Consider a liquid which fills a uniform `U` - tube uniform `U`- tube, as shown in figure , up to a height `h`. Find angular the frequency of small oscillation of th eliquid in the `U` - tube.

Answer

`sqrt((g)/(h))`

Explanation

Suppose that the liauid is displaced sligtly from equilibrum so that its level rise in one arm of the tube, while it is depressed in the second by the same amount,x.

If the density of the liquid is `rho`, then the total mechanical energy of the liquid column is
`E = (1)/(2) {A (h + x) rho + A (h - x) rho } ((dx)/(dt)) ^(2)`
`+ [A (h + x) rho g (h + x)/(2) + A (h - x) rho g (h - x)/(2) ]`
`= (1)/(2) (2 Ah rho) ((dx)/(dt)) ^(2) + (1)/(2) (2 A rho g) (h^(2) + x^(2))` (i)
After differentiating the total energy and equating it to zero, one finds acceleration is `- omega^(2) x`
The angular frequency of small oscillation, `omega`, is
`omega = sqrt((2 A rho g)/(2 A h rho)) = sqrt((g)/(h))` (ii)