My task is to compute the moment of inertia and the radius of gyration of a constant-density tetrahedron defined by $x,y,z\ge0$ and $\frac xa + \frac yb + \frac zc \leq 1$ about the x-axis.

I know that the moment of inertia, $I$, of an object is defined by: $I=\int_RdI$ over some region of integration R. Here, I'd let R be the volume of the tetrahedron. Then, $dI = r^2dM$ for mass element $dM=ρ(x,y,z)dV$ for some density constant density $ρ(x,y,z) = k$. So, we should have:

$I = \int_0^c \int_0^{b- \frac bcz} \int_0^{a- \frac aby - \frac acz} ρ(x,y,z)r^2dxdydz$

$r^2$ should represent the distance from any point in the tetrahedron to the x-axis, so we can (I think) let $r^2=y^2+z^2$. Hence:

$I = k\int_0^c \int_0^{b- \frac bcz} \int_0^{a- \frac aby - \frac acz}(y^2+z^2)dxdydz$.

From here, I would compute $I$ and then I would be able to calculate the radius of gyration, $r_g$, where $r_g=\sqrt{\frac IM}$.

This integral looks unnecessarily complicated and I'm not sure if I set it up correctly. Any idea where I might have gone wrong?

  • $\begingroup$ Should that be $z/c$? $\endgroup$ – Frpzzd Oct 25 '18 at 19:16
  • $\begingroup$ yeah that's a typo. $\endgroup$ – Jackson Joffe Oct 25 '18 at 19:17
  • 2
    $\begingroup$ The integral isn't really that complicated. Split it into two integrals and then integrate $y^2$ and $z^2$ separately. $\endgroup$ – Frpzzd Oct 25 '18 at 19:19
  • $\begingroup$ Why "unnecessarily"? The reasoning is correct and all formulas are in their place. Do the integral and set $M=abc/3$ $\endgroup$ – Rafa Budría Oct 25 '18 at 19:21
  • $\begingroup$ Actually, the density is $k=M/(\frac{abc}{3})$. The moment of inertia is a tensor. The one you're calculating is $I_{xx}$. See en.wikipedia.org/wiki/Moment_of_inertia $\endgroup$ – minmax Oct 25 '18 at 19:47

The set up

$$I = k\int_0^c dz \int_0^{b- \frac bcz} dx\int_0^{a- \frac aby - \frac acz}(y^2+z^2)\,dy$$

seems correct to me, the integration is maybe quite long but it shouldn't be complicated since we are dealing with polynomials integration.


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