Step 1: FBD
Draw a free-body diagram for the plate.
Step 2: Kinetics - Work/energy
Write down the kinetic energy and potential energy expressions for the plate, along with the work done on the disk by non-conservative forces. The instant center, IC, is fairly easy to locate for Position 2 - you are able to write down the KE for the plate as Tplate = 0.5*IICω2, if you choose to do so. The work done by F is equal to U = FsD, where sD is the distance traveled by corner D of the plate.
Step 3: Kinematics
The easiest approach here is to use the location of the IC for the plate.
Step 4: Solve
Solve your equations from Steps 2 and 3 for the angular speed of the plate.