Pendulum suspended from Horizontal rotating hoop

[Sswift]

Homework Statement

A massless hoop is suspended horizontally and is free to rotate about a vertical axis through its center with a constant angular velocity (omega). Attached to the edge of the hoop is a simple pendulum that is restricted to oscillate in only the radial direction. find the Lagrangian of the system.

M= mass of pendulum
L= Length of Pendulum
∅= angle the pendulum makes with the vertical
ω=angular velocity of hoop
R= radius of hoop

Homework Equations

L= T-U
T= Kinetic energy
U=Potential Energy

The Attempt at a Solution

T(hoop)=(1/2) I(hoop)*ω^2
But I is dependent upon the mass of the hoop and since the mass of the hoop is 0 so is I(hoop)->T(hoop)=0

T(pendulum)=(1/2)M[L*(d∅/dt)]^2+(1/2)I(pendulum)*ω^2
T=1/2M[(L*d∅/dt)^2+(R+Lsin∅)^2*ω^2
U=-MgLcos∅

L=(1/2M[(L*d∅/dt)^2+(R+Lsin∅)^2*ω^2)+MgLcos∅

Is this right?

 

[haruspex]

Looks right to me.