# Product of Inertia for common 3D Solids [closed]

I could not find a table with common products of inertia online. Even my textbook only contains common mass moments of inertia. It seems tedious to have to calculate and integrate each time to obtain the product of inertia if there exists formulas for common shapes that can be immediately plugged in. I would like to know if such a table exists?

• As it turns out, I haven't done a physics problem involving inertia for over 30 years now, so any tedium is long forgotten. However, you might try a mechanical engineering text, or remember the common shapes you have done already. Dec 23, 2019 at 14:53
• Do you mean moment of inertia? If you search on that term (and not "product") you should get lots of hits. Here's Wikipedia's list, which is probably not the most extensive out there. Dec 23, 2019 at 16:14
• Typically people use CAD software to estimate MMOI of various designs and materials. Also please edit the question and add an example of a shape that you are interested in and can't find a reference for. Dec 23, 2019 at 19:56

You will not find tables of products of inertia because it is not necessary to know them. The moments of inertia for any rigid body about any axis can be calculated from the 3 principal moments of inertia, using a rotation of co-ordinate axes and application of the Parallel Axis Theorem.

The principal moments of inertia are measured in the body co-ordinate system in which the origin is at the centre of mass and the co-ordinate axes are the 3 orthogonal directions about which the body can rotate without exerting any torque on the axis. This means that the products of inertia are zero in this co-ordinate system, and the inertia tensor has only diagonal elements. It also means that the angular momentum is in the same direction as the angular velocity.

It is perhaps remarkable that, regardless of the distribution of mass in the rigid body :

1. it is always possible to find a set of 3 zero-torque axes which are mutually orthogonal; and
2. the 3 principal moments of inertia are sufficient to determine the moment of inertia about any axis.

A corollary of #2 is that many rigid bodies (an infinite number in fact) can share the same set of 3 principal moments of inertia, and are therefore dynamically equivalent. For example a circular disk has the same principal moments as a trefoil lamina.

To calculate moments of inertia about axes other than the principal axes, see : What is the moment of inertia of a cylinder about a skew axis? and Compute an object's inertia around an arbitrary axis, using its known values for the x, y and z axes.

For an efficient algorithm for computing the elements of the inertia tensor see How to find the inertia tensor (or other mass properties) of a 3D solid body represented by a triangle mesh, Jonathan Blow and Atman J Binstock, 9 July 2004; Last updated 25 November 2004.