Lagrangian of a rotating disc with mass attached

1. The problem statement, all variables and given/known data
A uniform disk of mass 2M, radius R, is mounted on a frictionless horizontal pivot through its principal axis. The disk has an additional point-mass, M, fixed to a point on its circumference.

(a) Give the Lagrangian for this system.

(b) Find the frequency of small oscillations of the weight about its lowest point.

The disk is now placed on its rim on a at surface and set in motion such that it can roll without slipping.
(c) Show that the period of small oscillations has increased by a factor of √3/2 relative to the answer obtained in part (b).

2. Relevant equations
L=T-V
EL equation
T=0.5Iω2 for rotation about a principal axis.

3. The attempt at a solution
I have completed a) and b) to obtain an angular frequency of oscillation of √(g/2R) about equilibrium (although this may not be right).

Now I’m having a lot of trouble with c). This must be with determining the KE of the system. I define the angle θ between the line from the centre vertically down to the point of contact with the ground and the line from the centre to the mass.

The PE U=-mgRcosθ as before.

The KE has two contributions. One due to rotation and translation of the disc, and one due to rotation and translation of the mass.

For the disc, T=MR2(dθ/dt)2 – I think this is ok as the angle the disc rolls through matches with the angle the mass moves through and the pure rolling condition holds.

For the mass, the problems start. I though it would be best to write down it’s position, differentiate and find (dx/dt)2+(dy/dt)2 from this to get it’s total KE. So [x,y]=[Rsinθ+R(dθ/dt)t,-Rcosθ]. [dx/dt,dy/dt]=[R(dθ/dt)cosθ+R(dθ/dt)2t+R(dθ/dt),R(dθ/dt)sinθ]. Squaring the components, adding and multiply by M/2 gives the KE of the mass.

Now this gets very messy, and two attempts to work through the algebra haven’t led to anything that cancels nicely. So I believe my KE is wrong (more specifically my [x,y]) so if anybody could point out why I would be grateful, thankyou.

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