# Determining the principal moment(s) of inertia and KE of a cuboid

I am a bit perplexed with regards rotations about a certain axis.

Say we have a cuboid with lengths $$l_1$$, $$l_2$$, $$l_3$$ (not a cube, so $$l_1$$ does not equal $$l_2$$ does not equal $$l_3$$). I am okay with determining the inertia tensor matrix about the centre of mass and about a corner of the body (of constant density, I should add), but am perplexed as to how you go about calculating the inertia tensor, and subsequently the kinetic energy, for a rotation about the long diagonal that goes through a corner and the centre of mass. Oddly, I can't find information online concerning rotations about this particular axis, other than for a cube (which is somewhat simpler).

Any guidance would be appreciated.

• Wikipedia explains how to use the inertia tensor about a point to compute the scalar moment of inertia about an arbitrary axis passing through that point. Oct 20, 2020 at 2:43
• I think what I'm struggling with are determining the appropriate limits of integration for the instances where the body is rotating about its diagonal axis. Oct 20, 2020 at 8:54
• If the cube is centered at the origin and oriented along the $xyz$ axes, then you integrate from $-l_1/2$ to $+l_1/2$, etc. The point is that none of the axes you use to compute the tensor have to be the rotation axis. Oct 20, 2020 at 17:09

Is is not that difficult to integrate, it is more difficult to find the perpendicular distance $$r_\perp$$ from a point $$(x,y,z)$$ of the cuboid to the space diagonal of the cuboid.
For elegance, denote the cuboid lengths as $$a,b,c$$ and place it into a Cartesian coordinate system with the origin at one vertex such that $$x$$-axis is along length $$a$$, $$y$$-axis is along length $$b$$ and $$z$$-axis is along length $$c$$.
Say we are interested in the space diagonal from the origin $$(0,0,0)$$ to the point $$(a,b,c)$$. Now we need to recall that the distance of point $$A$$ from the line from $$B$$ to $$C$$ is given by $$d(A,BC) = \frac{\|\vec{BA} \times \vec{BC}\|}{\|\vec{BC}\|} = \frac{\|(A-B) \times (C-B)\|}{\|C-B\|}.$$ In our case, we are interested in the distance from $$(x,y,z)$$ to the space diagonal, which is $$r_\perp = \frac{\|(x,y,z) \times (a,b,c)\|}{\|(a,b,c)\|} = \frac{(bx-ay)^2+(az-cx)^2+(cy-bz)^2}{\sqrt{a^2+b^2+c^2}}.$$ Now the moment of inertia is simply \begin{align} I_{\text{diag}} &= \int_{\text{cuboid}}r_\perp^2\,dm \\ &= \frac{m}{abc}\int_{\text{cuboid}} r_\perp^2\,dV \\ &= \frac{m}{abc(a^2+b^2+c^2)} \int_{x=0}^a \int_{y=0}^b\int_{z=0}^c (bx-ay)^2+(az-cx)^2+(cy-bz)^2\, dz\,dy\,dx\\ &= \frac16m\frac{a^2b^2+b^2c^2+c^2a^2}{a^2+b^2+c^2} \end{align}
To verify that this result is indeed correct, we can proceed by the usual route as suggested in the comments. The moment inertia tensor of the cuboid around its center of mass is $$\mathbf{I} = \frac{m}{12}\begin{bmatrix} b^2+c^2 & 0 & 0 \\ 0 & a^2+c^2 & 0 \\ 0 & 0 & a^2+b^2\end{bmatrix}$$ and the moment of inertia around an axis with unit vector $$\hat{n}$$ through the center of mass is given by $$\mathbf{I}\hat{n}\cdot \hat{n}$$. In our case, the space diagonal has vector $$\hat{n} = \frac{(a,b,c)}{\|(a,b,c)\|} = \frac{(a,b,c)}{\sqrt{a^2+b^2+c^2}}$$ so $$I_{\text{diag}} = \mathbf{I}\hat{n}\cdot \hat{n} = \frac{m}{12}\frac1{a^2+b^2+c^2}((b^2+c^2)a^2+(a^2+c^2)b^2+(a^2+b^2)c^2) = \frac16m\frac{a^2b^2+b^2c^2+c^2a^2}{a^2+b^2+c^2}$$ which is of course the same result.