Thursday, October 4, 2012

A funny homework assignment

I've been having a bit of a dry spell on the physics over the last six months. In that time, I've started three new topics and gotten bogged down on each, unable to bring any of them to conclusion. First I started a series on induction motors. I wasn't really expecting to run into problems here. I have a couple of pretty good insights into this topic, and I thought it would make for some good articles. But as I got into it, I ran into a couple of surprises. First, in the process of carefully describing the field distributions in the rotor and the stator, I noticed an apparent error: according to the relative phases of the currents, the motor seemed to be working mainly by force of magnetic repulsion rather than attraction. I had never heard it described this way, and it seemed bizarre. But I couldn't find any other way of making things add up. I believe in the end I was correct about the repulsion business.

But that's not where I really got into trouble. While attempting to analyze the importance of the air gap, I noticed that when I drew the magnetic flux lines, the seemed to go around the rotor bars instead of through them. But if this were the case, how would you generate the very strong IxB forces needed to turn the motor? Try as I might, I couldn't explain this, and I still don't know the answer.

The next problem that I couldn't solve came from quantum mechanics. I figured out that I can analyze the physics pretty well when there's only one electron, but things get very dicey when there are two. I can break down some very simple cases, and I can do some cool things with approximations, but the fundamental essence of the physics remains elusive to me. I thought I was going to be able to solve a problem discussed by Feynmann in Vol. 3 of the Lectures: the scattering of two electrons from each other. It's basically the quantum mechanical version of the billiard ball collision, which is of course pretty much the starting point of classical mechanics. So it would be nice to really understand it. It's the spin states of the electrons that makes this problem especially significant, and I had recently been reviewing some pretty cool stuff about the basis states of some very simple two-electron systems, and I thought I ought to be able to apply this to the scattering problem. I butted my head against the wall for a few weeks and in the end I couldn't do it. It's still out there, and I'd like to figure it out one day.

The physics was going so bad I thought maybe I needed a change of pace, so I took up an old math problem: the question of the solvability of the fifth-degree equation. I knew I had some pretty good insights on this one, and I thought maybe if I forced myself to blog it out, I would be able to finally put all the pieces together. In fact, I made some progress, and I think I'm almost there, but I still can't put it all together. In fact, I came across this very good website recently according to which I seem to be at a very similar stage of thinking as was Lagrange, some forty years before Galois. So while my insights appear to be fairly sound, and quite original in the context of the way things are taught in university, the fact remains that I don't appear to have come up with anything the Lagrange didn't already know way back then. I'd still like to be able to write up the whole story in a way that puts things in perspective for a modern audience, but as with my other stalled topics...I'm stalled on this one too.

So what have I got for you today? Well, I've been getting into the math tutoring recently, and one of my students brought in a fascinating item from first-year stats. I call it fascinating because in my opinion, this little homework question sums up everything that is wrong with the education system today. I'm going to write out the question for you and see if you can figure out why I have a problem with it. After you've had a chance to think it over, we'll take it up when I return. Here is the question.

"You're going to a two-day conference and can't decide what shoes to pack. You own 5 pairs of flats, 7 pairs of heels, and 8 pairs of boots. To save time you decide to randomly pick your shoes. If you sample two pairs of shoes, one at a time, with replacement, what is the probability you will get a pair of heels and a pair of boots in that order"

Emily, my student, is taking first year stats at the University of Winnipeg, the same university that kicked me out of the Teacher Certification program last year. When she showed me this homework question, I almost couldn't believe my eyes. It is appalling to me on some very fundamental levels, which I promise to take up when we return. In the meantime, I wonder if my readers can figure out just what it is that I object to in this item?

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