Moment of inertia of a plate

• Lopina
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.

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
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/|\
/ | \ a
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a \ | /    x
\|/
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Homework Equations

Moment of inertia $$I=\int r^{2}dm$$

Perpendicular axis theorem $$I_{z}=I_{x}+I_{y}$$

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 $$I_{x}=I_{y}$$

Using a Perpendicular axis theorem I have $$I_{z}=2I_{x}$$

I need $$I_{x}=0.5I_{z}$$

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

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

But somehow, I think I'm wrong

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

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.

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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

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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.