Consider the double Atwood machine below:
I understand the classic approach to a solution for the acceleration of each mass involves setting the tension in the upper string equal to twice that of the lower string to achieve equilibrium in the lower pulley. The common reasoning is that "since the pulley is massless, it must be in equilibrium otherwise it would have infinite acceleration".
While I understand this reasoning must be correct, surely if the lower pulley is in equilibrium there would be no acceleration of m1 at all, which can't be correct? Therefore, under what conditions are the subsequent accelerations derived physically useful - are they just in the limit as the mass of real massive pulleys tend to zero?