# Moments of Inertia of a Flat Body

1. Mar 26, 2009

### 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.

Last edited: Mar 26, 2009
2. Mar 26, 2009

### 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.

3. Mar 26, 2009

### 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 $$\int \int \int r^2 \sigma(x,y,z) dx dy dz$$

4. Mar 27, 2009

### tiny-tim

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 …