Multi-Body Systems

Coupled systems of multiple masses connected by cords, rods, and pulleys.

General idea of the topic

In a system of several bodies joined by ropes and pulleys, the accelerations of the bodies are not independent: an inextensible rope imposes that the total length is conserved and relates their motions. With that constraint and one Newton equation for each body, the common acceleration and the tensions are obtained.

The topic advances from two masses joined by a rope over an ideal pulley to simple and compound pulleys, where the number of rope segments supporting the load determines what force has to be applied and what distance has to be pulled.

What the student is expected to understand

By the end, the student should be able to write the constraints of a system with ropes and pulleys, solve for acceleration and tensions and compute the mechanical advantage of a simple or compound pulley.

The student should also be able to explain why the advantage in force is paid for with distance pulled.

Recommended learning route

1. Two Masses Connected by a Rope: two bodies joined by a rope over ideal pulleys: constraint, common acceleration and tension.
2. Simple Pulleys: fixed and movable pulleys, and how the required force changes with the segments supporting the load.
3. Compound Pulleys: mechanical advantage, efficiency and the real force required.

Essential physical criterion

The essential criterion is to start from the constraint: which geometric relation ties the accelerations together. Then the second law of each body is written, with the tension as a common unknown if the rope is ideal. The result is checked with the equal-masses case, which must give zero acceleration.

Connection with real problems

• cranes, hoists and elevators with counterweight.
• workshop and climbing pulleys.
• lifting systems on construction sites.
• design of tackle and block-and-tackle systems.

Final goal of the topic

The final goal is to solve multi-body systems with ropes and pulleys, ideal or with losses, and compute accelerations, tensions and mechanical advantage.