# Is it possible to directly compute the maximum moment of inertia for an object

Given some arbitrary shape I can compute the moment of inertia about any axis without a problem by summing the inertia of each of the shapes making up the entire object. I also know the center of mass of the object.

Is it possible to directly compute the angles of the axis for the minimum and maximum moments of inertia of this object about the center of mass (or any other point for that matter) without using trial and error to solve the problem by continually rotating the axis until I find the answer?

If so, how?

Note: I am a computer programmer/engineer and although I took a lot of calculus and linear algebra in college, that was a LONG time ago.

thanks

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CWatters
Homework Helper
Gold Member
So to make this simpler lets just assume I am dealing with some sort of irregular shaped object but one of constant thickness so that we can just throw away the Z axis and just work in 2 dimensions.

Does that make the problem easier/directly solvable for the 2 scalar values of angle?

I have solved this problem on the computer by iteratively rotating though a series of angles, locating the angles associated with the min/max moments of inertia and then subdividing that angle until I get to the desired precision I need.

Usually when I do these trial and error solutions, my first thought is "Surely someone smarter than me could do this without trial and error." Again, I don't know if that is true in this case.

thanks

gneill
Mentor
You'll probably find your answer in the tensor version of moment of inertia, something I've read about but not had a reason to use or become intimate with. There are certainly principle axes and something called "the axis of figure" which is the axis of maximal moment of inertia.

So, perhaps do a bit of searching on "moment of inertia" + tensor" and see what pops up.

rude man
Homework Helper
Gold Member
Given some arbitrary shape I can compute the moment of inertia about any axis without a problem by summing the inertia of each of the shapes making up the entire object. I also know the center of mass of the object.

Is it possible to directly compute the angles of the axis for the minimum and maximum moments of inertia of this object about the center of mass (or any other point for that matter) without using trial and error to solve the problem by continually rotating the axis until I find the answer?

If so, how?

Note: I am a computer programmer/engineer and although I took a lot of calculus and linear algebra in college, that was a LONG time ago.

thanks
Not my comfort zone either, but I understand that if you form the expression for I for your object in your coordinate system with your choice of rotation axis, you can then determine the torque-free precession rate expression. Force that to zero and you get two solutions: one is the max I the other the min I axis ... as gneill mentioned I think this involves tensor algebra.