# Calculating the moment of inertia of a cylinder

## Homework Statement

Hi there.

Im trying to find the moment of inertia for a 2/3 empty cylinder, could anyone see if I solved this correctly?
Someone told me it's wrong, but I dont know why exactly, so any thoughts would be much appreciated.

(I added the question and the attempt to solve it in a picture)

The answer I got is 13/18 * M * r0^2.
Does it make sense that its more than the moment of inertia of a full cylinder? (1/2 * M r0^2)

## The Attempt at a Solution

#### Attachments

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cepheid
Staff Emeritus
Gold Member
Sapz,

I can find no error in your work, and have reproduced it for myself, so I'm pretty confident that your answer is correct. Furthermore, when you look at the moment of inertia of a thick-walled cylindrical tube around the z-axis here:

http://en.wikipedia.org/wiki/List_of_moments_of_inertia

and substitute in your values for r1 and r2 (inner and outer radii respectively) you also get this answer.

As for whether it makes sense when compared with a solid cylinder: I think it's not so easy to have an intuition for this. The whole point of this exercise is that the rotational inertia of a solid body depends not only on the mass, but also on how that mass is distributed around the axis of rotation. So, when you compare the two expressions, both in terms of "M", you have to keep in mind that "M" is distributed differently in the latter case: you have the same amount of mass, but all of it located farther from the axis. So I think it kind of does make sense that the inertia would be larger. Really though, you have to just do the math to be sure.

EDIT: If you just took the solid cylinder and hollowed 2/3 of it out, then M would be smaller than it was before. We didn't take this into account in our comparison of the two expressions for the moment of inertia above. We assumed that "M" had the same value in both expressions.

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Thank you very much cepheid!