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A cylindrical transparent vessel of height \( h\left(h \ll R_{\text {ves }}\right) \), where \( R_{\text {ves }} \) is the radius of the vessel, is filled with an ideal gas of molar mass \( \mu \) at a temperature \( T \) and a pressure \( p_{0} \). The dependence of the refractive index \( n \) of the gas on its density \( \rho \) obeys the law \( n=1+\alpha p \). The vessel is rotated at an angular velocity \( \omega \) about its axis. A narrow parallel beam of light of radius \( \Gamma_{\text {beam }} \) is incident along the axis of the vessel.
Determine the radius \( R \) of the spot on the screen placed at right angles to the vessel axis at a distance \( L \) behind the vessel, assuming that the change in the gas pressure at each point of the vessel due to rotation is small as compared with \( p_{0} \). The effect of the end faces of the vessel on the path of the rays should be neglected.
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