Liquid is filled in a U-tube of uniform cross-section to a total length $L$. If the liquid is slight

Question

Liquid is filled in a U-tube of uniform cross-section to a total length $L$. If the liquid is slightly displaced and released, what is the time period of its oscillation?

Professor PC Thomas Classes · Answer

When displaced by $y$, the restoring force is the weight of the liquid column of height $2y$. Restoring force $F = -(2y A \rho)g$. Mass $m = A L \rho$. $T = 2\pi\sqrt{m/k} = 2\pi\sqrt{(AL\rho)/(2A\rho g)} = 2\pi\sqrt{L/2g}$.