Two rods connected, with a sliding mass attached to a spring.lagrange

[mjmontgo]

A rigid T consists of a long rod glued perpendicular to another rod of length l that is pivoted about the
origin. The T rotates around in a horizontal plane with constant frequency ω. A mass m is free to slide along the long
rod and is connected to the intersection of the rods by a spring with spring constant k and a relaxed length zero. Find r(t)
where r is the position of the mass along the long rod from the intersection. [You will find there are three cases to
consider: depending on the size of ω2 compared to k/m.] There is a special value of ω; what is it and why is it special?

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Im basically having trouble setting up the lagrangian.
My kinetic energy so far is T=1/2mv^2+1/2Lω^2+1/2kx^2

The potential energy is throwing me through a loop as well
i have U=mghsin(ωt) although i know this is either incorrect or their is more to it.
I feel like i over think these lagrangian questions but I am never very sure. any pointers on the kinetic and potential energies would be appreicated. solving the lagrangian and analying the special cases of ω i should be ok with, although your input it always welcomed.

thank you in advance for your time and assistance.
If a diagram is needed i could provide a link.

 

[Drypht]

I am solving the same problem, and I cannot find a "comfortable" general coordinate for the mass. Also you are missing your spring potential term in the potential of the Lagrangian.