Final answer:
The equilibrium in a pulley system with a mass hanging on both sides is determined by the tension in the rope, which balances the weight of the hanging masses when there's no acceleration.
Step-by-step explanation:
In a pulley problem with a mass hanging on both sides, the system's equilibrium is determined by the tension in the rope. In an equilibrium state, the tension of the rope must equal the weight of the supported mass, assuming no acceleration is present in the system as stated by Newton's second law. If one of the masses on either side of the pulley starts to move, it will accelerate due to the imbalance and the acceleration due to gravity. The frictional forces and the mass of the pulley may have an impact on the system if they are significant, but in a frictionless and massless pulley scenario, they do not play a role in equilibrium. Moreover, the system's dynamics will change if the external conditions, such as the elevator's acceleration, are considered.