please
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The question you raised has to do with visualizing concepts, what to use in an equation for mass. If the entire system is moving at the same rate, then you need total mass. If only one piece is moving, then only one mass. What I think you failed to see on that centripetal acceleration problem is that it was an equilibrium problem, force of gravity was supplying centripetal force, and they had then to be equal. One mass was rotating, one was acting acting as a counter weight to supply that tension.
OK, now You are about to ask what about gravity on the rotating mass: ignore it, as it is "ninety" degrees to the direction of centripetal force, and does not "enter" the problem, so consider the force on the hanging mass to equal the centripetal force. (the truth is much more complex, I don't think you are ready for it).
IT seems to me you are overanalyzing these problems, avoiding the simple, and trying to apply a concept that does not fit the model. In general, that is onerous.
Here is a book I STRONGLY recommend to you. Visit BarnesNoble bookstore, or any college bookstore, and take a look at Schaum's Outline Series, College Physics (there are several variations, my favorite is Physics for Scientists and Engineers, but the may is probably beyond you). These are inexpensive, have readable text, and many solve problems that you can wade thru. Take a look at it first, the price is very inexpensive.