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I have a system diagrammed and explained in the image below.

enter image description here

Experimentally I believe the wheel will rotate around the pivot point where the cable is attached in a counter-clock motion if observed from above. However, theoretically why this would be so is confusing me. It seems to me that all linear forces (weight and tension) cancel and the only force not canceled is the torque force. But if I understand this torque force in the same way that I understand, say, a wrench being applied to a nut, then the the arm $\vec{r}$ and the force $\langle 0, -mg\rangle$ would only contribute to a rotational motion in the $yz$-plane.

Clearly the analogy must be bad and the part that's screwing it up is the fact that unlike the nut, this system has momentum. But I thought the momentum due to rotation was not of the system at the pivot where the cable was attached but rather was of the rotation of the wheel about its center. I don't see how that transfers over to momentum about the pivot where the cable is attached.

Am I fundamentally misunderstanding something here?

[Edit: As I think about this, maybe a better question to ask is, since the wheel spins around the pivot point where the cable is attached, there must be some force causing this motion. I assume it's a centripetal force since the motion seems roughly circular, so where does this force come from? Still, my thought is that it can't be the torque force because that's supposed act perpendicular to the plane of motion and here the torque is not in the right direction for that.]

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This gyroscope precession isn't very intuitive but you can understand in terms of rotational motion that you probably do have an intuition for.

As the torque from the supporting cable tips the wheel, the lower part of the wheel becomes closer to the vertical axis and the upper part becomes further away. As you bring something closer to the axis it speeds up since the rigid body forces bringing there will have a net component in the direction of motion. This is an aspect of angular momentum you probably have a feel for. It works because the angular momentum about the vertical axis is conserved, but you can think of it as a consequence of the rigid body forces if you prefer.

So the lower part of the wheel will be moving faster than the top part of the wheel. This drags the wheel in the precession direction.

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