Question #43921

A uniform vertical U-tube of inner radius $r$ contains a liquid of density $\rho$ and total length $L$. The inner surface of the tube has a special coating such that the contact angle $\theta$ of the liquid with the tube wall varies with the vertical height $y$ from the horizontal base of the U-tube according to the relation $\cos\theta = \cos\theta_0 - \alpha y$, where $\alpha$ is a positive constant and $\theta_0$ is the contact angle at the base. The surface tension of the liquid is $T$ and the acceleration due to gravity is $g$. The liquid column is given a small displacement and released. Assuming the liquid always wets the tube ($\theta < 90^\circ$) and ignoring viscous damping, what is the angular frequency $\omega$ of the resulting small oscillations?