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Lagrangian for a bead in a frictionless hoop

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A bead slides on a frictionless hoop. The hoop is spinning at rate w on vertical axis. Find Lagrangian. See attached file for full problem description.

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The solution solves for the lagrangian for a bead sliding along a frictionless hoop.

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Please see the attached file.
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bf Bead on Hoop rm

bf(a) rm

The system is one-dimensional since the position of the bead at any time is fully determined

by one coordinate which can be taken to be the bf angle rm $alpha$ between the direction

from the hoop center to the bead and the bf negative rm direction of the z-axis.
That is the coordinates of the bead at any time are
$$
eqalign{
x(t) &= ...

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