## What are moments and products of inertia?

Posted by Mark on April 8, 2009

I saw this question today on the internet at a forum, and nobody seemed to have an answer. Some people thought they knew what they were talking about, but had absolutely no clue. I have a clue, as I’m responsible for keeping track of these properties for over a hundred components on a developmental rocket program.

Question: Do the moments and products of inertia represent anything in the physical world?

Answer: In short, yes. But not intuitively.

It’s best to think of moments and products of inertia in terms of the inertia tensor, which is a mathematical concept – it’s a matrix of inertias for a given 3D object.

The moments are the diagonal values of the inertia tensor and the products are the off-axis values. Keeping the tensor in mind, you can think of an object’s moments of inertia to be the rotational equivalents of the general concept of inertia. Ie, as inertia is the resistance to change in motion, moment of inertia is an object’s resistance to change in rotational motion about some axis. As there are three dimensions, any physical object’s rotation can be defined about 3 axes. It is these axes that form the basis of the inertia tensor. If it took 4 axes to define rotation, the matrix would be 4×4, not 3×3.

In the real 3D world rotation is quite complex – for most real-life objects, you are not going to be able to find 3 axes about which the object will rotate in a stable manner. Ie, most objects don’t have mass distributed symmetrically about all 3 axes at once. Thus, you get off-axis moments of inertia, also known as products of inertia. POI’s can be thought of as measurements of dynamic imbalance of an object and basically represent asymmetrically distributed mass. Note that mass can be distributed symmetrically about one axis but not another, leaving you with products of inertia.

If you see an inertia tensor and the products are zero, there is no dynamic imbalance – the object will rotate perfectly about the defined axes. In such a case, you can call these the principle axes.

Ugh. This is complicated stuff and is often not covered in undergrad engineering or science. What I’ve written is only going to be a piece of the puzzle to understanding, but I hope it has helped. Let me know if any of you see errors in what I’m saying.

## Ben Wilder (@wilderben) said

We are studying moments and products of inertia in Engineering Mechanics: Statics class and I have been particularly baffled at the lack of a physical meaning given to the product of inertia. It seems that product of inertia is just a mathematical artifact.

For each meaning that was given, I found a situation that made it meaningless.

1. “It is the resistance to twisting”: No, because if you take the cross section of a symmetrical tube, flat-bar or i-beam, or even different sizes of any of these shapes, they all have product of zero about x and y axes through its centroid but have different resistance to twisting.

2. “It is the measure of unbalance” : No, because you could have a cross section of a z-bar that is balanced, but it could have a positive or negative product of inertia about x and y axes through its centroid; and a T-bar would have a product of inertia of zero no matter how high or low you set the x axis on the “T”.

Is there an intuitive explanation for product of inertia?

## Mike Sytsma said

The cross products of inertia themselves don’t have much easily decipherable physical meaning other than they indicate that the axes selected for your body don’t correspond to the principal rotational axes. As mentioned above a physical body will be seen to rotate around its principal rotational axes, but we tend to apply our own coordinate systems as we desire. However if you take the eigenvalues and eigenvectors of the inertia tensor, you’ll extract both the principal axes as well as the rotations to get there.