Parallel Axis Theorem for x and y Shift

Question

I have a quick question about the parallel axis theorem.

Let's say I have a box-like object rotated about the following pivot point

I was wondering, by applying the parallel axis theorem, would the moment of inertia about the pivot point be the following,

$$M = I_{cm} + M_d^2+M_h^2$$

where $I_{cm}$ is the moment of inertia about the box's centroid and $M$ is its mass.

Or, does the parallel axis theorem not hold true for two axis shifts?

Answer 1

Moment of inertia is based on both a point of reference and an axis of rotation.

So the answer to your question depends on the axis you're asking about.

For the moment of inertia around the horizontal axis, you only care about the vertical shift (and visa-versa for the vertical axis, of course):

$$I_{xx} = I_{cm} + M\cdot h^2$$

For any other axis, things get much more complicated, as suggested by @kamran's answer.

Answer 2

Parallel axis would apply if you first calculate the $I$ of the box about an axis perpendicular to a line connecting the pivot point to the center of mass of the box.

This I is not the same as the one fore a thin plate, the box has volume.

It gets a bit complicated, having to triple integrate in polar coordinate the $\text{d}v \cdot r^2$.