To answer the questions we left off with last week, we label last week’s solution as *I*_{z}, and denote with *I*_{y} the moment of inertia of the fractal about in-plane axis through one vertex, and *I*_{x} about an axis through the center of mass parallel to a side. Note that these three axes are all perpendicular.

Now, for *I*_{y}, we again divide into three smaller gaskets. The moment of inertia of each of these about an axis parallel to our axis through the piece’s center of mass is *i*_{y}, which, as with our dimensional analysis in the previous part, is proportional to our overall *I*_{y}; .

Next, we again use the parallel axis theorem. Two of our sub-segments have a center of mass to axis distance of , while the third is already on our axis; so we have

.

For the *x* axis, we could perform this breakdown method again, though finding the parallel axis distances would be less simple in this case. However, instead we can recall that since our Sierpinski gasket is a plane figure, we can use the perpendicular axis theorem; here, we have

,

so then

.

Now, how about the moment of inertia about an axis perpendicular to the plane of a Sierpinski carpet of mass *M* and side length *a*? How about the moment of inertia about a face-centered axis of a Menger sponge of mass *M* and edge length *a*?

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Tags: Fractal, Friday Physics, Moment of Inertia, Parallel Axis Theorem, Perpendicular Axis Theorem, physics

This entry was posted on October 22, 2010 at 12:10 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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October 29, 2010 at 1:44 am |

[…] Friday 141 By twistedone151 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 […]