I am talking about the very classical thermodynamics problem, where you are given a cylinder of length $D$. The cylinder is being rotated with one of the sides as axis. You're given the pressure in one side of the cylinder, angular velocity $\omega$, molar mass of gas, and now told to find out gas pressure on the other side. Now, if I solve this with Newton's law, and ideal gas law, the ans turns out to be proportional to $\exp(m \omega^2D/2RT)$, and this should be the answer. However, when I try to solve this with Boltzmann distribution function, the confusion arises to me. In the left corner of the cylinder the kinetic energy is 0, and in the right corner the kinetic energy is m $\omega^2D/2$. So the ratio of number densities of right side to left side should be $\exp(-m \omega^2 D/2RT)$, then the pressure should be proportional to $\exp(-m \omega^2D/2RT)$, but that doesn't make sense.
Where I am making the mistake?