The answer is very simple, and it is difficult to state concisely. In
summary, while riding in a straight line, the bicycle and rider
system is falling either to the left or to the right. The rider can
steer into a fall if the bicycle is moving, but she cannot do so if
the bicycle is not moving. With that extra degree of freedom taken
from her she must rely on a fundamentally different mechanism for
balance. A bicycle with a clamped headset cannot be steered, and it
cannot be ridden in a straight line by steering. It can be balanced
with considerably greater difficulty by shifting weight placed on the
pedals to counter the gravitational torque.
A moving bicycle is almost always falling either to the left or to
the right. The rider steers the bicycle in the direction of the fall.
Due to the geometry of a bicycle this moves the contact between the
front tire and the road in the direction toward which the bicycle is
falling. The reaction force of the road in this new position exerts a
torque about the line containing the contact points of both tires
with the road, which opposes torques in the direction of the fall.
Another way to analyze this is to recognize that in the turning
rider's frame there is a torque about the line of contact with the
road due to the centrifugal force, which opposes the gravitational
upsetting torque about that line. This description of the phenomenon
minimizes the importance of the effect of bicycle geometry on
stability, however. While the effect it real and it has the correct
sign, anyone who has ridden a bike with a radical head tube angle
will recognize how important steering geometry is to the ease of
riding a bike.
That's my best effort at answering the question in a short time. I
should emphasize that the bicycle is not a well understood system
despite the great amount of effort that has gone into modeling it.
For example, there is no first principles model of bicycle dynamics
which can be solved for the optimum head tube angle which can so
easily be determined by feel by a rider to be in a narrow region
around 72.5 degrees**. So how does one arrive at that angle? The
answer is still "by art", but perhaps someday it will be "by science".
Leigh
PS: By Googling I found a very nice essay on bicycle geometry by
Steven Miller, an undergrad in ME at SDSU: <http://www.madsci.org/
posts/archives/oct2000/971305467.Eg.r.html>. It looks good to me,
although from his drawing I infer that he prefers so-called "mountain
bikes" rather than to the more classical forms us old farts prefer. I
wrecked my shoulder in Cambridge in 1994 by going over the handlebars
of a borrowed "mountain bike" I rode for the duration of my stay
there. When I returned to North America I bought a mountain bike to
add to my stable. I doubt that I have put even 500 km on it in eleven
years. and yes, I did go over the handlebars again on my new bike,
though this time on a steep trail rather than on Trumpington Road.
* Note that the natural axis for analyzing this motion would seem to
be the line containing the contact points of both tires with the
road, but that line is not constant in direction. Its direction
changes with the direction of the front wheel.
** This is with "normal" values for other geometric factors such as
rake and wheelbase.