# How to Find Normal Mode Frequencies for Three Masses in an Equilateral Triangle?

• eventhorizonof
In summary, Three equal masses arranged in an equilateral triangle are connected by 'springs' with force constants 'k'. The coordinates of the masses are given, and the task is to find the normal mode frequencies. The potential energy is set up using the formula U = 1/2 * k (x_2 - x_3)^2, but there are more terms needed to fully describe the potential. There are six degrees of freedom, with two being translations and one being rotation. The remaining three are vibrational modes, including a "breathing" mode. Neglecting rotation, the potential can be expressed as U=\frac{k}{2}((\vec x^{(1)}-\vec x^{(2)})
eventhorizonof
Three equal masses arranged in a equilateral triangle are connected by 'springs' with force constants 'k'

the coordinates of the masses are:

mass 1 at [0, $$\sqrt{3}$$/2*L]
mass 2 at [L/2, 0]
mass 3 at [-L/2,0]

find the normal mode frequencies.

The only part i am having trouble with is setting up the potential energy i know what to do after I have the potential energy.

so far i have

U = 1/2 * k (x_2 - x_3)^2

i know there are more terms in the potential but i am having trouble projecting the deviations from equilibrium onto the diagonals.

well, there's 6 degrees of freedom, so there are 6 normal modes. 2 are translations as a whole and 1 is rotation as a whole. there are 3 vibrational mode, one of which is the "breathing" mode, and then there are 2 others.

neglecting all rotation what will the other terms of the potential be?

$$U=\frac{k}{2}((\vec x^{(1)}-\vec x^{(2)})^2+(\vec x^{(1)}-\vec x ^{(3)})^2+(\vec x^{(2)}-\vec x^{(3)})^2)$$

in your notation vector x 1 is the location of the first mass [x1,y1] , x 2 second mass [x2,y2],...

right?

yeah

## 1. What is "Classical Mechanics 3 Masses"?

"Classical Mechanics 3 Masses" is a branch of physics that deals with the motion of three masses under the influence of forces, such as gravity, friction, and tension.

## 2. What are the main principles of Classical Mechanics 3 Masses?

The main principles of Classical Mechanics 3 Masses are Newton's laws of motion, the law of universal gravitation, and the principle of conservation of energy and momentum.

## 3. How is Classical Mechanics 3 Masses different from Classical Mechanics?

Classical Mechanics 3 Masses focuses specifically on the motion of three masses, whereas Classical Mechanics studies the motion of a single object or system of objects.

## 4. What are some real-life applications of Classical Mechanics 3 Masses?

Classical Mechanics 3 Masses has many real-life applications, such as predicting the motion of planets in our solar system, analyzing the behavior of pendulums, and understanding the motion of objects in a simple pulley system.

## 5. What mathematical tools are used in studying Classical Mechanics 3 Masses?

Classical Mechanics 3 Masses uses a variety of mathematical tools, including calculus, vectors, and differential equations, to analyze the motion of three masses and calculate forces and accelerations.

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