Physics Friday 141

To answer the questions left lingering last week, we begin with the Sierpinski carpet:

We see that it is composed of eight smaller carpets, each with side length one-third that of the overall carpet; thus, if I is the moment of inertia of the total carpet of mass M and side length a, then each of the smaller carpets has moment about its center of
$i_{cm}=\frac{1}{8}\left(\frac{1}{3}\right)^2I=\frac{1}{72}I$ .
Now, we use the parallel axis theorem again. We see that four of the sub-carpets, the “edge” ones, have centers with distance from our axis of $d=\frac{a}3$ . The other four, the “corners”, have centers with distance $d=\sqrt{2}\frac{a}3$ .

Thus, we have
$\begin{eqnarray}I&=&4\left(i_{cm}+\frac{M}{8}\left(\frac{a}3\right)^2\right)+4\left(i_{cm}+\frac{M}{8}\left(\frac{\sqrt{2}a}3\right)^2\right)\\&=&4\left(\frac{I}{72}+\frac{Ma^2}{72}\right)+4\left(\frac{I}{72}+\frac{Ma^2}{36}\right)\\I&=&\frac{I}{9}+\frac{Ma^2}{6}\\\frac{8}{9}I&=&\frac{Ma^2}{6}\\I&=&\frac{3}{16}Ma^2\end{eqnarray}$ .

Now, for the Menger Sponge. We have 20 sub-units, each with edge-length one-third that of the overall sponge, so
$i_{cm}=\frac{1}{20}\left(\frac{1}{3}\right)^2I=\frac{I}{180}$ .
Finding the distances, we see that there are 12 which lie along edges parallel to our axis (comparable to the “corners” in the carpet case), and thus have distance $d=\sqrt{2}\frac{a}3$ . The other 8 are analogous to the carpet “edge” units, with distance $d=\frac{a}3$ . Thus,
$\begin{eqnarray}I&=&8\left(i_{cm}+\frac{M}{20}\left(\frac{a}3\right)^2\right)+12\left(i_{cm}+\frac{M}{20}\left(\frac{\sqrt{2}a}3\right)^2\right)\\&=&8\left(\frac{I}{180}+\frac{Ma^2}{180}\right)+12\left(\frac{I}{180}+\frac{Ma^2}{90}\right)\\I&=&\frac{I}{9}+\frac{8}{45}Ma^2\\\frac{8}{9}I&=&\frac{8}{45}Ma^2\\I&=&\frac{1}{5}Ma^2\end{eqnarray}$ .

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Tags: Fractal, Friday Physics, Menger Sponge, Moment of Inertia, Parallel Axis Theorem, physics, Sierpinski Carpet

This entry was posted on October 29, 2010 at 1:44 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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Physics Friday 141

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