# Moment of inertia for non homogeneous density.

1. Mar 6, 2012

### Teachme

Ok so I am trying to figure how I would find the moment of inertia for a special case. I have a 55 gallon barrel that is almost half way full and I am suppose to spin it roughly 5-10 rpm.

I know that to find the momement of interia of a hollow cylinder with thick walls is simply
I = 1/2M(r(1)^1+r(2)^2) yet this equation is too simple for this situation

I also know that
I = ρ ∫r^2 dV However this is for a homogenous density, which is not this case.

I am wondering what is the best way to find the moment of intertia for a non uniform density.

I have attached a picture for clarity.

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2. Mar 7, 2012

### tiny-tim

Hi Teachme!

I don't understand

if the barrel is rolling (in place) as shown in the diagram, then why would the fluid rotate at all (and so why would you need its moment of inertia)?

3. Mar 7, 2012

### Teachme

I think you are right. Since the fluid has no speed you are saying it does not add to the moment of inertia?. I am trying to find the total kinetic energy of the barrel and the fluid in the barrel when it is spinning 5rpm. I know that I need the moment of inertia to find the total kinetic energy... So how would I go about incorporating the the fluid inside the barrel effect my situation? Would I have to use torque for this? My final goal is finding what size motor i need to get it to spin. (not my question however).

Thank you again for your help, I appreciate it very much.

4. Mar 7, 2012

### tiny-tim

Well, the fluid will still slosh about a bit, so it'll have some kinetic energy, and also there'll be heating from the friction between the fluid and the barrel.

I think the only way you could find that is to carry out experiments, and actually measure either the power needed or (for example) the temperature rise in the fluid.