Why Do Principal Moments of Inertia Values Change with Different Origins?

AI Thread Summary
The discussion centers on the differing values of principal moments of inertia calculated for a system of four equal masses at the corners of a square, depending on the chosen origin. When using one corner as the origin, the moments are Ixx=mb^2, Iyy=3mb^2, and Izz=4mb^2, while using the center of mass yields Ixx=mb^2, Iyy=mb^2, and Izz=2mb^2. The difference arises from the mass distribution relative to the chosen origin, affecting the calculated moments of inertia. There is confusion about whether the moment of inertia tensor calculated from an arbitrary point will yield the principal moments of inertia when diagonalized. The discussion emphasizes the importance of the axis of rotation's location in determining the moments of inertia.
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Consider 4 equal masses at the 4 corners of a square of side b. First I took one of the corners as the origin and found the principal moments of inertia to be Ixx=mb^2, Iyy=3mb^2, Izz=4mb^2 after solving the secular equation. Again, I found the principal moments of inertia but now with respect to the center of mass as origin as Ixx=mb^2, Iyy=mb^2, Izz=2mb^2. Now my question is, why do I get different values of principal moments of inertia? I asked this was initially in the homework section but didn't get an answer so I am reposting it here. Let me give my thoughts on it. Intuitively, since the mass distribution is different with respect to different origins, the Principal moments of inertia are different. The eigen vectors corresponding to these principal moments would be different in the 2 cases which would mean that there are a number of principal axes which are not parallel which is not true. This means that the moments of inertia Ixx, Iyy and Izz that I calculated with respect to the corners are not the principal moments of inertia. Does my argument sound logical? I have tried to explain my question to the best of my knowledge, if my question is still not clear Please let me know.
 
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Are your axes of rotation going through the origin?

If so, then the moment of inertia will change since it is dependent on where the axis of rotation is.
 
Are your axes of rotation going through the origin?

Do you mean the center of mass? I have found it about 2 points-one is the center of mass and other is some arbitrary point. What is really creating a doubt in my mind is that, if I find the moment of inertia tensor about a point in the body other than the center of mass and then diagnolize it, will that give me the principal moments of inertia?
 

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