Calculating forces regarding angular momentum/gyroscopic forces?

[ryan31394]

Ok, say you have a wheel, of a given radius 'r' and a given mass 'M', spinning on an axle at a high RPM rate 'ω', and the axle is extended by a given distance 'd' on one end. The wheel is fixed in a way to prevent nutation and precession, so that it can only rotate along its spin axis 'x' and pitch on an axis 'y' perpendicular to the spin axis, with both axes parallel to the ground. While spinning, its momentum should 'resist' any changes in pitch where the wheel tries to leave its vertical position, the 'z' axis in this case, preventing it from 'tipping over', similar to a bike.

How would you calculate how much 'resistance' would be felt if you pushed upwards on the long end of the axle with a given force 'F'?

 

[Hertz]

What exactly would be causing the force? (Sorry, I'm not much help, I'm just as curious about this question as you :P )

After sitting here for a while thinking about it, I think I've reached a conclusion about one thing...
1. Angular momentum is conserved

Why? Well, since all the particles in the wheel are rigidly connected, any force on any particle consequently affects the entire system. Pushing up on one particle in the wheel effectively pushes up on every particle in the wheel (due to intermolecular forces). Thus, the bar pushing up on the wheel does not cause any net external torque so angular momentum is conserved.

That being said.. I'm inclined to say that the force required to lift the wheel in this context is invariant of ω. E.g. ω could equal zero and it would still be the same problem. Not too sure though, because intuition tells me otherwise :P

 

[ryan31394]

Yes, it would seem at first thought that the ω would not have any effect upon the force required to change the position of the axle. However, when you look at simple gyroscopes this thought is shown to be incorrect.

For instance, bicycle wheels use this concept to remain upright while in motion. While it is virtually impossible to remain upright while the wheels are not spinning, as you increase your speed it becomes far easier, to the point where you would need to fall off of the bike entirely, shifting the center of gravity far off center (therefore increasing the torque forces imposed upon the wheels), in order for the bike to tip over, but how far would you need to lean off center for the bike to tip over (with given values such as those stated above)?