What is the Moment of Inertia of a Hollow Sphere?

[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'?

 

[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?

 

[dextercioby]

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

Daniel.

 

[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.

 

[dextercioby]

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

Daniel.

 

[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!

 

[OlderDan]

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.

You can do it with integration, or just take advantage of the fact that calculating 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.

 

[OlderDan]

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

Daniel.

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.

 

[arildno]

Is a box hollow or filled, I wonder?

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

 

[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?

 

[EthanB]

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?

C'est parfait! Yes.

 

[flash]

Cool, thanks.

 

[dextercioby]

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.

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

Daniel.

 

[EthanB]

I use the word sphere. =)