How Do You Calculate and Rotate the Inertia Tensor for a 4-Particle System?

[darthmonkey]

Homework Statement

A)Find the moment of inertia tensor for a system of 4 particles of mass 1kg at A=(1,1,0), B=(1,-1,0) C=(-1,1,0) D=(-1,-1,0) in Cartesian coordinates.

B)Rotate the coordinates 30 degrees around the z axis and find the tensor in the new coordinates.

Homework Equations

Ixx=$\sum$y^2+z^2
Iyy=$\sum$x^2+z^2
Izz=$\sum$y^2+x^2
Ixy=$\sum$xy
Iyx=$\sum$yx
Izx=$\sum$zx
Izy=$\sum$zy

I'=RIR*

The Attempt at a Solution

I get Ixx=4, Iyy=4, and Izz=8. Anything multiplying z is 0 and all the xy components cancel out. So a diagonal matrix.

For B I just apply the z rotation matrix with 30,RIR*, but when I do this I get the same thing. This can't be right so I either calculated my starting matrix wrong or am rotating incorrectly.

 

[TSny]

Hi darthmonkey. Maybe the tensor components don't change under the rotation! Have you tried figuring out the new coordinates of the particles in the rotated system and then calculating the components of the tensor from these components?

 

[darthmonkey]

Haha, thanks. I got it to all work out.