# Vertical Spring System (Lagrangian)

• andris0110
In summary, the problem is that the equations of motion are not in their simplest form and it is unclear what the parameters mean. The student is trying to solve for the equations of motion for a vertical spring system and it is not clear what the parameters mean.
andris0110
Homework Statement
The problem states that for the vertical spring system below we give boundary conditions to the y positions of m1 and m2, so considering this I have to work out the equations of motion. The ring on the top is supposed to be moving only vertically.
Relevant Equations
Lagrange equations of motion
I am trying to solve this and get the equations of motion using the Lagrangian method.
I could do all the steps but the equations (especially the third one) seems..weird.
What am I doing wrong? Sorry if the equations aren't in their simplest form, they are pulled straight from Wolfram Mathematica.
I tried to explain my work in a little more detail below:

Thanks for the help in advance!

andris0110 said:
Homework Statement:: The problem states that for the vertical spring system below we give boundary conditions to the y positions of m1 and m2, so considering this I have to work out the equations of motion. The ring on the top is supposed to be moving only vertically.
Relevant Equations:: Lagrange equations of motion

I am trying to solve this and get the equations of motion using the Lagrangian method.
I could do all the steps but the equations (especially the third one) seems..weird.
What am I doing wrong? Sorry if the equations aren't in their simplest form, they are pulled straight from Wolfram Mathematica.
I tried to explain my work in a little more detail below:
View attachment 316474

View attachment 316475
View attachment 316471

Thanks for the help in advance!
Hi @andris0110. Welcome to PF. Since no one has yet replied, here are a few comments.

If you haven’t yet read the guidelines for posting, it’s worth doing so: https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/

There are some problems with what you have posted. Here are the first few I spotted before I gave up!

1. The original (complete/word-for-word-exact) question is missing. You have expressed the question in your own words. This may be OK but sometimes (often!) posters’ descriptions are incomplete and/or misinterpret the original question.

2. You say “the ring … is supposed to be moving only vertically”. But the ring appears to have the freedom to rotate about a horizontal axis through its centre while oscillating vertically. I’m guessing rotational motion is allowed and ##\theta## or ##\phi## is the angle of rotation.

3. The meanings of some parameters are unclear, e.g. ## \theta, \alpha_1 \alpha_2##. They should be shown on the diagram. (We can guess, but sometimes we'll guess wrongly.)

4. Are you using ‘x’ for both horizontal and vertical distances [Edit - and spring extension!]? That would be a dreadful source of confusion! Your potential energy equation (equation 6) seems to use ##x_1## and ##x_2## for the gravitational potential energy!

5. The kinetic energy term at the end of equation 5 appears to relate to rotational motion. But it makes no sense - it is dimensionally incorrect and doesn’t include the moment of inertia of the ring.

6. Rather than Wolfram Mathematica, you should formulate the various equations for yourself and include them in your posts. You should use LaTeX to format the equations here so we can read them easily. (You’ll get more replies/help if you create easy-to-read posts.) Here's the quick guide to LaTeX: https://www.physicsforums.com/help/latexhelp/. (This link is also below the bottom left corner of the edit window.)

I suggest taking a big step back. Your best bet might be to post a revised question/attempt with the above points addressed.

Edits: typo's corrected.

Last edited:
PhDeezNutz, topsquark, TSny and 2 others

## 1. What is a vertical spring system?

A vertical spring system is a physical system that consists of a mass attached to a spring that is suspended vertically from a fixed point. The spring allows the mass to oscillate up and down due to the force of gravity.

## 2. What is the Lagrangian of a vertical spring system?

The Lagrangian of a vertical spring system is a mathematical function that describes the system's dynamics and is used to derive the equations of motion. It takes into account the kinetic and potential energy of the mass-spring system.

## 3. How is the Lagrangian used in a vertical spring system?

The Lagrangian is used to derive the equations of motion for a vertical spring system, which describe how the position and velocity of the mass change over time. It is also used to calculate the system's energy and determine stable equilibrium points.

## 4. What is the significance of the Lagrangian in physics?

The Lagrangian is a fundamental concept in classical mechanics and is used to describe the dynamics of physical systems. It allows for a more elegant and concise formulation of the equations of motion, making it a powerful tool in analyzing and understanding complex systems.

## 5. Are there any real-world applications of a vertical spring system?

Yes, vertical spring systems are commonly used in various engineering and scientific fields. They are used in shock absorbers, suspension systems, and seismometers, among other applications. They also serve as a simple model for studying oscillatory motion and energy conservation.

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