Using the D' Alembert’s Principle determine the equation of motion for an Atwood’s machine with masses M and 3M

In classical mechanics, D'Alembert's principle is a powerful tool for deriving the equations of motion for systems subjected to forces. It states that the sum of the differences between the forces acting on the system and the inertial forces (which account for the system's acceleration) must be zero for the system to be in equilibrium.

Let's apply D'Alembert's principle to an Atwood's machine, where two masses, MM and 3M3M, are connected by a string passing over a pulley. The pulley is assumed to be frictionless and has a negligible mass, and the string is assumed to be inextensible and massless. The gravitational force acts downward on both masses, and the only other forces involved are the tension in the ______ ___ ________ ____ _________ ____ _________ _______ ____ ____.
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