Triumphant Return

I was released from the hospital yesterday afternoon, and, while I am still a bit nauseous, it's nice to be officially done with chemo.  While I was away, a few of you wrote in questions, either by email or comments, and I thought I'd take a moment to answer them.

Chris asked about this post, "How about getting your equations from the Lagrangian approach?"
Yes, that is a much better idea, and I may do it that way when I come back to this one.  I completely forgot about Lagrange, so thanks for the reminder (and shameful apologies to Matt Mewes if he's reading).  For those who don't know, the Lagrangian approach is an alternative to Newton's methods that is completely equivalent, but often is easier to work out in cases like this, where there are no dissipative forces, like friction.

Jim asked about this post, "Shouldn't velocity enter into the Crr term, thus changing the differentiation? I can't believe you get the same rolling resistance at 20m/s as at 1... And secondly, doesn't part of the force go into changing the wheels' angular momentum?  Do you need a -I*omega term on the right side of that equation?"
Rolling resistance is a bit dodgy, since the whole thing's an approximation anyway, but you're right that using a version that accounts for velocity would have been more accurate.  As far as the angular momentum, I think I'm ok neglecting that in favor of the overall linear momentum.  If you wanted to get into the actual construction of the chairs, you might need to consider it, but assuming everyone has the same design, I think we can do without.

Bob asked about this post, "I read your blog about noise cancellation and was wondering if you could explain how noise cancellation headphones work – I’m assuming they only deal with the problem in two dimensions?  Would it be possible to have two French horns properly situated in relation to an observer so if they blew the same note you would hear nothing? Or do the harmonics make this impossible?"
The key to noise cancelling is that sound is a pressure wave – a series of pockets of high and low pressure.  All you need to do to cancel a sound is absorb the high points, and fill in the low points.  A perfect way to do this is by using the very sound you're trying to cancel, but shifting it slightly so that its low points line up with the original high points, and vice versa.  This becomes more difficult as the frequency increases, since even small misalignments can increase, rather than decrease, the noise.  The timing required can really only be done electronically, so your proposed situation would require robot horn players using identical instruments.

Keep the questions coming; I'll be trying to post often this week to keep my mind off feeling sick.