Lagrangian Mechanics - Pulley System

In summary, the conversation discusses a diagram of a system involving a spring, masses, a pulley, and a string. The question asks for the vertical displacement of each mass and the Lagrangian of the system. The conversation then goes on to solve for the Lagrangian and address a misunderstanding in the solution.
  • #1
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Homework Statement


Here's a little diagram I whipped up in paint:
http://img83.imageshack.us/img83/7625/diagramcj4.th.jpg [Broken]

Sorry about my sucky art skills.

The wiggly line is a spring with spring constant mk and natural length d. The actual length of the spring is y. The two masses to the left have mass m, the mass on the right has mass 2m. The springs are attached by a light string which doesn't slip over the light pulley, which has a radius R which is free to rotate. The angle theta is the angle by which the pulley has rotated since the initial conditions. The red line indicates the position of all the masses at the initial conditions. The initial conditions are y=0, t=0, [tex]\theta[/tex]=0 and everything is at rest.

a) Write down the vertical displacement of each of the three masses from their initial positions in terms of theta and y.

b) Show that the Lagrangian describing the system is up to a constant given by:
L = 2mR[tex]^{2}[/tex][tex]\theta[/tex]'[tex]^{2}[/tex] -mR[tex]\theta[/tex]'y' + m/2 y'[tex]^{2}[/tex] + mgy - (mk/2)(y-d)[tex]^{2}[/tex]

where ' indicates a first derivative with respect to time- can't do the normal dots it seems.

Homework Equations



L = T - V

The Attempt at a Solution


Well for a) I think for the mass on the right its just -R[tex]\theta[/tex]
for the mass on the top left its R[tex]\theta[/tex]
for the mass on the bottom left its -[tex]\theta[/tex] - y

I assume that's right- correct me if I'm wrong!

for b) I can't get what I'm supposed to.
I do not get the -mR[tex]\theta[/tex]'y' term. Here's my thinking:

For the kinetic energy:
1/2mv[tex]^{2}[/tex] for each mass where v is the time derivative of their vertical displacement in a). This gives 2mR[tex]^{2}[/tex][tex]\theta[/tex]'[tex]^{2}[/tex] + m/2 y'[tex]^{2}[/tex]

For the potential energy: -mgy as the total loss in GPE is due to the mass on the bottom left going down by y, the other two masses cancel each other out. The last term is correct, just the potential energy of the spring.

So, my question: Where does the -mR[tex]\theta[/tex]'y' term come from?
hopefully this is fairly clear :/
Thanks (again) :D
 
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  • #2
lol. I'm an idiot. I finally see where its from. This will teach me to skip steps and do too much in my head.

The displacement of the third mass if [tex]R\theta + y[/tex]
The derivative of this wrt time is [tex]R\theta ' + y'[/tex]
This squared is [tex]R^{2}\theta^{2}' + y'^{2} + 2R\theta'y'[/tex]

I really feel like an idiot :)
 

What is Lagrangian Mechanics?

Lagrangian Mechanics is a mathematical formalism used in classical mechanics to describe the motion of a system of particles subject to constraints. It is based on the principle of least action, which states that a system will follow a path that minimizes the action, or the integral of the Lagrangian function.

What is a Pulley System?

A Pulley System is a simple machine that consists of a rope or belt wrapped around a wheel or axle. It is used to change the direction of a force, typically to lift heavy objects.

How does Lagrangian Mechanics apply to a Pulley System?

In a Pulley System, Lagrangian Mechanics is used to describe the motion of the pulleys and the objects they are lifting. It takes into account the constraints of the system, such as the tension in the rope and the mass of the objects, to determine the equations of motion.

What are the advantages of using Lagrangian Mechanics in a Pulley System?

One advantage of using Lagrangian Mechanics in a Pulley System is that it provides a more elegant and concise way to describe the system compared to traditional methods like Newton's Laws. It also allows for more complex systems to be analyzed, such as systems with multiple pulleys and objects.

Are there any limitations to using Lagrangian Mechanics in a Pulley System?

One limitation of using Lagrangian Mechanics in a Pulley System is that it assumes the system is in equilibrium, meaning that the objects are not accelerating. It also does not take into account factors such as friction and air resistance, which can affect the motion of the system.

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