The problem is problem 2 of the attached picture
The Attempt at a Solution
I got the problem wrong, and I am trying to understand what my teacher did. Particularly, on the right side of his solution, he took the θ acceleration to be zero, because there is no force in that direction, that is obvious. What is not obvious at all is the next step he took, d/dt(r^2ω)= 0, and from there he goes to equating the initial angular velocity and radius with the final angular velocity and radius. Everything else from the solution falls into place from there, but as far as I see, there is a PDE, which I have no idea how to solve, and I have no idea how he implied that time derivative from that information.
Any help would be greatly appreciated and thank you in advance, I plan to repay the 'karma' when I know how to solve some physics problems with confidence on this site.