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.

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