Bead on a Rotating Hoop: An Example of Lagrangian Mechanics

Photo: Rhett Allain. Just the start, the very early start, of a Lagrangian problem

If you don’t like Lagrangian mechanics, you won’t like this post. Oh, maybe you don’t like the Lagrangian but you want to LEARN to like it. OK, then you can stay. It’s sometimes useful to fully work out the solution for a problem like this. I’m going to go over ALL the step to solve this problem along with a numerical calculation in python. It’s going to be fun. You will have fun. You WILL have fun.

Oh, if you want a quick review of Lagrangian Mechanics — maybe this will help.

The Problem

A bead of mass m is on a wire hoop of radius R with zero friction. The hoop it oriented vertically and rotates about a vertical axis that passes through the center. Maybe this gif will help (but maybe not).

Image: Rhett Allain. I tried to make a bead on a spinning hoop. It just “kind of” worked.

The hoop rotates with a constant angular velocity (ω) and the bead can slide up and down the wire hoop.