Question

why does the rolling bicycle have greater stability than one at rest

Answer 1

A stationary bicycle falls over because it is at an unstable equilibrium.

Specifically, a rigid body standing on a surface is at an equilibrium if its center of gravity is above the convex hull of its support (the points where it contacts the surface). If this is not the case, the object is at a disequilibrium, and gravity will cause it to fall over until it reaches a new equilibrium.

An object at equilibrium can be either stable or unstable (or, on the borderline between the two, neutrally stable), depending on whether slightly pushing the object (thereby placing it temporarily at a disequilibrium) will cause it to return to the original equilibrium, or to move to a different one. In particular, a rigid body at equilibrium on a surface is stable (against small perturbations) if and only if its center of gravity is above the interior of the support region, so that slightly moving the center of gravity will not move the object out of equilibrium or more generally, the shape of the contact surface is such that slightly tilting the object will move the contact area in the same direction as the center of gravity, but further. (An example of this case would be e.g. an ellipsoid lying on its side on a flat surface.)

To stay upright, a moving bicycle must turn in the direction it leans towards, and it must turn quickly enough that its support will move sideways faster than its center of gravity.

A slow-moving, or stationary, bicycle will not be able to do this, and therefore is not stable.

One notable exception is that the person riding the bicycle can exert a limited amount of direct control over their center of gravity by shifting their weight and, depending on the caster angle, by turning the handlebars even while stationary. It's not a very strong effect, though, which is why it's very difficult to balance on a stationary bicycle.

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