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.

 

[alphy]

I would first like to commend you for your efforts in trying to understand and solve this problem. The moment of inertia for a non-homogeneous density can be a bit more complex to calculate, but it is certainly possible.

One approach would be to divide the barrel into smaller sections and calculate the moment of inertia for each section separately. This can be done by using the equation you mentioned, I = ρ ∫r^2 dV, where ρ is the density of each section and dV is the volume element. Then, you can sum up the moments of inertia for each section to get the total moment of inertia for the entire barrel.

Another approach would be to use numerical methods, such as Monte Carlo simulations, to approximate the moment of inertia. This involves randomly sampling points within the barrel and calculating the moment of inertia based on the density at each point. This method can provide a more accurate result, but it may be more time-consuming and require more computational resources.

Ultimately, the best way to find the moment of inertia for a non-homogeneous density would depend on the specific characteristics of the barrel and the level of accuracy required. I would recommend consulting with a physics or engineering expert for a more precise and tailored solution.