Plus, the pulley is suspended on a cord at its center and hanging from the ceiling.
You're given masses of the objects, mass of the pulley and it's radius. And your assumptions are that the string is massless and inelastic and that there is no friction between the string and the pulley. You have to find the tension in the string suspending the pulley (it's not drawn in this particular figure).
My conclusion was that, since there is no friction between the string and the pulley, the tension would have to be equal all along the string's length and the pulley would not rotate, it would slip and there would be no way to transfer the linear motion of the masses to the rotational motion of the pulley. (The resulting torque would be zero.)
So, according to me, the tension in the string suspending the pulley would simply be the tension along the rope multiplied by two.
But the solution includes a rotating pulley and its rotational inertia, and it gives a different answer. Where did I go wrong? And why? If not, how can I prove I'm right?