Torsional pendulum

1. The problem statement, all variables and given/known data
The same question was posted here before:

I’m struggling to come up with answers regarding the amplitude and energy in each case.

3. The attempt at a solution
For when the ring is dropped onto the disc when it is at rest, I can’t find a way to use the maths to tell me what would happen. θ=θ0cos(ωt+ψ) but I can’t justify θ0 not changing.

Obviously what happens to the energy follows from this (although I’m not sure how PE varies with amplitude exactly in this case so couldn’t say anything quantitative).

When the ring is dropped onto the disc moving at ωmax, my initial thoughts are that the angular momentum is conserved at that instant (because the torque has no time to change the angular momentum in that instant), i.e the angular momentum of the disc a tiny bit before and of the disc plus ring a tiny bit after are the same. Then the maximum angular velocity reduces to 2ωmax/3. Using t=0,dθ/dt=ωmax,θ=0 gives θ=[√(I/c)]ωmaxsin(√c/I)t so then the amplitude rises by a factor of √(3/2). I’m really not sure about my approach here though. Also I’m not sure if I could apply θ=[√(I/c)]ωmaxsin(√c/I)t to the first case (I don’t think I know anything about ωmax so couldn’t).

Then the energy would obviously reduce (but I can’t say anything quantitative).

If somebody could help me out that would be great, thanks!

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