The system on the left shown here is called an Atwood's machine. m$_1$ and m$_2$ are suspended at the ends of a string of negligible mass passing over a friction-less pulley as shown here. The system on the right just has a hanging mass m. What should m be in terms of m$_1$ and m$_2$ so that the tension in the top string is the same in both cases? Assume that the two systems have equal downward acceleration (a').

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