Moment of Inertia for 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.

 

[Achy47]

Hi there!

It sounds like you have a very interesting problem on your hands. To calculate the moment of inertia for your curved cuboid, you will need to use the parallel axis theorem. This theorem states that the moment of inertia of a body about any axis is equal to the moment of inertia about a parallel axis through the center of mass, plus the mass of the body times the square of the distance between the two axes.

In your case, the parallel axis would be the x-axis passing through the center of mass, and the axis you are interested in would be the y-axis passing through the top and bottom ends of the cuboid. To calculate the moment of inertia about the parallel axis, you can use the formula for the moment of inertia of a rectangular prism, since your curved cuboid can be thought of as a series of small rectangular prisms stacked together.

Once you have calculated the moment of inertia about the parallel axis, you can use the parallel axis theorem to find the moment of inertia about the y-axis. You will need to integrate the moment of inertia about the parallel axis over the entire length of the cuboid, taking into account the changing distance between the two axes as you move along the y-axis.

I hope this helps and good luck with your calculations! Don't hesitate to ask for further clarification if needed.