A liquid of density ρ  contained in a vertical U tube filled to a height h on each limb.  It executes simple harmonic motion in its limbs when a small disturbance is given.  Then the time period of its oscillation is

The restoring force responsible for oscillation is the weight of the liquid displaced.  Hence using the condition for simple harmonic motion we can say that acceleration  a=−ω2y.  Here  a=Fm. If the mass per unit length is μ  , then  F=μ(2y)g and m=μ(2h) .  Simplifying  we get acceleration =a=μ(2y)gμ(2h)=ygh,  here g is acceleration due to gravity   Comparing this with a=−ω2y,  we get      .ω2=ghwhere  ω=2πT. Hence the time period of oscillation is  T=2πhg