The classic way of solving these problems is like so: $T = mg - ma$ and $T = mg + ma$. However, according to Newton's third law, is this not wrong?
I know how to solve these problems, but the concepts are confusing to me. First off, apparently both tensions in the string are equal to each other. But I don't understand this, and according to Newton's third law, this should be false?
Assuming $T_1$ acts on $m_1$ and $T_2$ acts on $m_2$ Newton's third law should mean that $T_1 = -m_1g$ because the force of gravity from the box $m_1$ acts on the rope, and the rope reacts with equal and opposite force.
But according to this same logic, $T_2 = -m_2*g$
And since the tension in the rope must be equal at all points, and also since $m_1 \ne m_2$ this means that this is false.
Does Newton's third law not apply to Atwood's machines? Or am I confused?