# Calculating moment of inertia of a filled cylinder

1. Sep 28, 2009

### asinewave

Hi all,

I was wondering how to calculate the moment of inertia of a cylinder that is filled with a substance and is rolling down an inclined plane.

I know that I = int(r^2 dm), and I know what r is... but I gather that dm may be a complicated function of mass density-position and some volume differential.

I was wondering if someone can explain to me how this works ... I may have to find the density of the substance, right? Is that the only way I can find it?

Also, I understand that I may very well end up with an approximation, but that' sfine.. as long as its reasonably accurate. I want a numerical answer (even if it is a reasonable approximation).. not an exact integral.

Thanks!

2. Sep 28, 2009

### Gear300

If the density is uniform, then the moment of inertia is that of a solid cylinder (what changes is the amount of mass, which may change the value of the inertia, but the expression is the same).

3. Sep 29, 2009

### Bob S

A sine wave-
If the liquid has zero viscosity, it will not rotate as the cylinder rolls down the inclined plane. So although it contributes mass to the cylinder, it contributes nothing to the moment of inertia.
Bob S

4. Sep 29, 2009

### asinewave

Hi all,

Thanks for the replies. What if the liquid has a nonzero viscosity?