Glass tube

A glass tube of a circular section of $S$ was bent in a shape shown, has one arm vertical and the other inclined at an angle $\theta.$

A light piston that can slide without friction in the vertical arm is connected at the lower end of a vertical spring of force constant $k$ , the upper end of which is connected to a fixed support. When a mass $m$ of a non-viscous liquid of density $\rho$ is poured in the tube, the level of liquid in the inclined arm stays higher than the piston as shown in the figure.

If the d period-of small oscillations of the liquid is in the form $\alpha \sqrt{\frac{m}{k+\rho g S(\beta+ \sin \theta)}},$ submit your answer as the sum of constants $\alpha + \beta$ .

Assume there is no friction between the pisto. And the tube and acceleration due to gravity is $g$ .