C4.5 Parallel-Axis Theorem

Recall in the previous section that setting a different reference axis will yield you a different moment of inertia (MoI). What if you already know the MoI with respect to one axis, but you would like to find out the MoI with respect to another axis?

Well this is what the parallel-axis theorem is all about. It’s a theorem to help us transfer the axis for our MoI. The formula is pretty simple:

Transferring axis in the x- and y- directions for moment of inertia

C4.5 Parallel-Axis Theorem

Recall in the previous section that setting a different reference axis will yield you a different moment of inertia (MoI). What if you already know the MoI with respect to one axis, but you would like to find out the MoI with respect to another axis?

Well this is what the parallel-axis theorem is all about. It’s a theorem to help us transfer the axis for our MoI. The formula is pretty simple:

Note:

The Ī_x-x and Ī_y-y are the MoI’s with respect to the axis that passes through the centroid (centroidal axes).
d_x and d_y are the distances from the axes of interest (x-x and y-y) to the centroidal axes (x̄-x̄ and ȳ-ȳ).
Note that d_x is not the distance alone the x-axis. In fact d_x is actually in the same direction as the y-axis. That’s because it’s measuring the distance between 2 different x-axes (x-x and x̄-x̄).

This topic is a very important one, and historically students struggle with this topic. But not to worry, let’s look at a simple example to illustrate this concept.

Note:

The Ī_x-x and Ī_y-y are the MoI’s with respect to the axis that passes through the centroid (centroidal axes).
d_x and d_y are the distances from the axes of interest (x-x and y-y) to the centroidal axes (x̄-x̄ and ȳ-ȳ).
Note that d_x is not the distance alone the x-axis. In fact d_x is actually in the same direction as the y-axis. That’s because it’s measuring the distance between 2 different x-axes (x-x and x̄-x̄).

This topic is a very important one, and historically students struggle with this topic. But not to worry, let’s look at a simple example to illustrate this concept.