# Products of Inertia with 3 rods

1. Apr 7, 2005

### Claire84

Products of Inertia with 3 rods....

Hi there, I was hoping someone could give me a hand with a question I'm working on. We first of all had to find the moment of inertia of a uniform rod of mass m and length a about an axis through one end that makes and angle alpha with the rod. I got this, but for the next section we have 3 uniform rods OA, OB and OC, of mass m and length a, that are rigidly joined at 0. The angle between OA and OB is pi/4, and OC is perpendicular to the plane of OA and OB. We have to determine the principal moments of inertia and the principal axes of the system.

So I figure we've to start with constructing the inertia tensor. I've got my moments of inertia about the x-axis, y -axis and z - axis (I presume you just calculate the moment of inertia of each rod about say the x -axis and then add them together to get the total moment of inertia of the system about the x axis?). Anyway, I'm a bit stuck when it comes to the products of inertia (the one example in our notes is a cube, which was much easier). Just so you can get what I'm picturing, I have OA along the x -axis, OB at pi/4 to it and pi/4 to the y -axis and the rod OC along the z -axis.

For F I got it equal to zero (F is sigma myz), but for G (sigma mxz) I wasn't sure how to go about it. For road OA and rod OC I got it to be zero because for OA z = 0 and for OC x = 0, but how do I go about it for OB because of the way it is at an angle? This is probably really basic but I've been off uni on Easter hols for about 2 and a half weeks and my brain appears to have shut down. :yuck:

Thanks for any help!

Claire