Moments of Inertia of a Flat Body

AI Thread Summary
To calculate the moments of inertia for a flat rectangular body, treat it as having negligible thickness and use a mass-per-area approach. The moments of inertia can be determined using double integrals, specifically ∫∫ r²σ(x,y) dx dy, where r is the distance from the axis of rotation. The principal axes correspond to the axes of symmetry of the rectangle, which are typically aligned with the rectangle's sides. It's important to note that all three moments of inertia for the rectangle are non-zero, even though one may be significantly smaller due to the flat nature of the body. Understanding these principles will help in solving the assignment on moments of inertia.
hamjam9
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How do you take the moments of inertia of a flat body? I know howto take it if it's a 3d body. And the 2d case should be really simpe,but I'm too stupid to figure it out. Can you help me? For example.. Say we have a body that's a rectangle of mass m on |x| < a, |y| < b..? Thanks so much.
 
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Hi hamjam9! :smile:

I assume you're given a mass-per-area instead of a density?

For a "flat" body (no such thing, really :rolleyes:), you just treat it as if it has a very very small thickness. :wink:
 
Or perhaps the thickness doesn't matter. Could it be unit thickness? As long as the density doesn't vary with the z-coordinate of your example.

In any case, I usually find it easier to do a double integral \int \int r^2 \sigma(x,y) dx dy than a triple integral \int \int \int r^2 \sigma(x,y,z) dx dy dz
 
hamjam9 said:
Hey tiny-tim. Thanks for answering my question about moments of inertia. I thought no one would answer it lol. I'm so sorry to PM you but I'm really desperate. I'll be honest, I have assignment due tmrw, and I can't get this one question done !:frown:

"Find the principal axes and moments of inertia of a flat homogeneous rectangular body (|x| ≤ a, |y| ≤ b) of mass m (uniformly distrubuted)."

The question asks to find the 'principal axes' and 'momentS of inertia'. Not just the 'moment' of inertia, but moments.

Now I know I should be taking my origin to be in the dead center of the rectangle, but I'm unsure about how to state what the principal axes are, and then how to get the 3 moments of inertias (I know 1 of those 3 is actually zero, since it's flat). If you could help me that would be so great. I'mso sorry to PM you man.

Hi hamjam9! :smile:

No, none of the moments of inertia is zero.

Look at the PF library on moment of inertia :wink:

Any axis of symmetry is a principal axis.

Any body has a different moment of inertia about every axis …

this question asks you for the ones about the principal axes. :smile:
 

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