|MadSci Network: Physics|
Well, the problem, as it has been set up, is an impossible situation, and let me explain why. If you turn the fork (let us say you turn it counter-clockwise as viewed from above), the torque on the spinning wheel will be such as to drive the wheel to move its angular momentum vector upward. This motion is clearly not being allowed since by the setup of the problem, we are insisting that the wheel be locked into the fork by a frictionless axle and furthermore the frame of the bike is not allowed to accept any of the angular momentum. Therefore, there must be an equal and opposite torque, provided by a normal force from the bike frame, such that the angular momentum is constant (since you are not allowing the wheel to transfer its angular momentum elsewhere). Therefore, according to the setup of the problem, there is no way to couple the angular momentum of the spinning bike tire to any other object, and therefore, by conservation of angular momentum, the bike tire can not change the direction of its spin. It should be made clear that there is no "natural precession" of a bike tire in the absence of outside forces acting on the bike tire. So why can you turn a bike tire on the ground? Because in that case, the bike tire's angular momentum is coupled both to the frame of the bike and to the Earth. Therefore, angular momentum can be transferred to other items and the bike tire can turn.
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