Calculating Work Done on Pulley by Falling Block

[r_swayze]

The pulley in the illustration is a uniform disk of mass 2.40 kg and radius 0.220 m. The block applies a contant torque to the pulley, which is free to rotate without friction, resulting in an angular acceleration of magnitude 0.180 rad/s2 for the pulley. As the block falls 0.500 m, how much work does it do on the pulley?

The illustration is of a pulley with a rope hanging a block down vertically.

Here is my attempt:

work = torque x angular diplacement

If the block falls then the pulley turns by the same amount, right?

so arc length = .500m

theta = arc length / radius = .500 / .220 = 2.27 radians = angular displacement?? (is this right?)

To find torque, I need F x r

r = .220

How do I find the Force?

Do I use the mass = 2.4 kg and multiply by tangential acceleration?

 

[rl.bhat]

Yes. It is tangential acceleration.
How to find out the tangential acceleration?

 

[r_swayze]

tangential acceleration is angular acceleration times radius

but what about angular displacement? Did I do that right?

 

[rl.bhat]

Your angular displacement is correct.
Now the torque = moment of inertia*angular acceleration.