Torque and Force problem?

Question

A string of negligible mass is wound around a uniform disk of radius R and mass M. The disk is released from rest with the string vertical and its top end tied to a fixed bar. Find the acceleration of the center of mass of the disk.

I know you have the 2 forces acting on the disc, mg acting down, and the force of the torque caused by tension of the string on the disc. I know F=ma and Sum of torques=0 and I for a disc is 1/2(MR^2). My problem is what to do next and thus solving this problem. Thank you for your help.

Answer 1

One way is to consider the inertia of the disk as an equivalent mass, I/R^2, which, added to M, provides an "effective mass". Then the acceleration A = gM/effective mass. Basically this says that part of the force gM accelerates M and the remainder of the force accelerates the disk's rotation. You can check the result by computing the disk's rot. acceleration as A/R, then multiplying by I to get the torque, then compute Fr as torque/R and Ft as MA, and Fr and Ft should sum to gM.