I saw this question yesterday.
OP found a way to solve it using conservation of energy, and wants to investigate how the acceleration can be calculated using forces and torques.
OP also posted their method of solution, but got a wrong answer.
I realized that OP did not take into account the acceleration due to rotation of the mass on the ring, and the expressions of the normal force and friction are wrong.
Can we reopen this question?
It seems that this is not a "do my homework" problem (OP already solved it using another method, and showed their work on another (more interesting) method), and it requires some careful thought.
If we reopen it, then of course, the answer should not solve the whole problem, but rather, point out the mistakes OP made (they need to properly consider the acceleration of the rotating mass in order to calculate the friction and the normal force).