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A vertical U-tube of uniform cross-secti...

A vertical U-tube of uniform cross-section contains water upto a height of 30 cm. show that if the water in one limb is depressed and then released, its up and down motion in the two limbs of the tube is simple harmonic and calculate its time-period.

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Figure shows a U-tube of uniform cross-sectional area A. Let the liquid be depressed through the distance y in a limb, the difference of levels between two limbs will be 2 y as shown in figure

The liquid now oscillates about the initial positions.
Excess pressure on whole liquid = (excess height of the liquid column) (density) (g)
`= 2y xx 1 xx g` (as density of water `= 1`)
Restoring Force on the liquid = Pressure x area of cross-section `= 2ygA`
Due to this force the liquid acclerates and if its acceleration is a, we have `ma = 2ygA`
Do to this force the liquid accelerated and if its acceleration is a, we have `ma = -2ygA`
`rArr (2 xx 30 xx A) a = - 2ygA rArr a = - (g)/(30) y`
`T = (2pi)/(omega) = 2omegasqrt((30/g)) = 2pisqrt((30/980))` [as `omega = sqrt((g)/(30))`] `= 1.098` second
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