Floating beads in parabolic wires

A bead of mass mm slides freely along a frictionless wire in the shape of the curve y=x2y = x^{2} rotates around the yy-axis with a constant angular velocity ω\omega. There is a constant downward gravitational acceleration gg.

Suppose the bead maintains a constant height above the ground so that it "floats." How many such heights are possible?

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