Calculating the moment of inertia

[goomer]

I know that for for solid disks, the inertia is equal to (1/2)mr^2 and for hoops is just mr^2. This only works for if the mass is evenly distributed around though. So what about when the mass isn't equally distributed? How would you solve for the moment of inertia of a hollow disk if a small metal ball inside of it?

 

[gneill]

I know that for for solid disks, the inertia is equal to (1/2)mr^2 and for hoops is just mr^2. This only works for if the mass is evenly distributed around though. So what about when the mass isn't equally distributed? How would you solve for the moment of inertia of a hollow disk if a small metal ball inside of it?

You could determine the moment of inertia of the objects separately and add them. The parallel axis theorem may be of use if the object's individual moments of inertia are not co-aligned on the axis of rotation.

 

[goomer]

You could determine the moment of inertia of the objects separately and add them. The parallel axis theorem may be of use if the object's individual moments of inertia are not co-aligned on the axis of rotation.

We haven't learned about the parallel axis theorem yet, so I don't think I need to use that.

I forgot to mention that there is a rim inside the disk that keeps the ball at a constant distance away from the radius. Sorry! Would you still calculate the moments of inertia separately? How do you do that?

 

[gneill]

We haven't learned about the parallel axis theorem yet, so I don't think I need to use that.

I forgot to mention that there is a rim inside the disk that keeps the ball at a constant distance away from the radius. Sorry! Would you still calculate the moments of inertia separately? How do you do that?

Yes, calculate the moments of inertia separately. The moment of inertia of a spherical ball of mass M about a point a distance R from its center is just like a point mass M located at radius R. That is, MR². The moment of inertia of your other object depends upon exactly what it is; is it a hollow disk or a hoop?

There are tables of Moments of Inertia for various objects to be found on the web if you google for it. If you want to determine them from first principles then you'll have to do the calculus. It involves performing an integration for each mass element dm over the volume of the object, determining the moment of inertia for each dm and summing them all up. Your text should have an example or two.