Moments of Inertia of a Flat Body

[hamjam9]

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.

 

[tiny-tim]

Hi hamjam9!

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

For a "flat" body (no such thing, really ), you just treat it as if it has a very very small thickness.

 

[Cantab Morgan]

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

 

[tiny-tim]

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 !

"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!

No, none of the moments of inertia is zero.

Look at the PF library on moment of inertia

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.