To start with, draw a force diagram for the hanging mass. You should then be able to use Newton's Second Law to create an expression relating the tension in the string to the mass of the hanging mass and its linear acceleration as it's falling. Ignore the spinning disk for now. (Your expression should contain T, m, g, and a. Don't forget the force diagram!) How does the tension in the string relate to the torque on the disk and ring? Just think about the spinning disk now. (Your expression should contain $\tau$, T, and r, the radius of the axle that the string is wrapped around.) How does the torque on the axle relate to the angular acceleration of the platform as it spins faster and faster? (Include $\tau$, I, and $\alpha$.) We have two different accelerations in this situation---the linear acceleration of the falling mass and the angular acceleration of the spinning disk and ring. How are those two accelerations related? Remember, the string spools linearly off the axle as it turns, giving it a tangential acceleration related to the spinning disk. (Include a, $\alpha$, and r.) Combining all of the above equations should allow you to write an expression for the moment of inertia of the disk and ring in terms of the hanging mass, the radius of the axle, the angular acceleration of the platform, and acceleration of gravity. (You want an expression in the form I = some function of m, g, r, and $\alpha$).
