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Inertia tensor formula for point masses in rigid assembly?

Physics Asked on August 18, 2021

Suppose I have $N$ 1kg point masses in a massless rigid assembly such that the center of mass of the assembly is at the origin and point mass i is at $(x_i, y_i, z_i)$.

The inertia tensor of the assembly is given by:

$$ I = sum_{i in 1..N} F(x_i, y_i, z_i) $$

The function $F$ takes 3 scalars and returns a 3×3 matrix. What is the formula for $F$?

I think its something like: https://hepweb.ucsd.edu/ph110b/110b_notes/node21.html

But I’m not sure what $r$ is in terms of $x_i$, $y_i$, $z_i$?

One Answer

$$ F(x,y,z) = begin{vmatrix} y^2+z^2&-xy&-xz -xy&x^2 + z^2&-yz -xz&-yz&x^2+y^2 end{vmatrix} $$

Answered by Andrew Tomazos on August 18, 2021

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