# Can the second law of motion for rotation, tau=I alpha, be used for any axis? Is there any case that acceleration alpha is not in the direction of applied torque tau ?

Can the second law of motion for rotation, $\stackrel{\to }{\tau }=I\stackrel{\to }{\alpha }$, be used for any axis?
Is there any case that acceleration $\stackrel{\to }{\alpha }$ is not in the direction of applied torque $\stackrel{\to }{\tau }$?
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Kali Galloway
It can indeed be used for any axis. But keep in mind that I is a 3x3 matrix. Using different axes requires you to transform this matrix, so it corresponds to your set of choice. This can sometimes prove to be quite difficult as I can be rather complex.
The acceleration will always be in the direction of the total torque acting on your object. The moment of inertia cannot be negative as it is calculated as follows:
$I=\rho \left(r\right){r}^{2}dV$
With $\rho \left(r\right)$ being the density of the object and r the distance to the pivot point. These are all positive. Now we can conclude that in no situation would it be possible to have a negative relation between $\stackrel{\to }{\alpha }$ and $\stackrel{\to }{\tau }$. Of course, the angular velocity does not need to be in the direction of the total torque.