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

.

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 . The other four, the “corners”, have centers with distance .

Thus, we have

.

Now, for the Menger Sponge. We have 20 sub-units, each with edge-length one-third that of the overall sponge, so

.

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 . The other 8 are analogous to the carpet “edge” units, with distance . Thus,

.

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

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