Bifilar Pendulum w/ off-centred CM, need verification for derived eq.

ChessGM
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Homework Statement
For an essay I'm looking at the MOI about the central diameter of a composite object (2 flush concentric cylinders, the external cylinder is shorter lengthwise and can slide along the longer one) with a centre of mass that is not on the central diameter. This is done through the external cylinder, and placing it some distance from the centre, and timing its period in a bifilar pendulum.

I have an equation but I'm not sure if its correct, if anyone could take the time to come up with it and cross-reference it that would be appreciated.
Relevant Equations
The "standard" equation is 2 x pi x r x sqrt(IL/mg), but now it does not work because it assumes a centred centre of mass.
Here are some diagrams:
1735683470349.png

1735683493564.png

The mass of the rod is M, and the external cylinder is m. This is what I came up with:
1735683545670.png
 
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Can you show your math work that led to that result? I'll send you a message with tips for posting math here at PF using LaTeX.
 
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