- #1

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## Homework Statement

Calculate the moment of inertia of a cube which rotates along an axis along its diagonal.

## Homework Equations

Moment of inertia definition: [itex]I = \int \rho (\vec{r}) \vec{r} ^2 dV[/itex]

Angular velocity vector; [itex]\vec{\omega}=\omega (\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})[/itex]

## The Attempt at a Solution

My biggest problem is on finding the position vector for the volume element. My try at it goes like this:

[itex]

\vec{r} = \vec{p}- \vec{p}·\vec{\omega} \frac{\vec{\omega}}{\omega}

[/itex]

Where [itex]\vec{p}[/itex] is the point of interest. With this we calculate the difference between the point's vector and a vector with the direction of [itex]\vec{\omega}[/itex] and magnitude equal to that of the component of [itex]\vec{p}[/itex] along [itex]\vec{\omega}[/itex]. This should be the shortest vector from the axis of rotation to the point. Am I right?

I think, however, that there is an easier way to solve this. Isn't there any shortcut similar to the Steiner Theorem, only for rotates axes?

Thank you.