Moment of inertia for non homogeneous density.

[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.

Thanks for reading.

 

[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)?

 

[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.

 

[tiny-tim]

… 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.

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.