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So people are familiar with the idea of holding a spinning bicycle wheel while on a stool (whose seat can spin). You then tilt the spinning wheel and lo and behold you start to spin on the stool.

Ok fine, I get that (conservation of Angular momentum).

So the real question is: if I have 2 wheels mounted on the same axis and they are spinning in opposite directions, does the angular momentum cancel out? Do you still spin if you are tilting them while on the stool?

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Yes of course, the momenta cancel out. This was used by EH Jones in the 70ties when he tried to build a UnRidable Bicycle. One of those URBs he built had a second front wheel spinning backwards. With this URB he was able to show that gyroscopic action of front wheel is not relevant for riding a bicycle. –  Georg Nov 24 '11 at 17:58
    
Thanks the EH Jones reference was helpful (even found a reprint of the 1970s article in Physics Today). I guess what I'm struggling with is intuition. If you've ever spun a bicycle wheel (or a simple gyroscope) while holding it and turned it over and such, you can really feel the resistance. Its quite counter intuitive to suggest that by adding a second wheel spinning in the opposite direction, that you will no longer feel that resistance. –  Fraggle Nov 26 '11 at 14:32
    
Right, that is something everybody should experience personally. Before WWII gyroscopes with a Eiffel tower as a stand were sold as a gadget. It is a shame that nothing similar is available today (AFAIK) –  Georg Nov 26 '11 at 14:45
    
Correction, there is a Gyroscope available from Tedco. –  Georg Nov 26 '11 at 15:31
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Yes, the angular momentum is a vector, and two of them can sum to zero.

I'm unclear from the question if the wheel axes are vertical or horizontal.

Assuming they are horizontal, then you're talking about gyroscopic precession. In that case, if you tilt the common axis, one will try to precess in one direction, and the other will try to precess the other way, and the two forces will cancel out.

(If you want to understand gyroscopic precession, what works for me is not to think of the wheel as a single piece. I think of it as a circle of independent weights attached by strings to the hub. Then it is clear how an attempt to turn the axis produces an actual turn at 90 degrees because all you can do is deflect the weights in their "orbits".)

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Thanks, although I'm not sure if it matters whether the axis (there is only one, they are both mounted on the same axis), is vertical or horizontal. Gyroscopic precession in my understanding is something else, due to a torque induced by gravitational and normal forces on the gyro. But lets not get derailed on that point. I guess bottom line is, you won't spin on the stool, and if you are standing on firm ground you won't feel any resistance as you try to rotate the wheels? Right? –  Fraggle Nov 24 '11 at 15:41
    
sounds like experiment time to me :-) –  Nic Nov 24 '11 at 16:04
    
@Fraggle: I think that's right, but as Nic says, it sounds like a fun experiment! I think you won't get any resistance, but the common axle will feel the strain of the two gyroscopic precession forces fighting each other. –  Mike Dunlavey Nov 24 '11 at 16:50
    
If I had access to a machine shop or other such things, I might be able to build the right apparatus. However, I don't, so I'm not sure I'll ever get to trying this. –  Fraggle Nov 26 '11 at 14:23
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