Moment of inertia of a plate

In summary, the conversation discusses the calculation of the moment of inertia for a square-shaped homogenous plate with a diagonal axis of rotation. The attempt at a solution involves using the perpendicular axis theorem and results in a final answer of I_{x}=\frac{m*a^{2}}{12}. The conversation also touches on the interesting fact that this formula can be applied to any two perpendicular axes through the center of the plate.
  • #1
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Homework Statement


Calculate the moment of inertia of a straight homogenous plate with mass m shaped like a square where the axis of rotation goes through the diagonal of the plate.

Code:
       ^
       |y
       |
      /|\
     / | \ a
-------|------>
   a \ | /    x
      \|/
       |
       |


Homework Equations



Moment of inertia [tex]I=\int r^{2}dm[/tex]

Perpendicular axis theorem [tex]I_{z}=I_{x}+I_{y}[/tex]

The Attempt at a Solution



This is what I've come up with, but I don't know if I'm right.

Being this a square, I've concluded that [tex]I_{x}=I_{y}[/tex]

Using a Perpendicular axis theorem I have [tex]I_{z}=2I_{x}[/tex]

I need [tex]I_{x}=0.5I_{z}[/tex]

I have [tex]I_{z}=\frac{m*\left(a^{2}+a^{2}\right)}{12}=\frac{m*\left(a^{2}\right)}{6}[/tex]

And then I just put it in [tex]I_{x}=0.5I_{z}[/tex] and get [tex]I_{x}=\frac{m*a^{2}}{12}[/tex]

But somehow, I think I'm wrong :biggrin:
 
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  • #2
Well, why do you think you're wrong?

Also, is a the side length of the square, or is it the "half-diagonal" of the square? It's not clear from your ASCII-art diagram :biggrin:
 
  • #3
Here's the new picture
I hope it's better than ASCII one

http://img15.imageshack.us/img15/6522/pictureaor.jpg [Broken]

The reason why I think I'm wrong is the following. If I rotate the square on the upper picture 45° in any direction around z-axis then the x-axis no longer lies on a diagonal of the square. When I try to calculate the moment of inertia of such square plate (x-axis is the axis of rotation), I get the same solution as I get when the x-axis is on the diagonal.
 
Last edited by a moderator:
  • #4
Your answer is correct, (and well worked out).
You have discovered an interesting fact, which is more than a mere coincidence.
Your working could be applied to any two perpendicular axes in the plane of the plate
through its center!

If I had to criticize, I would say it is a pity that the formula for Iz is
usually considered trickier to derive than the answer you were asked for.

David
 
Last edited:
  • #5
Ty for your help, David.
It sure brightens things up a bit for me
 

1. What is moment of inertia of a plate?

Moment of inertia of a plate refers to the resistance of a plate to rotational motion around an axis. It is a measure of the distribution of mass around an axis and is dependent on the shape and size of the plate.

2. How is moment of inertia of a plate calculated?

The moment of inertia of a plate can be calculated using the formula I = (1/12) * m * (a^2 + b^2), where m is the mass of the plate and a and b are the dimensions of the plate perpendicular to the axis of rotation.

3. What factors affect the moment of inertia of a plate?

The moment of inertia of a plate is affected by its shape, size, and mass. Plates with larger dimensions and greater mass have a higher moment of inertia. The distribution of mass also plays a role, with more mass located further from the axis of rotation resulting in a higher moment of inertia.

4. How is moment of inertia of a plate used in engineering?

Moment of inertia of a plate is an important parameter in engineering, especially in structural design. It is used to calculate the resistance of a plate to bending and torsional forces, which is crucial in determining the stability and strength of a structure.

5. How can moment of inertia of a plate be changed?

The moment of inertia of a plate can be changed by altering its shape, size, or mass. For example, increasing the mass or the dimensions of a plate will result in a higher moment of inertia. Additionally, the distribution of mass can also be adjusted to change the moment of inertia.

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