Pendulum consists of a rod of mass m attached to a light rod

[cler]

Homework Statement

A pendulum consists of a uniform rod of mass m and length l hanging from the bottom end of a light rod of length l which top end is fixed to the ceiling. (see file attached)
System moves in a vertical plane. Find equations of motion.

Coordinates of the center of mass (X,Y)
angles θ and ψ of the light rod and the rod of mass m with the vertical respectively.

Homework Equations

Lagrangian method

L=T-U
U=mgY
T=mV²/2 + IΩ²/2

V is the velocity of the center of mass respect to a system at rest which origin is the top end of the light rod.

The Attempt at a Solution

X=l/2sinψ + lsinθ
Y=-l/2cosψ -lcosθ

|V|²= l²/4\dot{ψ} + l²\dot{θ}² + l²\dot{ψ}\dot{θ}cos(ψ-θ)

I relative to the top end of the rod of mass m I=ml²/3
ω=\dot{ψ}

then i will plug this into L= T-U and the fin the Euler- Lagrange equations.
but i am not sure about I and ω.

I am confused. My first attempt was to choose same X,Y,V but I relative to the center of the rod of mass m I= ml²/2 and ω=\dot{ψ}+\dot{θ}

 

[member 553083]

but it was not right.Is my first attempt wrong? How should I choose I and ω?![enter image description here][1] [1]: https://i.stack.imgur.com/fUOZ6.png