Stuck on separable equation relating to moment of inertia

[IneedPhysicsss]

Homework Statement

(a) Consider a cylindrical can of gas with radius R and height H rotating about its longitudinal axis. The rotation causes the density of the gas, η, to obey the differential equation

dη(ρ)/dp = κ ω2 ρ η(ρ)

where ρ is the distance from the longitudinal axis, the constant κ depends on the properties of the gas, and ω is the angular frequency of rotation. Solve this (separable) equation and use the result to set up the integral for the moment of inertia of the gas in the can with respect to the longitudinal axis. Evaluate the integral either by integrating in parts or by using a computer.

(b) A solid hemisphere of radius R sits with its bottom flat face on the x-y plane. The hemisphere is uniformly charged with total charge Q. Find the electric potential at the center of the flat face, V0. What would happen to V0 if you added an identical hemisphere just below the first one such that it completed it to a full sphere? How is this reflected in your calculation of V0?

https://www.physicsforums.com/file:///page1image11176

Homework Equations

For now I'm only concerned with part a

The Attempt at a Solution

I started by trying to solve the separable equation by getting my p's on one side and the n(p)'s on the other so:
dn(p)=kw^2 p n(p) (dp)
dn(p)/n(p)= kw^2 p (dp)

∫(1/n(p))(dn(p))=∫kw^2p (dp)
log(n(p))=(1/2)p^2kw^2

so that's where I'm at. The moment of inertia for a cylinders longitudinal axis is I=1/2MR^2 but I'm not really sure how to use the info to apply that. Any help or tips is appreciated. Thank you

 

[Dr.D]

Consider just part (a) to begin. Separate the equation and solve it for the density function, eta. The apply the integral definition for the MMOI, using this density function in the definition.

 

[IneedPhysicsss]

So would the new step then be n(p)=5p^2kw^2

and then i plug in n(p) for the M and get I=2.5p^2kw^2R^2?

 

[Dr.D]

Gee, I don't know. Is this your problem or mine? If you believe you have done the problem correctly, then check your work and move on.

 

[IneedPhysicsss]

I should have also clarified for the n(p) the (p) is written as a subscript of the n so I've been treating it as a single variable not eta times p

 

[IneedPhysicsss]

I wouldn't be asking if I thought I had done it correctly. I asked the question because I assumed i messed up since the problem seems
too easy

I'm not sure eta and M are interchangeable. I'm thinking I have to first relate the density to Mass per volume. Since the volume of a cylinder is pir^2H I would have

eta=M/pi R^2 Hand end up with I= 5pi p^2 kw^2R^2H R^2

 

[Dr.D]

The problem statement said

The rotation causes the density of the gas, η, to obey the differential equation

dη(ρ)/dp = κ ω2 ρ η(ρ)

eta is the mass density. In the usual terminology, M is the total mass, equal to the integral of eta over the volume.

 

[IneedPhysicsss]

I see, thank you