Constraints and phase transitions: Phase transition in infinite compound Atwood's machine

A compound Atwood's machine with N+1 different masses, e.g. mk≔αk−1m, is considered. It is shown that at the large N limit and at a critical value, αc=1∕4, system admits a phase transition. The critical exponent is obtained. In the low α phase, surprisingly the effective mass of a compound Atwood's machine becomes zero. The more realistic compound Atwood's machine with massive pulleys and non-slipping strings may also exhibit similar phase transition.