Imagine a situation where a long masless stick is inserted through a ring, and fix one of the ends of the stick. Then, the system is given an impulsive force such that it starts rotating with an initial velocity $\omega_0$, the ring will start moving outwards, and thus, the system will experience a negative torque. I want to find the conditions when the system finally comes to (angular) rest: i.e: distance from centre, time elapsed, final velocity of the ring. I have gotten the following two equations using angular momentum change, and simple free body diagrams, and I'm now stuck. $$\ddot r= \dot \theta^2 r$$ $$r \ddot \theta = 2 \dot \theta \dot r$$ How do I further reduce these?

Note: Initial distance from axis can be taken as $r_0$

Imagine a situation where a long masless stick is inserted through a ring, and fix one of the ends of the stick. Then, the system is given an impulsive force such that it starts rotating with an initial velocity $\omega_0$, the ring will start moving outwards, and thus, the system will experience a negative torque. I want to find the conditions when the system finally comes to (angular) rest: i.e: distance from centre, time elapsed, final velocity of the ring. I have gotten the following two equations using angular momentum change, and simple free body diagrams, and I'm now stuck. $$\ddot r= \dot \theta^2 r$$ $$r \ddot \theta = -2 \dot \theta \dot r$$ How do I further reduce these?

Note: Initial distance from axis can be taken as $r_0$