Calculating Moment of Inertia for Sierpinski Triangle - Tips and Tricks

In summary, the moment of inertia of Sierpinski's triangle about an axis through its center and perpendicular to the triangle can be found using a self-similarity argument and the parallel-axis theorem. It is equal to the mass of the triangle multiplied by the square of its height divided by 9.
  • #1
climbhi
Okay this problem sounded neat, but I'm stuck on it. How would one go about finding the moment of inertia of Sierpinski's triangle about an axis through its center and perpindicular to the triangle? Any thoughts?
 
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  • #2
Here's a http://physics.harvard.edu/undergrad/prob9.pdf to the problem so you can see what I'm talking about. This ones really throwing me for a loop. I thought maybe that you could use a symetry argument and say that it would be the same as for a regular triangle just with a smaller mass. But rethinking that I'm pretty sure its wrong. Then I thought that if you did what the problem statement described down to infinity you wouldn't have any mass left at the end which would make the moment zero. But again this seems wrong to me. This is really beginning to bother me. Can anyone help me out?
 
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  • #3
Well, part of the trick is that the object has infinite mass density (because of the zero volume).

I think you need to use a self-similarity argument to find the moment of the whole in terms of the moment of the individual parts, and coupled with the scale factor should yield a system you can solve.

Hurkyl
 
  • #4
Okay, you lost me there...
 
  • #5
Let I be the moment of inertia around the center of the fractal.

Consider now just one section of the fractal. One section has one third of the mass and is scaled down by a factor of two, so the moment of inertia of one section about its center is (I / 12); multiply by one third because of the mass change, and by one half squared because of the scaling.

By the parallel-axis theorem, to find the moment of inertia of one section about the center of the whole fractal, we add the mass of the section times the square of the distance between the axes. The mass of the section is (m / 3) because it's one third of the whole. The distance between the axes is (insert some geometrical reasoning) one third of the height which is (sqrt(3) / 2) * l, so the distance is l / (2 sqrt(3)), so the moment of inertia of one section is

(I / 12) + (m / 3) * (l / (2 sqrt(3))^2
= (I / 12) + (m / 3) * l^2 / 12
= (I / 12) + m l^2 / 36

The moment of inertia of the whole should be the sum of the moments of inertia of the three sections, so:

I = 3 * ((I / 12) + m l^2 / 36)
= I / 4 + m l^2 / 12

3 I / 4 = m l^2 / 12
I = m l^2 / 9
 

Related to Calculating Moment of Inertia for Sierpinski Triangle - Tips and Tricks

1. How do I calculate the moment of inertia for a Sierpinski Triangle?

To calculate the moment of inertia for a Sierpinski Triangle, you can use the parallel axis theorem. First, determine the moment of inertia for an equilateral triangle with the same side length as the Sierpinski Triangle. Then, use the formula I = Icm + md2, where Icm is the moment of inertia for the equilateral triangle, m is the mass of the Sierpinski Triangle, and d is the distance between the center of mass of the Sierpinski Triangle and the center of mass of the equilateral triangle.

2. What is the center of mass for a Sierpinski Triangle?

The center of mass for a Sierpinski Triangle is located at the center of the largest equilateral triangle within the Sierpinski Triangle. This can be found by dividing each side of the equilateral triangle into three equal parts and connecting the midpoints to form a new equilateral triangle. The center of mass is located at the intersection of the three lines connecting the midpoints.

3. Does the size of the Sierpinski Triangle affect its moment of inertia?

Yes, the size of the Sierpinski Triangle does affect its moment of inertia. As the size of the triangle increases, the distance between the center of mass of the Sierpinski Triangle and the center of mass of the equilateral triangle also increases, resulting in a larger moment of inertia.

4. Can I use the same formula for calculating moment of inertia for other fractal shapes?

No, the formula for calculating moment of inertia for a Sierpinski Triangle cannot be directly applied to other fractal shapes. Each fractal shape has its own unique formula for calculating moment of inertia, depending on its geometric properties.

5. Are there any shortcuts or tricks for calculating moment of inertia for a Sierpinski Triangle?

One possible shortcut for calculating moment of inertia for a Sierpinski Triangle is to use the formula I = (3/80)mL2, where m is the mass of the Sierpinski Triangle and L is the length of one side. This formula is an approximation and may not be as accurate as using the parallel axis theorem, but it can provide a quick estimate of the moment of inertia for larger Sierpinski Triangles.

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