Okay, so for the record, I've solved this system considering each object as an individual system but what I can't comprehend is why I can't consider the hanging pulley and the 2 hanging masses as a system with a combined mass of (m1+m2) and total force acting as [(m1+m2)g-T] and then solve for the tension by equating accelerations. This always yields an incorrect answer and I've serious troubles justifying that to myself. Some help would really do me good and help me gain a better perspective. The picture encloses what I wanna consider as a system just so it's clear. Thanks. (Friction is absent, strings and pulleys are ideal)
Answer
You can consider the hanging pulley and the masses hanging from it as a system of mass $m_1+m_2$. However, analysing the forces on this sytem does not tell you about the motion of individual parts of the system. It only tells you about the acceleration of its centre of mass. Often this is not useful.
If the system is rigid, then the acceleration of the centre of mass is the same as the acceleration of each part of the system. (This is also the case if $m_1= m_2$, even if the two masses are in relative motion.) If the system is not rigid, these accelerations are not the same. To find out how the separate parts of the system move, you have to consider internal forces. Ultimately this is the same as analysing the forces on each part of the system separately.
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