# Moment of Inertia of a Sphere

1. Nov 28, 2006

### flash

The moment of inertia of a sphere rotating about the centre is (2/5)mr^2, but what if it has a hollow 'core'?

Last edited: Nov 28, 2006
2. Nov 28, 2006

### EthanB

You'll need to be more specific. Is it a shell with negligible thickness? Are we considering an inner radius? Do you know how to integrate to find moment of inertia?

3. Nov 28, 2006

### dextercioby

A (2-)sphere is hollow to begin with. So your question doesn't make too mush sense.

Daniel.

4. Nov 28, 2006

### flash

Sorry, I should have been clearer. There is an inner radius involved, so the thickness is not negligable. I know the basic idea of integration to find the moment of inertia, r^2dm, but haven't done much of it.

5. Nov 28, 2006

### dextercioby

This problem is classical on PF. You should use the search option and i'm sure you'll get satisfied.

Daniel.

6. Nov 28, 2006

### arildno

Okay, so you want to find the moment of a spherical SHELL (can you accept that wording, dexie?).

Now, described in spherical coordinates, set up the limits of integration for the three variables first!

7. Nov 28, 2006

### OlderDan

You can do it with integration, or just take advantage of the fact that calculting the moment of inertia is just an addition problem. The moment of inertia of a solid sphere (known) is the sum of the moments of inertia of a smaller inner sphere plus the moment of inertia of a concentric outer spherical shell.

8. Nov 28, 2006

### OlderDan

Therefore the volume of a sphere is zero. Even mathematicians resort to "common usage" when it serves their purpose. I've never heard anybody say "the volume of the region bounded by a shpere of radius R" is . . . .

We of course are making the same mistake when we talk about the area of a circle. Using its formal definition, it has none.

Last edited: Nov 28, 2006
9. Nov 28, 2006

### arildno

Is a box hollow or filled, I wonder?

And what about the prolate spheroids and the parallellepipedes in the world?

10. Nov 28, 2006

### flash

Thanks for all the replies. I only know the mass of the outer part of the sphere. Here's what I think I will do: Find the density of the outer part, calculate the moment of inertia of the solid sphere with this density and subtract the moment of inertia of the inner sphere with this density. Will that work?

11. Nov 28, 2006

### EthanB

C'est parfait! Yes.

12. Nov 28, 2006

### flash

Cool, thanks.

13. Nov 29, 2006

### dextercioby

Of course you didn't hear that, simply because they use the word "ball".

Daniel.

14. Nov 29, 2006

### EthanB

I use the word sphere. =)