 #1
 3,812
 92
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
$$\tau=\frac{mgl}{2}\sin\thetaMgl\sin\theta$$
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!
Attachments

4.4 KB Views: 336

26.9 KB Views: 333

27.4 KB Views: 356