Rotation of axes and transformation equations

The moments of inertia of an object depend upon the axes of reference. Therefore, if we rotate the axes, we will get a different value for the moment of inertia, too. To work out the relationship between the rotation of axes and the moment of inertia, let's consider the following illustration:

Here is some data about the diagram:

• x' and y' have been rotated CCW about O at alpha degrees

• AB=DC, AB||DC||x'

• AD=BC, AD||BC||y'

Now that we are familiar with the geometry of the diagram, we need to find the lengths of OB and BK to work out the moments of inertia about the x' and y' axes.

A useful equation related to moments of inertia: I_x' + I_y' = I_x + I_y

We can use these equations to find the I_min/max values, too. To do this, we'll take the derivative of I_x' (for max value) in respect to the principle angle, alpha, and set it to zero:

Now we can present this graphically:

Note that R was simply obtained by the use of Pythagoras' theorem: A² = B² + C².

Substituting the two expressions above into our equations for moment of inertia about x' and y' axes and using the new value of R from the triangle:

For the maximum value we would add the result of the square root and vice versa.