Well, I don't mean to sidetrack from the current discussion but I would offer another perspective on why a bicycle 'turns' when it is tipped over. I don’t disagree
with the material above but would suggest someone go look at a bike and carefully inspect the pivot mechanism
above front fork.
Specifically; inspect the angle
at which the front fork assembly passes through the frame. I think you’ll find that on most bicycles and motorcycles that angle is just off-vertical (having a small amount of ‘backward lean’ of the rotational axis of the rotating assembly).
This off-vertical-angle-of-axis is what causes the front tire of a bicycle to ‘turn’ automatically when the bicycle is tipped to either side. Draw a line directly along this axis and you’ll find that it hits the floor somewhat in front of where the tire is centered on its contact with the floor.
When the front tire is manually turned it actually pivots from an axis that is off-center of the tire’s contact point. The tire therefore “pushes its heels out” when it tries to turn
(it pivots from a point in front of its actual contact center causing the back portion of the contact zone to be skidded a bit). This off-center relationship means that when the bike is tipped
the top of the tire will want to ‘fall’ to the side on its own because its CM is also above the actual axis of rotation. The front tire will therefore turn to the side of its own accord.
This is easily provable: Simply hold a bicycle vertically by hand and tilt it to the side. The front wheel will turn.
(Not all bikes have this feature, though I’d bet 99.9% of modern street bikes do. Not sure about ‘trick bikes’ or Mountain Bikes. Very old bicycles didn’t have this feature and were very hard to ride because of the lack. Also, handle bar configurations can muck this up by revising the CM location in relation to the axis.)
This off-vertical axis feature makes it much easier to coordinate redirection of the front wheel with tipping motions. Essentially, the front tire ‘steers’ automatically to support tipping actions of the rider. Designed properly (axis angle, frame height and length, tire diameter, handlebar, etc taken into account) a tipped bicycle will automatically try to ‘turn’ a slightly tighter radius
than necessary to laterally support the bike against Centrifugal Force for a given angle of tilt and a given speed.
This design feature creates bikes that ‘try not to fall over’. The more they tip over at typical speeds; the tighter the front tire tries to ‘over-steer’ into the turn thus righting the bike again. You can see this when a bike is launched down a small hill without a rider. The bike seems to repeatedly self-correct for a while (until it goes too slow or hits something).
All this applies directly to skis also. I suspect this is the design element that makes ‘Beta’ skis so cool. A ski can be designed such that for a given skier’s performance level its sidecut will try to turn a tighter radius than the probable skier-tipping-pattern will normally require for lateral balance at the probable speeds designed for.
OK, I know that seems a bit of a stretch… but have you ever wondered why certain skis seem to ‘turn really well’ for you but not your friend? Why a certain ski seems to ‘turn by itself’ or seems to ‘come right back under you’ so easily? It’s a matter of the ski’s sidecut/flex pattern being well-matched to your own CM height, typical degree of angulation in your style and the typical speeds at which you make you typical turn radii.