Finding the force of constraint--compound pendulum on spring

[anf3]

Homework Statement

From Fetter and Walecka 5.1:[/B]
Consider the compound pendulum in FIg 28.1 (mass M, moments of inertia Iij relative to the center of mass, which is a distance L from the point of support Q) but with Q attached to the bottom of a vertical spring (force constant k) and constrained to move vertically. The top of the spring is fixed to a rigid support.
(a) Construct the Lagrangian for this system in terms of the generalized coordinates η (the downward displacement of Q) and φ. Derive Lagrange's equations.
(b) Obtain the same equations from the general principles of rigid-body dynamics concerning the motion of the center of mass and rotations about it (don't forget the force of constraint)
https://lh3.googleusercontent.com/8K4utS2xNx0v9MV9P7Nhg-wnYFxzLVknSHNo1OI7pbzBpL8B9LqbpT9ODbnKp2jlMtDT0WYQzJF5y3cdueMftuaZSBG3PoR0WbzK-fOI0deYmKe9JVqxgevpVgBKvf1hR5FCJsYnVJVKlWYggq3Pg3Juy0vTQD5MGmEZSssJOUq1uD7tjEELzDu24Obg0F2FOepDH3dBFUBxxx4ucVHUE7up5g16qkq1k8TtvmZZLUNv7d2qC9Tu3hvCHB5yH-s7AJTBXLTwNyzRdEVQWyev69i-H19gyhFElV-nEKtbkb_-zTmHWBd37ZSYCzN6Wi0hoJ2xTh5MxLnIgs2wdiAeIxmzN9lnsx4g7wbIsPgbQMcxm4v2y-mjoSCTOVmeRx-ABsuKK0cnnPS-N14xfpuoemzzdquUsRvLhcKCKcxTTov6g6b7qhb3JpJpuXZPPABni7Zia3QcF3ZZ5mkIrHYL1ilAb7W-4jots6SbihTd61WIAPavW7tZUCg2b2jGF0-D7rIDhA-ok53B-8l_0LWdhICHj8mHcSV2SWTvTdjIAvQnavDxO2-F6QAlHM8O7w1gJiVjD7O4Npz5GlocePLAUpLDxnWPQVsJQTpDx_LYFSVZFIlHQg=w345-h612-no

Homework Equations

Euler's equations:
I₁ dω₁/dt =ω₂ω₃(I₂-I₃) +Γ₁^(e) and similar permutations of indexes
where
Γ₁^(e)= torque⋅e^{^1}

The Attempt at a Solution

I've done part (a) getting η^⋅⋅= g- k/m η and φ^⋅⋅ = -mgl sin(φ)/(I_33,cm+ml²)

For part(b)
I initially used the euler equations and the torque about the point Q Γ^ext_Q=R×mg= - mlg sinφ, which results in the equation I had from part (a). However since the question specifically asks to use rotations about the center of mass I'm not sure if I would be allowed to do this.

However bigger issue is in trying to find the torque about the center of mass I don't know how to go about finding the force of constraint from the suspension point that contributes. Nor am I sure of what the ωs which are needed to use the Euler equation would be in a reference frame centered on the center of mass, since the rotation is taking place around Q and not cm. I imagine I may have to use the relations of each reference frame in terms of the other? It seems as those the torque about the center of mass is needed in order to find the η EOM.

Γ_cm^spring = R×(-kη)= -lkη sin(φ) e^{^[\SUP]₃}

 

[BigJDubs]

[/SUP] (since the spring is only in the direction of the z axis)Γcmgravity = R×mg = -mlg sin(φ) e^[\SUP]3[/SUP]Would the net torque about the center of mass be:Γcmnet = Γcmspring + Γcmgravity = -lkη sin(φ) e^[\SUP]3[/SUP] - mlg sin(φ) e^[\SUP]3[/SUP]= (-mlg-lkη)sin(φ)e^[\SUP]3[/SUP] and if so then would the equation of motion for η bem(ddη/dt²) = -mlg-lkη)sin(φ)Which is the same result as before, but I am not sure if this is the correct way to go about it.