Please ignore my preceding email. My analysis of the half-circle humps is incorrect (as the picture makes clear if you look at it critically for half a second). I'll be happy to have input from any of you who feel inclined to compute the wobble, but I just realized that the problem (in addition to being too specific to be all that interesting) is easier than I thought, and I'm confident I can solve it on my own. Jim Jim On Wed, Jul 8, 2015 at 5:14 PM, James Propp <jamespropp@gmail.com> wrote:
I'm nearly finished with the square-wheeled tricycle essay, especially the annoyingly technical Extras section ( http://mathenchant.org/wheel-extras.pdf), in which I use high school math to work out how much wobble you get from various non-catenary roads (to make up for the fact that most of the relevant math is college level).
One nicety I neglected is that in computing the wobble, it's not obvious that it suffices to compare how high the center of mass of the wheel is when it's symmetrically balanced atop a hump, versus how high the center of mass of the wheel is when it's symmetrically wedged between two humps. That's because it's not a priori clear that the height of the center of mass of the wheel will change in a monotone fashion as one goes between the two positions ("atop" vs. "wedged").
Assuming I did the work correctly, for the case of the flat road and the quarter-circle road, the center of mass is slightly higher for the " wedged" position, while for the case of the half-circle road, the center of mass is slightly higher for the "atop" position. The monotonicity property I desire is clear for the case of a flat road, but I don't see why it should be true for the half-circle road or the quarter-circle road (and maybe it's not even true!).
Can anyone help me out? As always, the essay will acknowledge everyone who provided assistance along the way. The current draft is posted at http://mathenchant.org/wheel-draft.rtf .
Thanks,
Jim Propp