Hi Posthumane, that's a really intriguing set of observations. I've always found this issue of what turns a bike quite puzzling, Until now, the argument about the differential speed on the two sides of the patch has been almost the only one that has made any sense to me. Now, I'm not sure. There is one other, but I won't muddy the waters here with it, at least yet.
Your own explanation of what does the job is not one I've heard before. In its own terms, it sounds very plausible, though I'd like to think about it some more and do a few observations of my own. In the meantime, I'll give you my first thoughts on these matters.
Your first observation, with regard to scrub, had me going for a moment, but I don't think it supports your general argument. To keep things simple, I'll start with your unicycle. Unless I misunderstand you, I don't think a tyre would have to turn about the point in the centre of the contact patch - unless the unicycle were literally turning on the spot. It would turn about the centre of the unicycle's turning circle. Disgarding for the moment any distortion in the tyre at the point of contact, there would be no more scrub from a single turning tyre than there would be slippage from say a pair of bevel gears, or from a cone or conical section rolling round on a flat, horizontal surface.
On a bike, which, as you say, has a significant wheelbase, there would be more of an issue with 'scrub'. But I can't see it being a significant one. In any realistic curve a bike might have to make the length of arc between the tyres would not be great in comparison to the radius of the turn, so that any scrub could be mopped up by tyre slip - as you yourself observe.
Your second, empirical point about the bicycle tyre is more imediately convincing, and at this point, you've mostly sold it to me.
The turning force arising from the contact patch is, as you say, clearly related to the tyre's radius of curve. Therefore, on the face of it, if the different rates of turn across contact patch were responsible for turning the vehicle, you would expect a bicycle to have much less ability to turn than a motorbike. However, it also occurs to me that the turning force on the patch is not just related to tyre radius, but to the radial angle of the contact patch. The contact patch on a bicycle tyre is much larger in relation to tyre section than it is on a motorbike, and the differential forces would therefore be greater. Even so, it doesn't sound likely that this would make enough difference to account for the similar handling, but sometimes when you do the maths, the results can be surprising.
I really don't know. What do you think?
Looking at your own explanation of why a bike turns for a moment, the main doubt I would have would be similar to the one you have introduced about the sufficientcy of the contact patch on a bicycle. If, as you say, on a left hand turn, the front wheel steers to the left of the centreline (even though there is a small countersteering element to the right) would this ever be sufficent to explain a sportsbike's ability to turn sharply? It's a matter of degree again. It may be that it does. Again, I don't know. I'll have to get out on my bike, and check this one out as far as I can empirically.
I suspect the answer to all this lies somewhere in the behaviour of your thin disc. Will think some more.
In the meantime, thanks for your thoughts. I feel a lot more uncertain about this issue than i have for a while - but that is no bad thing. However, if the contact patch is not the answer, then I hope your explanation is!!!
Cheers
Boy, am I glad this wasn't another issue about centrifugal force.