# Calculating angular acceleration when moment of inertia tensor is singular

I am attempting to simulate the motion of a rigid body composed of point masses in a 3D coordinate system. In doing so, I need to calculate the angular acceleration of the rigid body due to external torque as such

$\boldsymbol{\alpha} = \mathbf{I}^\mathrm{-1} \boldsymbol{\tau}$

where $\boldsymbol{\alpha}$ is the angular acceleration, $\boldsymbol{I}$ is the moment of inertia tensor, and $\boldsymbol{\tau}$ is torque in the body frame. However, when I compose a rigid body of two point masses, the moment of inertia tensor is rank-deficient and thus non-invertible. In this case, how would the angular acceleration be calculated? Would a pseudoinverse suffice?

## 1 Answer

What you're describing can only happen if all the point masses are on a line, and there is a component of torque trying to spin them about that line. Well, if point masses are all on a line, then it is meaningless to ask how fast they are spinning about that axis. Classical point masses have no spatial extent: They are either moving or not moving; they cannot spin about their own axis. So the state where they are not spinning about this line is the exact same physical state as the state where they are spinning at 10000RPM about this line. The rotational velocity about this line is not measurable or meaningful.

Anyway, you should find that the component of $\alpha$ along a singular axis has no effect on the motion. Therefore, I expect that a pseudoinverse would work fine. (But I'm not sure.)