How to Model Pulley Systems

Modeling pulley systems like elevators and weight machines make great use of a mathematician’s toolkit

A pulley system is a system of objects which are connected via inelastic cables or strings which pass over smooth pulleys, usually wheels on fixed axles. Most commonly, pulley systems that we find in everyday life have two heavy objects connected via a steel cable. The cable passes over a smooth pulley at the top of the system and the objects move up and down vertically on either side of the pulley.

A common example is a lift/elevator system, where the lift is on one side of the pulley and its counterweight is on the other side. Another common and very similar example is a weight machine, where the human pulls a cable over a pulley against a counterweight.

Pulley system problems can be neat and interesting to solve using the toolkit of mathematics that we commonly learn about in high school. I recently found a lovely little problem of this form that had a very satisfying solution. To solve this problem we will need to know Newton’s laws, some basic equations of motion, and a little bit about energy and momentum. We will also need to be able to think systematically about the problem in order to model it with mathematics.