# Water column resonance

**1. The problem statement, all variables and given/known data**

There is a cylindrical vessel of cross-sectional area 20cm^2 and length 1m is initially filled with water to a certain height as shown. There is a very small pinhole of cross-sectional area 0.01cm^2 at the bottom of the cylinder which is initially plugged with a pin cork. The air column in the cylinder is resonating in fundamental mode with a tuning fork of frequency 340Hz. Suddenly the pin cork is pulled out and water starts flowing out of the cylinder. Find the time interval after which resonance occurs again.

**3. The attempt at a solution**

The setup is equivalent to a closed organ pipe. The fundamental frequency is given by nv/4L.

Let the initial height of water column be h1. At second resonance let the height change to h2.

h1-h2=v/4*340

The velocity of water coming out of hole is given by [itex]\sqrt{2gx}[/itex]

Thus, rate of decrease of water column = 0.01v/20.

[itex]\dfrac{dx}{dt} = \dfrac{0.01\sqrt{2gx}}{20} [/itex]

If I integrate this from h1 to h2, I am left with the expression [itex]\sqrt{h_1}-\sqrt{h_2}[/itex].

I already know the difference between the heights but not the difference between their square roots. How do I calculate it?

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