Oscillations in a U-tube
Take 15 minutes to prepare this exercise.
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A detailed solution is then proposed to you.
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The oscillations of an incompressible fluid is studied in a small cross section of the U-tube.
It is assumed that the free surfaces remain in the rectilinear and vertical portions of the tube.
It is desired to determine the oscillation period of the liquid in the tube.
First method : by applying Bernoulli's theorem in a regime depending on the time.
The cote of the free surface of fluid in the right branch of the tube is denoted .
The fluid is incompressible, the speed at any point is :
It integrates the Euler equation of a current line :
Either, noting the liquid length :
The period is then :
Second method : by performing an energy balance.
The kinetic energy of all of the fluid is : ( is the U-tube cross section)
The potential energy is divided into three parts : it is assumed that at time , the drop was (the left side is lowered relative to the equilibrium position).
So (the center of mass is in the middle of the column of a considered fluid) :
By derivation :
We find the same equation.