How to Find Eigenvalues and Eigenvectors of Masses on Springs in a Circle?

In summary, the conversation discusses a problem involving three masses connected by springs in a circle and confined to move in that circle. The task is to find the eigenvalues and eigenvectors of the system, and there is uncertainty about how to approach the question. Some suggestions include using F=ma on each mass and potentially ending up with separable equations for θ and φ, which can then be solved to find the SHM.
  • #1
technial
3
0
Hello, I am a having a problem finding the solution of the follow:

3 equal masses are connected by springs in a circle, and are confined to move in the circle. Find the eigenvalues and eigenvectors of the system.

I'm really at a bit of a loss here, I don't know whether you begin by assuming SHM occurs and get answers from there or something completely different. The question is from a general paper so you can use anything to get an answer! Any help would be greatly appreciated.
Thanks everyone.
 
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  • #2
technial said:
… I don't know whether you begin by assuming SHM occurs and get answers from there or something completely different …

Hello technial! :smile:

Call the spring lengths θ φ and (2π - θ - φ).

Do F = ma on each mass, and you should end up with an equation in θ'' φ" θ and φ which I suspect (i haven't worked it out) will be separable into two x'' = -kx equations with x (I'm guessing) = θ ± φ.

So no need to make assumptions :wink: … the SHM should come out of the elementary F = ma equations.

What do you get? :smile:​
 

FAQ: How to Find Eigenvalues and Eigenvectors of Masses on Springs in a Circle?

What are eigenstates of masses on springs?

Eigenstates of masses on springs refer to the specific states or configurations of a mass or object attached to a spring that allows it to oscillate or vibrate at a specific frequency.

What is the significance of eigenstates in studying masses on springs?

Eigenstates play a crucial role in understanding the dynamics of masses on springs as they represent the natural frequencies of oscillation and can help determine the behavior of the system under different conditions.

How are eigenstates of masses on springs calculated?

Eigenstates are calculated using mathematical equations and formulas, such as the eigenvalue equation, which take into account the mass, spring constant, and other factors of the system.

What are the applications of eigenstates in the real world?

Eigenstates of masses on springs have various real-world applications, such as in designing and analyzing mechanical systems, understanding the behavior of molecules in quantum physics, and even in musical instruments.

Can eigenstates change over time?

Yes, eigenstates can change over time if there are external forces or factors that affect the system, such as changes in mass or spring stiffness. However, in a stable system with no external influences, the eigenstates will remain constant.

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