General approach to find principal axes of rotation?

[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?

 

[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.

 

[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 $J_A$.

 

[Mind----Blown]

thanks!