The Moment of Inertia Tensor for a Triangle
Let’s start with a crash course on the moment of inertia tensor. Suppose you have some 3D rigid object (like a block of wood, but not like a block of jello). If you want to model the motion of this object, you have two options:
- Option 1: Treat the object as a bunch of smaller objects (a real object would be made of SO MANY objects). For each of these tiny objects, calculate the total force (due to interactions with other tiny objects) and then find the change in momentum (using the momentum principle).
- Option 2: Calculate the total force and total torque on the rigid object. Then use the net force to find the motion of the center of mass and use the net torque to find the changes in rotational motion.
Option 1 just uses the momentum principle. It looks like this:
We aren’t going to use this, but it’s important for the angular momentum principle. This is what we would use for option 2.
Yes, τ is the net torque and L is the angular momentum. Here’s the important part. We define the angular moment as the product of the moment of inertia (I) and the angular velocity (ω).
It turns out that in order to get everything to work, the moment of inertia is a tensor. It’s the elements of this tensor that we are going to calculate. The tensor looks like…
