Moment of inertia of curved cuboid

[kylem2122]

Hi Physics Forums!

Moment of inertia question for you:
I have a cuboid, like the first one in this link

http://en.wikipedia.org/wiki/List_of_moments_of_inertia

only it is curved around the y (w in pic) axis such that the x (h in pic) axis touches the top and bottom ends of the cuboid and it is a parabolic curve that satisfies the following:
z=r*(1-(x^2)/((c/2)^2))
where r is the distance from the origin to the center of the cuboid and c is the chord length (in the x direction)
I need to find what the moment of inertia components would be, and I'm stumped. Any help would be greatly appreciated.

 

[Curl]

I can't picture what you're describing. If you can provide a sketch then I can probably figure it out.

If the shape is a 2D "extrusion" then it is sufficient to find the inertia of the 2D slice which should be a simple region in R2.

 

[kylem2122]

Thanks for the reply. I have made a simple 2D sketch for you. The y-axis is coming out of the page. Technically it is more like a flat plate than a cuboid but I would like to find the xx yy zz components of inertia if possible.

 

[Curl]

Ok this is simple. For Iyy do double integral of x^2 + z^2 dA on the region -c/2<x<c/2 and 0<z<z(x). This gives moment of inertia about the origin.

 

[kylem2122]

Okay, and I think the Izz will stay the same as a regular cuboid, right? What about Ixx? And for z(x) I use r*(1-(x^2)/((c/2)^2)) right? And should it be dm instead of dA for mass moment of inertia?