Solving Lagrangian Mechanics: Rod and Bean System Explained

In summary, the problem involves finding the lagrangian for a system consisting of a rod sliding along a vertical pole, with a bean sliding along the rod. The kinetic and potential energies for the rod are calculated, but the bean's velocity is more complicated as it is in an accelerating frame of reference. It can be found by considering the relative velocity of the bean with respect to the point on the rod underneath it, and adding it to the velocity of that point on the rod. This absolute velocity can then be used in the lagrangian for the system.
  • #1
Kate R
2
0
I'm stuck on a problem with lagrangian mechanics.

Here's the problem;

One end of a rod slides along a vertical pole while the other end
slides a long a horizontal pole. At the same time a bean slides a long
the rod. Find the lagrangian for the system.

And this is what I worked out so far;

The kinteic energy for the rod would be;

T = 1/2* M Vcm^2 + 1/2* Icm(theta dot)^2

Where M is the mass of the rod
Vcm is the velocity for the center of mass
Icm is the moment of intertia for center of mass; Icm = M/12*(L/2)^2
L is the length of the rod
theta is the angle between the vertical pole and the rod

The potential energy for the rod would be;

V = MgL*cos(theta)

So far I think it's ok because the velocity of the rod is relative to a fixed intertial frame, but I don't know what to do with the bean.

The beans velocity would be the veclocity of the rod plus the beans velocity relative to the rod, right?

I would be very greatful if someone could give me a little help with
this.
 
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  • #2
Yes, jusst add the two velocites vectorially.
 
  • #3
And that is easier said than done (at least to me).The bean is in an accelerating frame of reference that is rotating around it's own axis AND in a circle (with the redius L/2).
So is it not really three velocities? The beans velocity, the velocity of the rod around it's own axis and the velocity of the rod in the circle

And how do I find the beans velocity relative to the fixed frame of reference?
 
  • #4
Kate R said:
And that is easier said than done (at least to me).The bean is in an accelerating frame of reference that is rotating around it's own axis AND in a circle (with the redius L/2).
So is it not really three velocities? The beans velocity, the velocity of the rod around it's own axis and the velocity of the rod in the circle

And how do I find the beans velocity relative to the fixed frame of reference?

Picture it this way. At any instant of time, let B denote the point on the rod right under the bean, A denote the rod.

(i) v_B = v_A + v_(B/A), where v_B is vel of the point on the rod under the bean, v_A is the vel of the bean, and v_(B/A) is due to the rotation of the rod.
(ii) Now, the relative velocity of the bead wrt to the rigid point on the rod under it is always going to be in a direction along the rod. So, the absolute velocity of the particle is going to be v_B + v_(P/B), where v_B as before, is the vel of the point on the rod under the bean, and v_(P/B) is the vel of the particle with respect to the point (which is constrained to move along the rod.

You need to use this absolute velocity in the Lagrangian of the system.
 
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1. What is Lagrangian Mechanics?

Lagrangian Mechanics is a mathematical framework used to describe the motion of a system of particles or rigid bodies. It is a variation of classical mechanics that uses a generalized coordinate system and the principle of least action to determine the equations of motion.

2. How is Lagrangian Mechanics used to solve the Rod and Bean system?

In the Rod and Bean system, Lagrangian Mechanics is used to determine the equations of motion for the rod and the bean, which are connected by a string and can move freely in a horizontal plane. By using the principle of least action, the equations of motion can be derived and solved to determine the positions, velocities, and accelerations of the rod and bean at any given time.

3. What are the advantages of using Lagrangian Mechanics to solve the Rod and Bean system?

One of the main advantages of using Lagrangian Mechanics to solve the Rod and Bean system is that it provides a more intuitive and simpler approach compared to traditional methods such as Newton's laws of motion. It also allows for the inclusion of constraints and external forces in the system, making it a more versatile tool for solving complex systems.

4. Are there any limitations to using Lagrangian Mechanics for the Rod and Bean system?

While Lagrangian Mechanics is a powerful tool for solving the Rod and Bean system, it does have some limitations. It is most effective for systems with a small number of degrees of freedom and does not account for dissipative forces such as friction. Additionally, it may not be suitable for systems with highly nonlinear behavior.

5. What are some real-world applications of solving the Rod and Bean system using Lagrangian Mechanics?

The Rod and Bean system is a simplified model of more complex systems, such as pendulums and double-pendulums, which are commonly found in physics and engineering. By understanding and solving the Rod and Bean system using Lagrangian Mechanics, we can apply the same principles to analyze and design more complex systems in various fields, including robotics, aerospace engineering, and biomechanics.

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