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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
and length
and
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
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:

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
with a potential energy

- 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
with potential energy written in terms of the angles, cross-section A and
.

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

**A**contains a perfect liquid with mass density**L**plusWe displace the liquid from its equilibrium position with a distance

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:

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

- 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

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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