Frequency of small oscillations

In summary, the conversation is about finding information on relating the frequency of small oscillations to the second derivative of potential energy. The context is a problem involving two masses connected by a string and the goal is to find the frequency of small oscillations. The solution involves using the approximation method of \omega^{2}=\frac{1}{M_{eff}}\frac{ \partial^{2}U_{eff}}{\partial r^{2}}\mid_{r=r_0}} where ro is the stable point and Meff = M+m. The conversation also discusses the use of Taylor series to find the effective mass and stiffness of a system for small oscillations.
  • #1
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Does anyone know where I can get some information on how you can relate the frequency of small oscillations to the second derivative of potential energy. I saw this done recently in a qualifying exam level problem but I do not remember learning this method and it is not in my classical dynamics book. See below if you want a more extensive context for this question.

I solved a problem recently where you were given two masses m and M connected by a string. The first mass was set rotating on a frictionless table. The string passed through a hole in the center of the table allowing the second mass to hang vertically under gravity. I was asked to:

1) Set up the Langrangian and derive eqns of motion.
2) Show that the orbit is stable with respect to small changes in orbit.
3) Find the frequency of small oscillations.

I was able to do the first two without any problem but got stuck on the third. The d.e. was too messy to solve by hand in order to acquire the freqency. I looked at the solution and they used the approximation:
[itex]\omega^{2}=\frac{1}{M_{eff}}\frac{ \partial^{2}U_{eff}}{\partial r^{2}}\mid_{r=r_0}}[/itex]
where ro is the stable point and Meff = M+m
Where can I get more information discussing this approximation method?
 
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  • #2
If it's any help, for a simple spring and mass system

PE = (1/2)Kx^2

d^2(PE)/dx^2 = K

And w^2 = K/M.

So the same result would follow for small (linearized) oscillations of any system I suppose, if you expanded the PE and KE as Taylor series to get the effective mass and stiffness of the system about the equilibrium condition.
 

What is the definition of frequency of small oscillations?

The frequency of small oscillations, also known as the natural frequency, is the rate at which a system oscillates around its equilibrium position when subjected to a small disturbance.

What factors affect the frequency of small oscillations?

The frequency of small oscillations is affected by the mass, stiffness, and damping of the system. The greater the mass and stiffness, the lower the frequency, while the greater the damping, the higher the frequency.

How is the frequency of small oscillations calculated?

The frequency of small oscillations can be calculated using the equation f = 1/(2π√(k/m)), where f is the frequency, k is the stiffness of the system, and m is the mass of the system.

What is the relationship between frequency of small oscillations and amplitude?

The frequency of small oscillations is independent of the amplitude of the oscillations. This means that the system will oscillate at the same frequency regardless of how large or small the amplitude of the oscillations is.

Why is the frequency of small oscillations important in scientific research?

The frequency of small oscillations is important in scientific research because it helps scientists understand the behavior of various systems, such as pendulums, springs, and electronic circuits. It also allows for the prediction of future oscillations and the design of stable systems.

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