# Why Does the System Have a Zero Frequency in the Normal Mode Oscillations?

In summary, the problem involves two equal masses connected by two massless springs and constrained by a frictionless pivot. The goal is to find the normal mode frequencies of oscillation about stable equilibrium. The Lagrangian, equations of motion, equilibrium point, and normal mode frequencies have been determined. However, one of the frequencies is zero, which suggests that the system is displaced from equilibrium and does not oscillate or diverge. This could potentially be explained by the displacement of the center of mass.

## Homework Statement

Two equal masses are connected by two massless springs of constant k and nat. length l. The masses are constrained by a frictionless tube on a pivot, (also massless) so that they remain colinear with the pivot. The pivot subtends angle theta with the vertical. The 1st mass is at distance r1 from the pivot (therefore the 1st spring has length r1) and the second mass is at r2 from the pivot.
We need to find the normal mode frequencies of oscillation about stable equil.

## The Attempt at a Solution

I have found the Lagrangian, equations of motion, equilibrium point and normal mode frequencies. The only problem is that I get one frequency of zero, and I don't understand physically what is going on (this is not an oscillating solution but a constant solution). This appears to imply that the system is displaced from equilibrium and just stays at the new position (i.d does not oscillate about equil. or diverge from equil.), although physically this seems not to make sense.

I would think that the zero frequency solution is just for when there is no initial displacement.

The zero frequency comes from displacement of the center of mass.
In 3D you get 6 zero frequencies corresponding to 3 displacements of the C.O.M and 3 rotations about the C.O.M.

## What are normal mode frequencies?

Normal mode frequencies refer to the different vibrational frequencies that a system or object can exhibit. These frequencies depend on the physical properties and geometry of the object.

## How are normal mode frequencies calculated?

Normal mode frequencies are typically calculated using mathematical equations and models, such as the equations of motion or the finite element method. These calculations take into account the mass, stiffness, and damping of the object to determine its natural frequencies.

## Why are normal mode frequencies important?

Normal mode frequencies are important because they can provide valuable information about the physical properties and behavior of a system. They can also be used to identify potential resonances or weaknesses in a structure.

## What factors can affect normal mode frequencies?

The main factors that can affect normal mode frequencies are the physical properties and geometry of the object, as well as any external forces or disturbances acting on the object. Changes in temperature, pressure, or material properties can also affect these frequencies.

## How are normal mode frequencies used in real-world applications?

Normal mode frequencies have a wide range of applications in various fields such as engineering, physics, and chemistry. They are used to analyze and design structures, study the behavior of materials, and identify the presence of defects or damage in objects. They are also important in fields such as seismology and acoustics, where they can help predict and mitigate potential hazards and hazards.

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