= U-tube liquid oscillation
{title2=$\omega_0=\sqrt{2g/l}$}
In a uniform U-tube containing incompressible liquid of <density> $\rho$, cross section $A$ and total moving length $l$, let one surface fall by $y$ while the other rises by $y$. The height difference is $2y$, and <potential energy> relative to equilibrium is $U=\rho gAy^2$. The whole moving <mass> is $\rho Al$, so $K=\rho Al\dot y^2/2$. <Newton's second law> or <energy> conservation gives $\ddot y+(2g/l)y=0$ and $\omega_0=\sqrt{2g/l}$. This assumes small excursions that keep the free surfaces in the uniform vertical limbs.
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