A cylindrical space station rotating with angular velocity \(\Omega\) contains an atmosphere with molecular weight \(M\) and temperature \(T\). Show that if \(P_{0}\) is the atmospheric pressure at the rotation axis, the pressure at radius \(ho\) is \(P=P_{0} \exp \left(M \Omega^{2} ho^{2} / 2 R T\right)\), where \(R\) is the ideal gas constant. If the station has a radius of \(100 \mathrm{~m}\), an effective rim gravity \(10 \mathrm{~m} / \mathrm{s}^{2}\), and an oxygen atmosphere at temperature \(300 \mathrm{~K}\), what is the ratio of the rim pressure to that at the rotation axis? Would this difference be important to inhabitants who travel from the rim to the axis?