Moments of Inertia: Practical Applications & Integration

[Sonny]

I am currently learning about how to calculate moments of inertia of various shapes. I can calculate them fine using the Parallel Axis Theorem, but I am having some difficulty trying to understand the overall concept.

Can somebody give me some practical applications of MOEs? I am also having some difficulty relating them to my understanding of integrals. I use no integration in my present calculations, since most of the shapes I am evaluating are composites of rectangles, triangles, circles, etc..

Would integration make even these simple shapes easier to calculate?

 

[Michael D. Sewell]

Sonny,
Moment of inertia is also known as rotational inertia. That in itself says a lot. The Inertia of a body is it's resistance to acceleration. I'm sure that you are quite familiar with the concept of inertia from your study of rectilinear motion. The moment of inertia is the magnitude of, and the location of the inertia of a rotating body (it's resistance to angular acceleration).

As far as the practical use of this concept is concerned, think about this:
When you accelerate in you car, how many parts of that car undergo angular acceleration? What is the moment of inertia of each of these parts? How much force does it take to impart a given rate of acceleration to your car taking all of this into account?


A careful examination of your work will reveal that you are already using integral calculus in determining the moment of inertia for the shapes that you mentioned.

I hope this helps you,
Mike

 

[Doc Al]

The parallel axis theorem allows you to calculate the rotational inertia about any axis once you know the rotational inertia about the center of mass. In order to find the rotational inertia of an object about its center of mass you would need to use integration, as Mike says. Of course, for standard shapes you can just look up the answers. (Someone else has done the integration!)

Rotational inertia is the rotational analog to mass. Newton's 2nd law for rotational motion is: α = (∑Τ)/I, where α is the angular acceleration, Τ is the torque, and I is the rotational inertia about the axis of interest. Just like mass tells you how a body will respond to an applied force (using Newton's 2nd law), rotational inertia tells you how a body will respond to an applied torque (a twisting force).

 

[Sonny]

Thanks Guys,

I guess inertia being resistance to acceleration was what I was forgetting. And, in retrospect, it seems quite obvious that calculus is used to come up with the formulas for 'rotational inertia' about a centroid.

Cheers.

 

[Michael D. Sewell]

it seems quite obvious that calculus is used to come up with the formulas for 'rotational inertia' about a centroid.

Sonny,
Good for you for seeing this! Sometimes your prof will tell you this sort of thing and you may miss it because it goes by in the wink of a eye. Sometimes some of the pieces of the puzzle are left out by a prof (gee I thought I mentioned that) and you have to try spot them on your own. It can happen, even to a very good prof.

Doc Al,
Thank you for rounding out my (too brief) explanation.
Good job,
Mike