Lagrangian mechanics: a spring and a bar

In summary, the conversation discusses finding the Lagrangian equations for a system consisting of a massless spring attached to the ceiling and a homogenous bar attached from its other end to the lower end of the spring. The bar is free to move on the xy-plane and the goal is to find the Lagrangian equations for the system. The conversation also mentions the use of kinetic and potential energy equations and the inclusion of gravity as a potential energy.
  • #1
atwood
9
0

Homework Statement


A massless spring (spring constant k) is attached to the ceiling. It is free to move only in the y-direction. A homogenous bar of mass m and
length l is attached from its other end to the lower end of the spring, and the bar is free to move on the xy-plane.

A fancy picture for those who appreciate:

+<--y-->-
|~~~--

|=ceiling, ~~~=spring, --=bar

Find the Lagrangian equations of the system.

Homework Equations


[tex]\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i}-\frac{\partial L}{\partial q_i}=0[/tex]

The Attempt at a Solution


I suppose there's no gravitational force present, as there's no mention of it, although the spring is attached to the ceiling. But this
problem still stuns me.

First, there's the spring related energy of the system.
[tex]L=T-V=\frac{1}{2}m\dot{y}^2-\frac{1}{2}ky^2[/tex]

However, in this case the velocity doesn't depend only on y (the location of the moving end of the spring), but it also depends on the
movement of the center of the bar, as it's free to rotate around the end of the spring. Rephrasing the previous:

The center of mass of the bar:
[tex]y_b=y+\frac{l}{2}cos\theta[/tex]
where theta is the angle away from the vertical
[tex]\dot{y_b}=\dot{y}-\frac{l}{2}\dot{\theta}sin\theta[/tex]

[tex]L=T-V=\frac{1}{2}m\dot{y_b}^2-\frac{1}{2}ky^2
=\frac{1}{2}m\left(\dot{y}-\frac{l}{2}\dot{\theta}sin\theta\right)^2-\frac{1}{2}ky^2[/tex]

Secondly, there's the kinetic energy of the bar. The spring is massless so it has no kinetic energy, so the kinetic energy in the previous
equation is the translational energy of the bar. Additionally there's rotational kinetic energy (R) but no potential energy.

[tex]L=R=\frac{1}{2}J\omega^2[/tex]

Here
[tex]J=\frac{1}{3}ml^2[/tex]

[tex]\omega=\dot{\theta}[/tex]

So [tex]L=(T+R)-V=\frac{1}{2}m\left(\dot{y}-\frac{l}{2}\dot{\theta}sin\theta\right)^2
+\frac{1}{2}J\dot{\theta}^2-\frac{1}{2}ky^2[/tex]

I've been told that there's an error, but I haven't been told where it is. Of course my equations look awful, but knowing that doesn't help.
 
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  • #2
Your first part looks fine to me; for the second, i wouldn't go for a rotational energy but instead simply get an x-component of kinetic energy where x = l*sin(theta)/2... the rest should follow (you have to focus on the center of mass, where the net force is at work; it would be good to know if you're supposed to include gravity, though...)
 
  • #3
atwood said:
I suppose there's no gravitational force present, as there's no mention of it, although the spring is attached to the ceiling. But this
problem still stuns me.

I'm pretty sure you are supposed to take gravity into account. Otherwise they would not have said that the spring is attached to the ceiling, but that all the stuff is located on a table with neglectable friction or something like that.
 
  • #4
I agree. It also gives the only usable potential energy...
 

1. What is Lagrangian mechanics?

Lagrangian mechanics is a mathematical formalism used to describe the motion of systems in classical mechanics. It was developed by Joseph-Louis Lagrange in the late 18th century and is based on the principle of least action, which states that the path a system takes between two points in space and time is the one that minimizes the action, a quantity that combines the system's kinetic and potential energies.

2. How is Lagrangian mechanics different from Newtonian mechanics?

Lagrangian mechanics is a more general and powerful approach than Newtonian mechanics. While Newtonian mechanics is based on the concept of force and the laws of motion, Lagrangian mechanics is based on the principle of least action and does not explicitly use the concept of force. It also allows for the use of generalized coordinates, making it more suitable for complex systems with many degrees of freedom.

3. What is the role of a spring in Lagrangian mechanics?

A spring is often used as an example in Lagrangian mechanics to demonstrate the application of the principle of least action. It is a simple system with only one degree of freedom, making it easier to understand and calculate. The spring's potential energy is described by Hooke's law, which relates the force applied to the displacement of the spring from its equilibrium position.

4. How does a bar fit into Lagrangian mechanics?

A bar, or a rigid body, can also be described using Lagrangian mechanics. In this case, the system's kinetic energy is described by the rotation of the bar around its center of mass, and the potential energy is described by the force of gravity acting on the bar's center of mass. The Lagrangian equations of motion can then be used to determine the bar's motion and rotation.

5. What are some real-world applications of Lagrangian mechanics?

Lagrangian mechanics has many practical applications in physics and engineering. It is commonly used in the study of celestial mechanics, such as the motion of planets and satellites, as well as in the design and analysis of mechanical systems like robots and spacecraft. It is also used in quantum mechanics and field theory to describe the behavior of particles and fields.

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