Homework Help: Final angular velocity of a mass on a pulley

1. The problem statement, all variables and given/known data
Use Newton's 2nd Law, sum of torques, and kinematic equations to determine the angular speed of the spool shown in the figure below. Assume the string has a negligible mass, and it turns without slipping. Use g=10 m/s² for acceleration due to gravity.



2. Relevant equations
Ʃτ=rxF=Iα
F=mA
ω_f²=ω_i²+2αΔθ

3. The attempt at a solution
Sum of torques:
Ʃτ=rxf
=(.6m)(3kg*10m/s²)(sin 90°)=Iα
--> 18 kgm²/s²=Iα
α=18/[.5(5)(.6²)]
α=20 rad/s²

But plugging this into the kinematic equation to solve for ω_f doesn't give me the right answer (I know this because I calculated ω_f based on conservation of energy first (ω_f=11.01 rad/sec), so I think I may have messed up somewhere up until this point. Any suggestions?

 

Thank you! I knew it was something having to do with the tension. And I wasn't thinking of doing forces on both objects, more of just doing it as a whole.