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Here's how I use center of rotation:

In this question, you need to find out the angular acceleration of the circular body.
I'll use torque=moment of inertia*angular acceleration in a differente' fashion.

Write the torque about 'P' (the center of rotation is ALWAYS denoted by the letter 'P'):
Torque=m*g*(2R)

Now whats the moment of inertia about 'P'? The rigid body contributes M*(K^2+R^2). But how much MOI does the mass 'm' contribute? It contributes m*(2R)^2. Note that 2R is the distance of the point 'T' from 'P'. i.e. I'm assuming 'm' to be at 'T'. Why? I don't know why but this ALWAYS works.

Now, we write:
m*g*(2R)=(M*(K^2+R^2)+ m*(2R)^2)*alpha.

Now find alpha.