’s interesting to note that the full scope of it’s meaning may appear to require an understanding of moment of inertia and torque–whereas for linear momentum, there really isn’t much of a hangup for students. This could be evidenced by asking students to give examples of something that has “little” or “a lot” of angular momentum before linear momentum is brought up (you’ll likely get confused looks). So, would a full understanding of linear momentum be lost if introduced prior to forces and Newton’s laws of motion? Or does it just mean that students intuitively know what mass and force are before they step into the classroom?

I’d like to see a survey done that asks a few hundred expert/professional ice skaters (who haven’t had physics class) to define angular momentum in their own words. Based on their experiences, they might give a better colloquial definition than a seasoned physics teacher!

• Andy "SuperFly" Rundquist says: