Lagrangian Mechanics: Speed Run Problems (with VPython Animations)

Photo: Rhett Allain

Just for fun, I’m going to set up some physics problems for Lagrangian mechanics. This will give you some situations to practice with.

For each situation, see if you can do the following:

• Determine the generalized coordinate (or coordinates).
• Write the Lagrangian.
• Get a differential equation of motion (or equations). That is, get an expression for the acceleration of the generalized coordinates.

Oh, I made up some of the names for these situations.

Q1: Circle Pendulum

A vertical disk of radius r rotates in a circular motion with an angular velocity ω. A pivot point is on the edge of the rotating disk and from that a massless string with a radius R. At the end of the string is a mass (m). It’s basically a pendulum on a rotating wheel.

Q2: Half Atwood With Spring

A mass, m1 is on a horizontal frictionless table. On one side it’s connected to a spring (with a spring constant k). On the other side is a string that runs over a pulley to a…