Computing Inertia Tensor

figure 1

Today, we will be computing the Inertia Tensor of figure 1.  The black circle represent a mass. In this problem, they all have the same mass. First, we need to set a coordinate axis.  We can set the point were all the poles connect as the origin, with the x-axis point east, the y-axis pointing north and the z-axis coming out of the screen.  In order to compute this tensor, we need the following equation:

Next,  we need to define the length of the rods.  In this case, they all have the same length. We will call this variable a. Now we can calculate this tensor.  The i and j  values from the equation above represent the position in our matrix, meaning which section we are trying to calculate.  For example, in a 3x3 matrix, the top left number will be I11. The one directly to its right will be I12, and so on.  After plugging the Iij into each portion of our matrix, we get the following matrix:

We can check our work by looking at each axis.  If we rotate around just the x-axis, we will see there are two masses that effect our rotation.  This is the same for the y-axis.  If we rotate around the z-axis, we will see all four masses effect the rotation, which is what we found.