Period of Pendulum: Spinning Disk Forces & Inertia Moment\

[LCSphysicist]

Homework Statement

Find the period of a pendulum consisting of a disk of mass M and radius R fixed to the end of a rod of length l and mass m.How does the period change if the disk is mounted to the rod by a frictionless bearing so that it is perfectly free to spin?

Relevant Equations

t = r*f = I*theta

If the bottom disk is free to spin, will it necessarily spin? all the forces in this disk don't produce a torque.
I don't know why we disregard it's inertia moment when it is free to spin (It's intuitively set, but I can't see it mathematically)
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[etotheipi]

If the bottom disk is free to spin, will it necessarily spin? all the forces in this disk don't produce a torque.

I think you nearly answered it yourself; if there is no torque on the disk about the centre of mass of the disk, can the disk undergo rotational acceleration about its centre of mass?

As for the other part, I think it might be helpful for you to write $\vec{L} = \vec{L}_{rod} + \vec{L}_{disk}$, and $\vec{\tau} = \frac{d\vec{L}}{dt}$ for the whole configuration. König's theorem states that the angular momentum of a body equals the angular momentum of its centre of mass plus the angular momentum relative to the centre of mass - using this, can you decompose the angular momentum of the disk further?