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Finding moment of inertia of irregular object with calculus?

  1. Nov 23, 2015 #1
    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 [itex] I = \int r^2dm [/itex]. 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
     
  2. jcsd
  3. Nov 24, 2015 #2

    Nugatory

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    Staff: Mentor

    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}##.
     
  4. Nov 24, 2015 #3
    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!
     
  5. Nov 24, 2015 #4

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    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).
     
    Last edited: Nov 24, 2015
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