# 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 $\sqrt{2gx}$

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

$\dfrac{dx}{dt} = \dfrac{0.01\sqrt{2gx}}{20}$

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

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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