Final angular velocity of a mass on a pulley

[austindubose]

Homework Statement

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.

Homework Equations

Ʃτ=rxF=Iα
F=mA
ω_f²=ω_i²+2αΔθ

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?

 

[Doc Al]

Do not assume that the force exerted on the spool equals the weight of the bucket. You need to figure out the tension in the string. (Analyze forces on the bucket as well as on the spool.)

 

[austindubose]

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.