Finding moment of inertia of irregular object with calculus?

[cmkluza]

For a mathematics project I'm trying to figure out the moment of inertia for a propeller. I'm told that it is possible to find the moment of inertia of irregular objects through calculus, so I'm determined to figure it out using calculus.

I plan on using a 3D modelling program (since I don't have actual propellers at the moment) to construct a propeller, or just using a 3D model of a propeller, so I can get exact measurements of it. I can also use density (found on the Internet) and volume to deduce the mass. I figure I can separate a propeller into an idealized, hollow, cylindrical center (easy to calculate) and the blades.

So, where do I start? I've seen some mathematics on Wolfram, but I'm not following it. Wolfram shows a function ρ(r) as the density of the object, but it doesn't show where ρ(r) comes from; how would I write the density of an object as a function of its radius? Others show it more simply as $I = \int r^2dm$. What should I use, and how should I go about collecting the input data for whatever method I use? Also, I understand where the second function I mention comes from, but if I were to use something like the Wolfram function, could anyone tell me where it comes from, or how it is deduced?

Thanks for any help!

Edit: Missing info/formatting

 

[Nugatory]

Wolfram is is using the convention that vectors are written in boldface (and the magnitude of the vector is written in ordinary italics). Thus, the function $\rho(\mathbf{r})$ is the density at the point identified by the vector $\vec{r}$ - it will be a constant if the density is uniform.

The $\mathbb{d}m$ form is just a different way of writing the same thing, because $\mathbb{d}m=\rho(\mathbf{r})\mathbb{d}V$ - either way, it's the mass in the infinitesimal volume located at $\vec{r}$.

 

[cmkluza]

Wolfram is is using the convention that vectors are written in boldface (and the magnitude of the vector is written in ordinary italics). Thus, the function $\rho(\mathbf{r})$ is the density at the point identified by the vector $\vec{r}$ - it will be a constant if the density is uniform.

The $\mathbb{d}m$ form is just a different way of writing the same thing, because $\mathbb{d}m=\rho(\mathbf{r})\mathbb{d}V$ - either way, it's the mass in the infinitesimal volume located at $\vec{r}$.

So, is the function that includes vectors including all vectors from the axis of rotation to the point that they signify? I still don't quite understand it. Could you give me an example of it used in a simple situation? How could I go about using that to calculate the moment of inertia for a propeller, or other such irregular object?

Thanks!

 

[Nugatory]

So, is the function that includes vectors including all vectors from the axis of rotation to the point that they signify? I still don't quite understand it. Could you give me an example of it used in a simple situation? How could I go about using that to calculate the moment of inertia for a propeller, or other such irregular object?

A simple example: $\rho(\mathbf{r})=\alpha\frac{R-r}{R}$ for for all vectors $\vec{r}$ such that $r<R$ and zero everywhere else describes a ball of radius $R$ whose density decreases linearly from $\alpha$ at the center to zero at the surface. To calculate the moment of inertia of that ball you'd integrate $r^2\mathbf{d}M=r^2\rho(\mathbf{r})\mathbf{d}V = r^2\alpha\frac{R-r}{R}\mathbf{d}V$ across the entire volume of the ball.

The problem you'll have will be finding a simple and easily integrated function that describes a complicated shape you've put together with 3D modelling software. Often engineers working with real objects have to resort to numerical integration (hopefully with some support from the software).