I Gyroscopic precession

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I was watching this and I get it that when you rotate the wheel, the torque due to its weight will cause it to revolve parallel to the ground instead of perpendicular to the ground. I know that the faster you spin the wheel, the less likely the lever from the wheel to the pivot will be slanted. What I can't figure out is why will it be slanted? I mean the torque(perpendicular to the angular momentum of the rotating wheel) due to the weight of the wheel will change the angular momentum of the rotating wheel to its direction but I see no reason for the wheel to still "fall" if it is not spin fast enough.

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