Imagine a bead, free to move along the length of a horizontal rod, pivoted at one end. The system is initially at rest, with the bead at some distance from the end. Now, a constant torque is applied to the rod. I want to find the equation of the motion of the bead.
My attempt: (The mass of the bead is $m$, and the moment of inertia of the rod is $I$ and the applied torque is $\tau$)
Writing the force equation along the radial direction in the frame of the rotating rod, I get $$\ddot r = \omega^2 r$$
Writing external torque as the rate of change of angular momentum, I get $$\tau = 2m\omega r \dot r + mr^2 \alpha + I \alpha$$
And using the work kinetic energy theorem, I get $$2\tau \theta= I \omega ^2 + mr^2 \omega^2+ m \dot r ^2$$
I tried a lot, but couldn't figure out how to simplify these equations. I'm totally stuck, and a nudge in the right direction would be really helpful.
