A pendulum consists of a light inextensible rod $OA$, length $b$, freely pivoted at $O$, attached at $A$ to the rim of a uniform disc of radius $a$ and mass $m$. The disc is free to rotate about $A$. The system moves in a vertical plane containing $O$.
Can you help me to set Lagrangian?
I found that
$$\vec r_A=b\sin\theta\vec i+b\cos\theta\vec j$$
$$\dot{\vec r_A}=b\dot\theta\cos\theta\vec i-b\dot\theta\sin\theta\vec j$$
I am not sure if this is correct for point $G$ ($G$ is center of mass of a dics): $$\vec r_G=(b\sin\theta+a\sin\phi)\vec i+(b\cos\theta+a\cos\phi)\vec j$$
This is all I've done for now. I would appreciate some help and hints for position vector of the point $G$ and forces acting here.