Solve Coupled Oscillator Homework: Find 2 Eigenfrequencies

In summary, we can find the eigenfrequencies of a thin hoop with a small mass attached to it by setting up the Lagrangian of the system and using the Euler-Lagrange equations.
  • #1
roeb
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Homework Statement


A thin hoop of radius R and mass M oscillates in its own plane with one point of the hoop fixed. Attached to the hoop is a small mass M constrained to move (in a frictionless manner) along the hoop. Consider only small oscillations, and show that the eigenfrequencies are blah blah blah (two eigenfrequencies).

Homework Equations


The Attempt at a Solution



My difficulties are in setting up this problem. I believe that I am picturing the system correctly, but I can't quite figure out how to do it. I need to find the Lagrangian of the system first, but I am having a hard time with the kinetic energy part.The oscillation of the hoop if I am not mistaken will be like that of a pendulum.

Hoop: T = 1/2*Iw^2 = 1/2 m * R^2 * [tex]\omega ^2[/tex]
Small Mass: T = 1/2 mR^2 [tex]\theta '[/tex]

I have set up theta as the angle between the center of the hoop and the position of the small mass. The problem is that I don't quite know how to get the second generalized coordinate -- I am assuming there are two generalized coordinates because this is a coupled oscillator problem and I am given two eigenfrequencies.

I am tempted to say that [tex]\omega = R \theta ' [/tex] but that doesn't yield a correct answer and I don't think it's right to begin with...

Any help?
 
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  • #2


First, let's define the two generalized coordinates for this system. Let x be the displacement of the small mass along the hoop, and let θ be the angle of the hoop with respect to its equilibrium position. Now, we can write the Lagrangian as:

L = T - V

where T is the kinetic energy and V is the potential energy. Let's start with the kinetic energy:

T = 1/2m(dx/dt)^2 + 1/2I(dθ/dt)^2

where m is the mass of the small mass and I is the moment of inertia of the hoop. Now, we need to express dx/dt and dθ/dt in terms of our generalized coordinates. We can do this by using the chain rule:

dx/dt = R(dθ/dt)

dθ/dt = θ'

where θ' is the angular velocity of the hoop. Substituting these expressions into our kinetic energy equation, we get:

T = 1/2mR^2(θ')^2 + 1/2I(θ')^2

Now, let's move on to the potential energy. The small mass has a potential energy due to gravity, given by mgh, where h is the height of the small mass above its equilibrium position. Since the hoop is fixed at one point, the height h is given by Rθ. The hoop also has a potential energy due to its rotation, given by 1/2kθ^2, where k is the spring constant of the hoop. Putting this all together, we get:

V = mgRθ + 1/2kθ^2

Now, we can write the Lagrangian as:

L = 1/2mR^2(θ')^2 + 1/2I(θ')^2 - mgRθ - 1/2kθ^2

To find the equations of motion, we can use the Euler-Lagrange equations:

d/dt(∂L/∂θ') - ∂L/∂θ = 0

d/dt(∂L/∂x') - ∂L/∂x = 0

Solving these equations will give us the eigenfrequencies of the system, which will be two oscillatory modes. I hope this helps!
 

FAQ: Solve Coupled Oscillator Homework: Find 2 Eigenfrequencies

What is a coupled oscillator?

A coupled oscillator is a system of two or more oscillators that are connected or linked together in some way, such as through a physical coupling or through a shared energy source. This results in the oscillators affecting each other's motion and creating a more complex motion overall.

What is an eigenfrequency?

An eigenfrequency, also known as a natural frequency, is the frequency at which a system naturally oscillates without any external forces acting on it. In the case of coupled oscillators, there are two eigenfrequencies corresponding to the two oscillators in the system.

Why is it important to find eigenfrequencies in coupled oscillators?

Finding the eigenfrequencies in a system of coupled oscillators allows us to understand the natural motion of the system and how the oscillators interact with each other. This information can help us predict and control the behavior of the system, as well as design and optimize systems for specific purposes.

How do you solve for the eigenfrequencies in coupled oscillators?

The eigenfrequencies can be found by solving the equations of motion for the coupled oscillators using techniques such as matrix diagonalization or the method of normal coordinates. These methods involve finding the characteristic equation and solving for the eigenvalues, which correspond to the eigenfrequencies.

Are there any real-world applications of coupled oscillators?

Yes, coupled oscillators can be found in a variety of systems and devices, such as pendulum clocks, musical instruments, and electronic circuits. They are also used in fields such as engineering, physics, and biology to model and study complex systems. Understanding coupled oscillators is important for designing and improving technologies and systems in various industries.

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