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## Homework Statement

Find the period of a pendulum consisting of a disk on 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?

## Homework Equations

## The Attempt at a Solution

If the rod is displaced by an angle ##\theta## as shown in the attachment 2, the torque acting about the fixed point P is

$$\tau=-\frac{mgl}{2}\sin\theta-Mgl\sin\theta$$

From small angle approximation, ##\sin\theta \approx \theta##. The moment of Inertia about P is

$$I=\frac{ml^2}{3}+M\left(\frac{R^2}{2}+l^2\right)$$

Let ##\alpha## be the angular acceleration, then

$$\alpha=-\frac{gl\theta}{I}\left(\frac{m}{2}+M\right)$$

Is this correct?

Moving to the second part of the question, I am thinking of assigning two angular velocities to the disk, ##\omega_1=\dot{\theta}## about P and ##\omega_2=\dot{\beta}## about CM of disk. Next I will write the expression for energy at any instant and differentiate it wrt time to find the time period. See attachment 3.

I came up with this: ##R\beta=l\theta##. Is this correct? Is my approach correct?

Any help is appreciated. Thanks!