Study of harmonic motion of a liquid in a V shaped tube

AI Thread Summary
The discussion focuses on analyzing the harmonic motion of a liquid in a V-shaped tube using the Lagrangian method. The user seeks to determine the kinetic and potential energies of the system, identifying gravity and hydrostatic pressure as key forces. It is clarified that in the Lagrangian formalism, only the total potential energy should be considered, which is primarily due to gravity, while hydrostatic pressure forces are internal and do not contribute to potential energy. There is a concern about potential double counting if both gravitational and hydrostatic pressures are included. The conversation emphasizes the importance of correctly identifying the contributing forces to avoid miscalculations in the analysis.
sumatoken
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Homework Statement
Study of harmonic motion of a liquid in a V shaped tube using the Lagrangian method.
Relevant Equations
What is the total potential energy of the system? and why is the restoring force considered to be only the one due to hydrostatic pressure?
A V-shaped tube with a cross-section A contains a perfect liquid with mass density
gif.gif
and length L plus
gif.gif
and
gif.gif
the angles between the horizontal plane and the tube arms as shown in the attached figure.

We displace the liquid from its equilibrium position with a distance
gif.gif
and without any initial velocity.

I'm interested in applying the Lagrangian method.

For that, I need to determine both the system's kinetic energy and potential energy.

Kinetic energy is by definition:
gif.gif


As for potential energy I need to know which forces to consider.

I can see three forces applied to the liquid:

- The force due to gravity
gif.gif
with a potential energy
gif.gif


- The normal force due to the tube reaction. This force will have a null work therefore no potential energy.

- The force due to hydrostatic pressure on the liquid by the portion displaced of length
gif.gif
with potential energy written in terms of the angles, cross-section A and
gif.gif
.

I did some research, and some solutions did not consider the potential energy due to gravity and considered the restoring force to be only the one due to hydrostatic pressure which I do not understand why.

Please feel free to correct me.

Mohammed
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You should not consider forces at all in the Lagrange formalism. What you should be considering is the total potential energy, which here is solely due to gravity. Any pressure forces are internal to the fluid (or orthogonal to the motion) and therefore do not affect the potential.
 
sumatoken said:
some solutions did not consider the potential energy due to gravity and considered the restoring force to be only the one due to hydrostatic pressure
Since it is gravity that leads to hydrostatic pressure, maybe considering both would be double counting.
 
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