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In some slides of a robotics course I found the following text:

Principal axis of inertia u is a unit vector along a principal axis if I . u is parallel to u There are 3 independent principal axes!

I have these questions about it:

  1. What is an axis of inertia?

  2. What in this context is I?

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    $\begingroup$ Moment of inertia $\endgroup$
    – NickD
    Apr 3, 2017 at 20:29
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    $\begingroup$ $I$ is the moment of inertia and the "principal axis of inertia" is often also referred to by "principal axis of rotation", which should clarify your first question $\endgroup$ Apr 3, 2017 at 20:41
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    $\begingroup$ -1. No research effort. $\endgroup$ Apr 4, 2017 at 1:16
  • $\begingroup$ While I could believe slide notes could omit it, I cannot believe a textbook would not introduce such an important variable.... $\endgroup$
    – Kyle Kanos
    Apr 4, 2017 at 10:15
  • $\begingroup$ +1 and VtL for the brave try to clear an unclear textbook. $\endgroup$
    – peterh
    Apr 5, 2017 at 19:45

1 Answer 1

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This seems quite related to your earlier question, and my answer to it.

In that earlier answer I illustrated why, if you take a "general" direction of rotation for an arbitrary object, the angular momentum will not be aligned with the angular velocity. However, for any object there are some special directions for which they will be aligned. These directions are often axes of symmetry of the object (although they don't have to be in the case of objects with no obvious symmetry). In the case I illustrated in my earlier answer, the long axis of the rod would be a "principal axis". Such an axis has the nice property that it's easy to calculate the angular momentum when you know the angular velocity.

In the context of your question, $I$ is once again the inertia tensor: that is, a tensor that allows you to compute, in general terms, the angular momentum of an object given the angular velocity. An inertia tensor can be decomposed into a rotation matrix and a diagonal matrix, the elements of which correspond to the principal moments of inertia. This is explained in the subtopic principal axes on the Wiki page for "moment of inertia".

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  • $\begingroup$ Do you know where I could find some exercises with moment of inertia? I think only that way I could understand better. $\endgroup$
    – bsky
    Apr 4, 2017 at 11:56
  • $\begingroup$ There are quite a few questions and answers on this topic on this site. See for example physics.stackexchange.com/q/272896/26969 and the links given there - or just look to the right -----> in the sidebar where there are several questions with "principal" in the title... $\endgroup$
    – Floris
    Apr 4, 2017 at 11:58

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