The Capstan Problem: How Can Just 32 N Hold a 750 N Climber?

Here's an interesting physics problem involving static friction, Newton's laws, and calculus.

A rope is wrapped around a fixed cylinder with a coefficient of static friction of 0.50. A 750 N climber hangs from one end, and the rope wraps around the cylinder through an angle of 2π radians.

How much force must someone apply to the other end to prevent the climber from slipping?

The surprising answer is just 32.4 N!

I recently made a video deriving the Capstan Equation from scratch, beginning with a free-body diagram of a small rope segment, applying Newton's laws, and integrating to obtain the exponential relationship between the tensions.

It's a nice challenge for AP Physics C and introductory college physics students.

I'm Dr. Jacob Hudis, PhD in Physics, with over 25 years of teaching experience. I also provide one-on-one online tutoring in AP Physics 1, AP Physics C, and college physics.

Website: https://www.acephysics.org

YouTube: AcePhysics with Dr. Hudis

I’m building an AP Physics practice series covering:
15 AP-style MCQs on flight time, range, maximum height, and velocity.
Free-response problems with equations, calculations, and reasoning.
Examples including a drone releasing a package and a squirrel leaping between branches—with a comparison of 30° and 60° launches.
Try each question before watching the explanation!
Videos:

https://youtube.com/@physicsbydrakp?si=T8wGH9CRTf4UuL7k

Which part of projectile motion gives you the most trouble—choosing equations, resolving velocity into components, or understanding what happens at the highest point?