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I was referring to this thread: https://www.physicsforums.com/threads/accelerating-atwoods-machine-problem.44305/ to solve a problem on accelerating atwood machines in which, naturally, one attempts to find the acceleration of each of the individual masses.

Although I now understand how to solve the problem thanks to that thread, I have difficulty identifying the issue with my first approach, where I solve for the accelerations as if the atwood machine were NOT accelerating and then adding the acceleration of the entire system to these accelerations. Doing this results in a different answer than the correct approach mentioned in the thread.

What is wrong with this approach? Thanks.

PS: My apologies if this should be in the Homework section. I was unsure where to put it since technically I know how to do the problem; I am just unsure of the general nature of changing our reference frame in this instance.