We have a pulley in the shape of a solid disk of mass M = 2.0 kg and radius R = 0.50 m. If we apply a constant force F = 8 N to a string wrapped around the outside of the pulley what is the pulley's angular acceleration? The pulley is mounted on a horizontal frictionless axle.
As usual, start with a freebody diagram.
The freebody diagram has the force F we apply, mg down and a normal force up from the axle. The pulley can only rotate, and the only force associated with a torque about the axle is the force F, so we often just draw that force on the freebody diagram.
Apply Newton's Second Law for Rotation, taking torques about the axcle through the center.
Στ = I α
Apply the equation torque = r F sinθ to get the torque from the force we apply. Using r = R, the pulley radius, and an angle of 90° we get:
R F = I α
What do we use for the moment of inertia? Looking up the expression for the moment of inertia of a solid disk rotating about its center we find that:
I = ½MR^{2}
This gives: R F = ½MR^{2} α
This gives: F = ½MR α
α  = 

= 

= 16.0 rad/s^{2} 
Once you know this, you can plug it into the constant angular acceleration equations to find things like the angular displacement and angular velocity after a certain amount of time has gone by.