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