Two rods connected, with a sliding mass attached to a spring.lagrange

In summary, the conversation is about a rigid T structure with a mass connected by a spring, rotating around a horizontal plane with constant frequency ω. The position of the mass along the long rod, r(t), is being discussed and there are three cases to consider based on the size of ω2 compared to k/m. There is a special value of ω that is being sought after and the conversation also involves setting up the Lagrangian and finding the kinetic and potential energies for the system.
  • #1
mjmontgo
10
0
A rigid T consists of a long rod glued perpendicular to another rod of length l that is pivoted about the
origin. The T rotates around in a horizontal plane with constant frequency ω. A mass m is free to slide along the long
rod and is connected to the intersection of the rods by a spring with spring constant k and a relaxed length zero. Find r(t)
where r is the position of the mass along the long rod from the intersection. [You will find there are three cases to
consider: depending on the size of ω2 compared to k/m.] There is a special value of ω; what is it and why is it special?

-----------------------------------------------------------

Im basically having trouble setting up the lagrangian.
My kinetic energy so far is T=1/2mv^2+1/2Lω^2+1/2kx^2

The potential energy is throwing me through a loop as well
i have U=mghsin(ωt) although i know this is either incorrect or their is more to it.
I feel like i over think these lagrangian questions but I am never very sure. any pointers on the kinetic and potential energies would be appreicated. solving the lagrangian and analying the special cases of ω i should be ok with, although your input it always welcomed.

thank you in advance for your time and assistance.
If a diagram is needed i could provide a link.
 
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  • #2
I am solving the same problem, and I cannot find a "comfortable" general coordinate for the mass. Also you are missing your spring potential term in the potential of the Lagrangian.
 

FAQ: Two rods connected, with a sliding mass attached to a spring.lagrange

What is the Lagrangian for this system?

The Lagrangian is a function that describes the energy of a system in terms of its generalized coordinates and their derivatives. In this case, it is given by L = T - V, where T is the kinetic energy of the system and V is the potential energy. The kinetic energy is given by the sum of the rotational and translational kinetic energies of the two rods and the sliding mass. The potential energy is due to the gravitational potential energy of the system.

How do you derive the equations of motion for this system using Lagrange's equations?

The equations of motion for this system can be derived using Lagrange's equations, which state that the time derivative of the partial derivative of the Lagrangian with respect to the generalized coordinates is equal to the partial derivative of the Lagrangian with respect to the corresponding generalized velocities. This results in a set of second-order differential equations that can be solved to determine the motion of the system over time.

What is the significance of the spring in this system?

The spring in this system provides a restoring force that acts on the sliding mass, causing it to oscillate back and forth between the two rods. This spring-mass system is a common model used to study simple harmonic motion and oscillatory behavior in physics.

How does the length of the rods affect the dynamics of this system?

The length of the rods affects the dynamics of the system by changing the moment of inertia and the distance between the two rods, which both contribute to the overall kinetic energy of the system. A longer rod will have a greater moment of inertia and a larger distance between the two rods, resulting in a larger kinetic energy and potentially different oscillatory behavior.

Can this system be extended to include more than one spring?

Yes, this system can be extended to include multiple springs and sliding masses, resulting in a more complex but still solvable system. This type of system is often used to model more complicated oscillatory behavior, such as coupled oscillators or systems with multiple degrees of freedom.

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