Finding the period of oscillation of a u-tube w/ water (almost solved)?

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SUMMARY

The natural period of oscillation of fluid in a u-tube manometer is derived using the energy method, establishing that the formula is t=2π(√(L/2g)), where L represents the length of the fluid column and g is the acceleration due to gravity. The discussion highlights confusion regarding the origin of L in the equation, specifically questioning its role in the relationship between angular frequency (wn) and gravitational acceleration. The solution involves equating potential energy to kinetic energy, leading to the derivation of the oscillation period.

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jaron_denson
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Homework Statement


Using the energy method( setting potential energy= kinetic energy) show that the natural period of oscillation of the fluid in a u-tube manometer is t=2PI(l/2g)^(1/2) where L is the length of the fluid column.
2. The attempt at a solution
2attempt.jpg


So as you can see I do not know where the L is coming from, although i just fuged it in when I wrote what wn^2=2g/L. Where is the L coming from? wn^2 should equal coefficents of x divided by coefficents of xprime (aka velocity).
 
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You might find this lecture interesting at just about the 21:00 minute mark.

 
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