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

Question

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

Infinity Learn · Accepted Answer

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  $$a=$$−ω$$2y$$ .  Here $$a=Fm$$ . If the mass per unit length is   μ, then $$F=$$μ(2y)g$$  and  $$m=$$μ$$(2h).  Simplifying  we get  $$a=$$μ$$(2y)g$$μ$$(2h)=ygh$$   Comparing this with $$a=$$−ω$$2y$$,  we get ω$$2=gh$$.where  ω$$=2$$$$π$$T. Hence the time period of oscillation is  $$T=2$$$$π$$hg

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

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