Lagrangian Mechanics Example: Motion of a Half Atwood Machine
I’m in the process of working through some mechanics examples that use the Lagrangian to find a solution. In case you missed it, here is my introduction to Lagrangian Mechanics.
For this physics example, I’m going to look at a half Atwood machine. That’s just a fancy name for two blocks connected by a string. One block (mass 1) sits on a horizontal frictionless table. The other block (mass 2) hangs vertically off the table and is connected to mass 1 with a string. Oh, the pulley at the edge of the table is massless and frictionless. The string is also massless (you can get these frictionless and massless objects at the local physics store).
Yes, you can solve this problem without using the Lagrangian — but that’s why it’s a great example. Since we already know the solution, it’s a great way to build up confidence.
Let’s get started. I’m going to break this into the important steps.
Degrees of Freedom and Generalized Coordinates.
I want to describe this whole system by picking some coordinates. But to do that, I need to know how many coordinates are even needed. In this case, there are two objects that are moving (mass 1 and mass 2). However, if mass 1 moves 1 cm to the right, then mass 2 must move 1 cm down. So, I don’t really need to…
