Solve your equations for the moment of inertia of the ring/disk/shaft/spool system as a function of the mass of the hanging weight, the acceleration of the hanging weight, and the radius of the spool. start with the equation containing the quantity you want to know, the moment of inertia of the ring/disk/shaft/spool system. identify the unknowns in that equation and select equations for each of them from those you have collected. if those equations generate additional unknowns, search your collection for equations that contain them. continue this process until all unknowns are accounted for. now solve those equations for your target unknown