# Principal axes/inertia tensor

problem:

three equal masses connected by light rods, the masses are positioned at (a,0,0), (0,a,2a) and (0,2a, a) now i work out all the products, and moments of inertia to get the inertia tensor, the thing that is baffling me is how you can have Ixx = 10ma^2, with all off diagonal products of x (Ixy, Ixz,) equal to zero. this clearly shows that one of the principal axes is the x - axis, but i dont see how this can be the case when the centre of mass clearly doesnt lie on the x axis(im picturing a triangle with a point on the x axis, and its base in the zy plane). unless this result simply means one of the principle axis is parrallel to the x axis?

its just kinda weird

help?

i actually get the right answers, and i understand that for the rest of the tensor (Iyy, Izz, Iyz) its a diagonalisation problem to find the other principle axes for this shape/orientation. its just i have issue with the physical meaning of what going on here with the x axis.

cheers

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if its not very clear, ill happily post stuff/files whatever to make it clearer.

I assume you're evaluating the tensor with reference to the coordinates with respect to which the masses have the positions you named. That means that the COM is not at the origin (as you noted), so you can't get a principle axis that goes through the COM - well, you could if one of them happened to run through the origin and the COM.

I think what you have is as you suspected, the axis going through your origin that is parallel to one of the principle axes through the COM - kind of like what you do when using the parallel axis theorem.

To put it another way, moments of inertia are always defined with reference to a center of rotation, and since you're working in the given coordinates, the results you get will be referenced to that center.

Hope that helped ... it didn't feel very coherent! Maybe someone else will offer a better explanation.