Three Masses - pulley, rotational and translational forces

[iluvcanucksfo]

An green hoop with mass mh = 2.4 kg and radius Rh = 0.18 m hangs from a string that goes over a blue solid disk pulley with mass md = 2.4 kg and radius Rd = 0.09 m. The other end of the string is attached to a massless axel through the center of an orange sphere on a flat horizontal surface that rolls without slipping and has mass ms = 3.3 kg and radius Rs = 0.25 m. The system is released from rest.

1)What is magnitude of the linear acceleration of the hoop?

I'm wondering how to solve this question, especially wondering how the tension forces would be related to each other throughout the system?

 

[Simon Bridge]

From what is written, the hoop is not rotating - so it is just a mass falling: you can do the free body diagram for that one.

The pulley rotates, the net torque on the pulley will have it accelerate consistently with the acceleration of the falling mass. Label the tensions on each side of the pulley differently.

The sphere is rotating, the tension acts through the center, but there is a force from friction (rolling without slipping) where there is contact with the table. Just think of it as a wheel being pulled by the axle - you will have course notes for that situation.