# Oscillation of a disc about a rod perpendicular to the disc but not through the centre.

A thin uniform circular disc of radius a and centre A, with density p, has a circular hole cut in it of radius b and centre B, where $AB = c < a−b$. The disc is free to oscillate in a vertical plane about a smooth fixed horizontal circular rod of radius b passing through the hole. Show that the period of small oscillations is $2\pi\sqrt{l/g}$, where $l = c + (a^4 -b^4)/(2a^2c)$.

I have the moment of inertia to be $I =p/2 pi(a^4 -b^4 +2a^2c^2)$, which looks correct. I don't know how to find the period of the oscillations - is there a formula I should know? Should I use conservation of energy of forces??

The oscillating disk is a compound pendulum. The period is $2\pi \sqrt{\frac{I}{mgh}}$ where $I$ is moment of inertia about the axis of rotation and $h$ is the distance of the centre of mass from the axis.