In an idealized scenario with all factors being equal, the acceleration of the three systems you mentioned would depend on the specific configuration and parameters of each system. Let's consider each one individually:
Atwood Machine: An Atwood machine consists of a pulley with a massless and frictionless axle and two masses connected by a light inextensible string that passes over the pulley. The two masses can be different, and their difference determines the net force and the resulting acceleration of the system. In an idealized scenario, neglecting any additional factors, an Atwood machine would accelerate at a constant rate determined by the mass difference and the total mass of the system.
Modified Atwood's Machine: It's unclear what modifications you're referring to in the modified Atwood's machine. The acceleration of the modified version would depend on the nature of the modifications made. If the modifications affect the mass difference or the total mass, they could alter the acceleration. Without specific details about the modifications, it's difficult to provide a definitive answer.
Rim: It's unclear what you mean by a "rim" in this context. If you are referring to a solid disc or a hoop-like object, the acceleration would depend on the forces acting on it. If a force is applied tangentially to the rim, it can cause angular acceleration, and the resulting linear acceleration would depend on the radius of the rim. Again, without further details, it's challenging to provide a specific answer.
In summary, the acceleration of each system depends on its specific configuration and any additional factors or modifications involved. Please provide more details if you have a specific setup or context in mind, and I'll be happy to provide a more precise answer.