A cylindrical container of length 2L is rotating with an angular speed $\omega$ about an axis passing through its centre and perpendicular of its length. Its contains an ideal gas molar mass M. The ratio of gas pressure at the end of the container to the pressure at its centre is $e^{\frac{M{\omega }^a{L}^b}{cRT}}.$ Neglect gravity and assume that temperature of the gas throughout the container is T. The value of a+b+c is _____ 
 

For an ideal gas $\frac{P}{\rho }=\frac{RT}{M}$  
$\rho =\frac{PM}{RT}$                                
Consider a cylindrical element of the gas at a distance x from the axis of rotation. Mass of element $=\rho Adx$
[A=cross section of the element]
Net force on the element towards centre is =AdP
This must be equal to the centripetal force 
$\therefore AdP=\left(\rho Adx\right){\omega }^{2}x$   
$dP=\rho {\omega }^{2}xdx$ 
$dP=\frac{PM}{RT}{\omega }^{2}xdx$ 
$\underset{P_0}{\overset{P}{\int }}\frac{dP}{P}=\frac{M{\omega }^2}{RT}\underset{0}{\overset{L}{\int }}xdx$ 
[$P_0$=pressure at centre, P=pressure at the end]
$\therefore \ln \frac{P}{P_0}=\frac{M{\omega }^2{L}^2}{2RT}$ 
$\therefore \frac{P}{P_0}=e^{\frac{M{\omega }^2{L}^2}{2RT}}$