# Combining Moment of Inertia Tensors

In a physics simulation of rigid bodies, if I have a cube with a known mass and moment of inertia tensor, and I attach it to another cube with a known mass and moment of inertia tensor such that its transform relative to the second cube is constant (put simply, they are stuck together completely), how can I compute the moment of inertia tensor of the resulting body? How would I compute it if I continued to add cubes in different places? Keep in mind that I'm not just stacking them, I could end up with a strange U or T or L shape, for instance. The cubes' rotation will all be identical, i.e. they will always be connected face-face, never vertex-face or edge-face or anything. They are all the same size, but may vary in mass. The faces will always connect in such a way that four vertices on one cube will touch four vertices on another cube (in other words, I won't have one cube poking out from behind another cube, they will line up nicely). The bodies have constant density.

Could I possibly use the Parallel Axis Theorem? Maybe find the center of mass of the combined bodies and finding the moment of inertia tensor for each body through that axis, and then somehow adding all the matrices together?

• Actually, @Qmechanic, it's not homework, I'm writing a game using a physics engine I wrote, it's a builder game, I need to be able to combine multiple rigid bodies. – Joshua Hyatt Nov 25 '14 at 23:18
• Hi Josh. Welcome to Phys.SE. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. – Qmechanic Nov 25 '14 at 23:32
• I see. But that would make it seem like every question I've asked, here or other Stack Exchange sites, would be classified as homework-and-exercises. It just seems too broad. – Joshua Hyatt Nov 26 '14 at 1:16

## 2 Answers

After doing some research, I found out that my last idea was correct: Find the combined center of mass, find the moments of inertia of each object through that point, and then combine the moments. The last part was what I was confused about, and as it turns out, combining moments of inertia tensors is as easy as adding them together.

The method of "find the moments of inertia of each object through that point" is not correct. We only need to find the inertia of each object through the origin of the absolute coordinate system and add them together. The inertia of the conbimed object will be obtained. Finally we can get the inertia with respect to the CG of the combined object by tensor translation method.

• Is this merely an exercise in pedantics or is there an actual difference in the maths and outcome? It sounds like the same but with extra steps. – AnnoyinC Jun 3 '20 at 10:34