The parallel-axis theorem allows you to use SIMPLE pieces to build up the moment of inertia of a COMPLEX body. Let's do an example to see how it works.

Start with a very simple object: a long, thin rod
of mass **m** and length **L**,
spun around its center.

What's the moment of inertia of this rod?

Now, suppose that I moved the rod away from the
axis of rotation by some small distance **x**,
like this:

What's the moment of inertia of this displaced rod, if I keep rotating it around the same axis?

So, what if I combine two rods, making a compound object?
Suppose I place the first rod centered on the axis, and the
second rod offset by a distance **x**.
What would the total moment of inertia of this 2-rod
combination be?

But why stop at just two rods? I can use this same technique to build up a large structure out of many rods, and compute its moment of inertia, too.

For example, I could place a whole bunch of rods next to each other, to make a square plate.

Let's call the total mass of this entire rectangular plate **M**.
It has surface area **A = L x L = L ^{2}**.
We can define the

Now, to figure out the moment of inertia of this
complex structure, I just consider one piece at a time.
Let me pick one thin little rod,
a distance **x** from the axis
and of width **dx**.

The mass of this little piece **dM** is

and the contribution of this piece to the moment of inertia must be

Okay, now it's your turn: figure out the moment of inertia of the entire square plate, adding up the contributions from all the little rods.

- Express the moment of inertia in terms of the
total mass
**M**and side length**L** - Does this value agree with the one in your textbook?

You can check your answers by looking at my own solution.

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.