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

In summary: Therefore, it is more efficient to only consider the potential energy due to gravity when using the Lagrangian method. In summary, when using the Lagrangian method to analyze a V-shaped tube with a perfect liquid, it is only necessary to consider the potential energy due to gravity and not the potential energy from hydrostatic pressure. This is because the force of gravity ultimately leads to hydrostatic pressure, so considering both would be redundant.
  • #1
sumatoken
1
0
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
1666995197923.png
 
Last edited:
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 

1. What is harmonic motion?

Harmonic motion refers to the repetitive back-and-forth movement of an object or substance around a central point or equilibrium position.

2. How is harmonic motion of a liquid in a V shaped tube studied?

The harmonic motion of a liquid in a V shaped tube can be studied by measuring the displacement and velocity of the liquid as it moves back and forth within the tube. This can be done using specialized equipment such as a laser displacement sensor or a pressure transducer.

3. What factors affect the harmonic motion of a liquid in a V shaped tube?

The harmonic motion of a liquid in a V shaped tube can be affected by factors such as the viscosity of the liquid, the shape and size of the tube, and the amplitude and frequency of the oscillations.

4. What applications does the study of harmonic motion of a liquid in a V shaped tube have?

The study of harmonic motion of a liquid in a V shaped tube has applications in various fields such as fluid dynamics, acoustics, and engineering. It can also be used to understand and improve the performance of devices that utilize harmonic motion, such as pumps and turbines.

5. What are some possible future developments in the study of harmonic motion of a liquid in a V shaped tube?

Possible future developments in this field include the use of advanced imaging techniques, such as high-speed cameras, to better visualize and analyze the motion of the liquid. Additionally, researchers may explore the effects of different tube shapes and materials on the harmonic motion of the liquid.

Similar threads

  • Introductory Physics Homework Help
Replies
6
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
666
  • Introductory Physics Homework Help
Replies
1
Views
4K
  • Introductory Physics Homework Help
Replies
3
Views
1K
  • Introductory Physics Homework Help
Replies
27
Views
3K
  • Introductory Physics Homework Help
Replies
11
Views
746
  • Introductory Physics Homework Help
Replies
18
Views
2K
  • Introductory Physics Homework Help
Replies
12
Views
6K
Replies
17
Views
1K
  • Introductory Physics Homework Help
Replies
14
Views
3K
Back
Top