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,

.

## Archive for October, 2010

### Physics Friday 141

October 29, 2010### Monday Math 140

October 25, 2010What are the lengths of the diagonals of a regular *n*-sided polygon with sides of unit length?

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### Physics Friday 140

October 22, 2010To answer the questions we left off with last week, we label last week’s solution as *I _{z}*, and denote with

*I*the moment of inertia of the fractal about in-plane axis through one vertex, and

_{y}*I*about an axis through the center of mass parallel to a side. Note that these three axes are all perpendicular.

_{x}Now, for

*I*, 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

_{y}*i*, which, as with our dimensional analysis in the previous part, is proportional to our overall

_{y}*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*?

### Monday Math 139

October 18, 2010Prove that for positive integers *k*, *m*, and *n*, that

.

Solution:

### Physics Friday 139: Moment of Inertia of a Fractal

October 15, 2010What is the moment of inertia of a Sierpinski gasket of mass *M* and side length *a* about the axis through its center and perpendicular to its plane?

Solution:

### Monday Math 138

October 11, 2010Show that .

Solutions:

### Physics Friday 138

October 8, 2010Here is a classic special relativity exercise: the “detonator paradox.” Suppose we have two objects, one shaped like a “T” and the other resembling a “U”

The T fits into the U, and comes just short of touching the “inside” of the U, where we have a pressure switch; this pressure switch is connected to the detonator for some explosives.

Now suppose we start with the T some distance away from the U, and send the T toward the U at a relativistic speed , so that *γ*=2. In the lab frame, the U frame, the U is at rest, and the T, moving at *v*, undergoes length contraction to half it’s rest length, leaving it clearly too short to reach the detonator.

However, in the T’s frame of reference, it is at rest and the U approaches at speed *v*. In this frame, the U is length contracted to half, and so the T can easily reach the switch, and trigger the explosion.

So, is there an explosion, or not?

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