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If you put the value of the tension ##2T = \dfrac{4m_1m_2}{m_1+m_2}g## and subtract it from the force due to gravity ##(m_1+m_2)g##, it...

Gravity is an external force. I suppose what you mean is that if ##m_1## were to accelerate "up" with ##g##, we'd need an agent to apply...

Oh am sorry; you made the correction. However, I find myself struggling with something else. How can the tension be ##T = 2m_1g##...

I failed to understand. Mass ##m_2\; (\rightarrow\infty)## is accelerating downward at ##g##. How is mass ##m_1## accelerating upward...

System : The Atwood's machine is well-known, and drawn alongside, where ##m_2 > m_1##. The (common) acceleration of the masses ##a =...