How do you turn corners?

[gauss]

I too gave the matter some more thought. Of course we can't think of a rolling unicycle with static equations, as there is a certain amount of momentum. Moreover, the angular momentum of the wheel ends up being very important. I am using a computer on which I can't make jpegs, so you are free of diagrams this time, but you may draw your own from this. If you don't understand what a vector is then maybe just skip to the last paragraph and hold off on the usual replies to this post about how some idealized situation doesn't apply to the real world.
Imagine a disk spining about the z axis. If you apply a moment about the x axis, That is you lean your unicycle. The result is not a rotation about the x axis, but rather a precesion about the y axis. The reasons for this are pretty complicated, I don't think I can explain it well, as I don't understand it well, but...

think of a point mass moving with a constant speed in the x-zplane (but not constant velocity). The partical has a momentum G =m v . I will try to use bold print to denote a vector quantity. If a force normal to this motion is applied it will cause a change in the momentum d G= d (m v ) in the direction of the force F. Newton's second law states that F dt = d G.
Here it gets a little complicated without diagrams, but you know the vector F is perpendicular to mv. So th adding them together gives a new vector describing the new direction of motion for the particle (that is velocity +change in velocity is the new velocity). Now we are expecting a precesion about the y axis, so let angle dtheta be the angle between the original velocity and the new velocity. the tan(dtheta)= Fdt/mv. If dtheta is small, (that is to say the change in momentum is small, then tan(dtheta)=dtheta. Solving for F we get F= mvdtheta. dtheta is an angle in the xz plane. so it is a revolution around the y axis. lets call it omega* j. Remember j is the unit vector in the y direction. omega is the rotation velocity in this direction (dtheta means how fast does the angle theta change which is velocity). so finally we have:
F = m omega x v
where "x" is cross product. Which makes sense considering the directions o f these vectors.
That was a logical case for point masses. This can be extended to rotations to get the same sort of thing considering newtons law as Moment = a change in angular momentum (M=dH) So now the original motion is a fast rotation about the z axis( instead of a motion of a point) a moument is applied about the x axis and the result from the cross product relation is a rotation in y.

I'm guessing that if you followed all this, particularly without diagrams then you already know what I am saying, maybe this was a refresher. If you didn't I will refer you to the Meriam text on dynamics where I snagged this, although any dynamic text will explain this phenomena one way or another. You can do an experiment yourself. Spin a unicycle wheel and hold it off the ground by the seat. Try to make it "lean" and you will see is twist into a turn in your hands.
-gauss