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.
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 . 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 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 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 is a useful concept, but, really, shouldn’t angular momentum for a simple particle be important, before jumping into rigid bodies?
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 . 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 .
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.