Pulley w/ 3 Masses Homework: Find Linear Acceleration

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The problem involves a system with a hoop, a solid disk pulley, and a rolling sphere, requiring the calculation of the linear acceleration of the hoop. The poster expresses confusion about incorporating the mass of the pulley into the calculations, as they have only learned about massless pulleys. They seek a step-by-step explanation of the solution process, particularly how to approach the problem if the pulley were massless. Understanding the dynamics of the system, including tension and rotational inertia, is essential for solving the problem. The discussion highlights the need for foundational knowledge in mechanics to tackle such multi-body problems.
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Homework Statement




An green hoop with mass mh = 2.4 kg and radius Rh = 0.12 m hangs from a string that goes over a blue solid disk pulley with mass md = 2.4 kg and radius Rd = 0.11 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 = 4.2 kg and radius Rs = 0.25 m. The system is released from rest.

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


Homework Equations





The Attempt at a Solution



I have no idea, would love someone to explain all the very elementary steps, we have never even looked at taking the mass of the pulley into account!
 
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How would you solve it if the pulley were massless?
 

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