# Adding inertia tensors together

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How would I go about adding inertia tensors of different rigid bodies together? I know it's not just a simple addition because what if the objects have different positions. I was thinking of something with the parallel axis theorem but I'm not sure. I'm trying to add simple bodies together to make one complex/composite rigidbody.

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Find the centre of mass of the composite body, then the moment of inertia of each bit about this centre is it's own moment of inertia + that contributed by the body treated as a point mass at it's centre. If you look at equation (4) on this page:

http://scienceworld.wolfram.com/physics/MomentofInertia.html

Then the moment of inertia of a point at (x, y, z) is given by the object mass * the matrix there.

If there are two bodies the total moment of inertia is

I = I1 + I2 + mass1 * (matrix with x1, y1, z1) + mass2 * (matrix with x2, y2, z2)

Of course all calcultions need to be in the same frame of reference so you need to fix your axes first. In such problems it's easiest to align them with axes of symmetry, or here along the line joining the bodies, if at all possible. This can dramatically simplify things as e.g. all the non-diagonal elements of the matrices are zero if two of (x, y, z) are zero, which happens when two objects are joined along an axis.

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