Pulley Systems (with Rope Constraints)

An ideal pulley is massless and frictionless. It redirects the rope without changing the tension. A massless rope transmits force uniformly: the tension $T$ is the same throughout the entire rope, even around pulleys.

Rope-Length Constraint

The key idea: if the rope is inextensible (doesn't stretch) and has fixed length, the motions of connected objects are related.

Consider a rope passing over a fixed pulley. If one end moves down by distance $d$, the other end must move up by the same distance $d$ to keep the total rope length constant.

Since the displacement magnitudes are equal but opposite, the velocity magnitudes are also equal but opposite. Similarly, the acceleration constraint is:

when using the same sign convention (e.g., upward positive) for both ends. The accelerations have equal magnitudes but opposite directions.

Forces on a Mass Over a Pulley

Consider a mass $m$ hanging from one end of a rope over a pulley:

We can draw the free body diagram for the mass: