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# Inertia matrix question watch

1. I was wondering where they got the R matrix from? As in the elements inside it? I am confused how they got this..
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2. (Original post by DOD1_)

I was wondering where they got the R matrix from? As in the elements inside it? I am confused how they got this..
I can't really read what you've posted too well - it's too small. However..

R is the 3D rotation matrix that transforms coords from one of the coord systems to the other. In this case, they are rotating about the x-axis, which accounts for the 1 in the first row/column. The remainder of the cos/sin stuff is the standard 2D rotation matrix.

The inertia matrix is a rank-2 tensor, and to find its components in a different coordinate system, you have to transform them according to a certain rule, which turns out to be equivalent (at least in this case - not sure about in general) to that pair of pre- and post- matrix multiplications, using the matrices (i.e. R and R^T) that transform back and forth between the coord systems.

The point here is that:

a) it is easy to write the inertia tensor in the x'y'z' system.
b) there is a standard transformation rule to find the components of the inertia tensor in the rotated xyz system, assuming that the origins are coincident.
c) we can then apply the parallel axis theorem to correct for the fact that the origins are offset from each other, rather than being coincident.

So we start off in an "easy" coord system, then use rules to transform the tensor into its correct form in our tricky, rotated and offset coord system.

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