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Fin_de_Siecle

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**The problem asks me to find the density of gas in a cylinder of radius R and length l rotating about its axis with angular velocity ω, there being a total of N molecules in the cylinder.**

**What I have done is shown as following**

I choose to look this scenario in a static frame with the cylinder, which is rotating at an unchanging angular speed ω. Then we can consider the field is a centrifugal field.

Thus the energy of a particle can be written as

$$E=E_0-\frac{1}{2}m\omega ^2 r^2$$ (1)

I haven't determined E_0 yet as I'm not sure about the thermal energy of the particle and the energy caused by the centrifugal field differs with the choosing of zero potential point.

After that, I wrote the probability as

$$dP=\frac{\rho dV}{N}=\frac{\rho \cdot 2\pi rdr\cdot l}{N}$$ (2)

where ρ is what I want to find.

My goal is to convert all r and dr into E and dE and finally compare it with Boltzmann distribution

$$P(E)=C\cdot e^{-E/kT}$$ (3)

where C is a constant.

However, I have met with some problems doing this.

From (1), it can be derived

$$E=-m\omega ^2 r$$ (4)

After I plug this into (2), I find P(E) in dP=P(E)dE is always negative. Thus is cannot be compared with the Boltzmann distribution.

How can I fix this?

Thank you.

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