# I General approach to find principal axes of rotation?

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1. Apr 27, 2017

### Mind----Blown

Suppose i have an equilateral triangle and i want to find the principal axes of rotation passing through one of the vertex. How can i do that? I am thinking along the following lines but i'm not too sure:

1)Since the equilateral triangle has symmetry about a median, that definitely is one principal axis.

2)Now, i want 2 axes such that those 2 axes and the centroidal axis which i found above are mutually perpendicular. The problem now, however, is that i don't have any "symmetry" to rely on. Sure i COULD think along this line now :

"if i rotate the triangle 360 degrees about one of the sides, i would return to the original configuration, so let me choose that as one of the axis, which leaves me with only one choice for the third axis and voila!"

But i am not too sure of my approach in 2nd point since it just doesn't seem right; rotating an object 360 degrees to get the original configuration isn't really a symmetry!

1) So, is there some fool-proof way i can use (and be 100% certain of being correct) to determine principal axes of rotation?

2) Maybe mathematical?

3) Also, how reliable is this symmetry approach i follow?

2. Apr 27, 2017

### phyzguy

There is a general method to find the principal axes:

(1) First, calculate the inertia tensor. This mathworld site gives a definition of how to calculate the tensor, but it is basically:

(2) Find the eigenvectors of the inertia tensor. The eigenvectors point in the direction of the principal axes, and the eigenvalues are the moment of inertia about these three axes.

3. Apr 28, 2017

### zwierz

First specify how mass is distributed in the triangleAnyway the following two theorems will be useful for you.

1) Let $S$ be a center of mass of a rigid body and $J_S$ be the operator of inertia about $S$. If $\ell,\quad S\in \ell$ is the principle axis for $J_S$ then for each point $A\in \ell$ the exis $\ell$ is the principle axis for $J_A$.

2) Assume that $\Pi$ is a plane of material symmetry of the rigid body and let $A\in \Pi$. Define an exis $\ell$ to be perpendicular to $\Pi$ and $A\in\ell$. Then $\ell$ is a principle axis for $J_A$.

Last edited: Apr 28, 2017
4. Apr 28, 2017

thanks!