Exploring Coupled Masses with Hooke's Law: Homework Equations and Solutions

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In summary, the conversation discusses finding the angular frequencies for each of the normal modes of vibrations in a system with 3 masses connected by springs of the same spring constant (k) and connected to solid walls via 2 more springs. Equations of motion for each of the masses are provided and the user later solved the problem.
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Homework Statement


There are 3 masses connected by springs of the same spring constant (k). The end masses are connected to solid walls via 2 more springs. Assuming simple harmonic motion find the angular frequencies ([tex]\omega[/tex]) for each of the normal modes of vibrations...

Homework Equations


The Attempt at a Solution


I just need help getting the equations of motion for each of the masses, I can't seem to get them right
 
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  • #2
What have you got so far? Posting your work may assist everyone in spotting out your mistake :smile:
 
  • #3
So far I have got:

Mass A:

ma[tex]_{A}[/tex] + kx[tex]_{B}[/tex] - kx[tex]_{C}[/tex] = 0

Mass B:

ma[tex]_{B}[/tex] + kx[tex]_{A}[/tex] - 2kx[tex]_{B}[/tex] + kx[tex]_{C}[/tex] = 0

Mass C:

ma[tex]_{C}[/tex] + kx[tex]_{A}[/tex] - kx[tex]_{B}[/tex] + 2kx[tex]_{C}[/tex] = 0

Thats the best I can do... :S

Sorry, the superscripts are meant to be subscripts XD
 
  • #4
Could you please describe or draw the system? I cannot deduce any system that matches your equations.
 
  • #5
hey, sorry for not putting up a diagram. didn't think to do that XD

It's okay though, I worked it out finally.

Thanks for trying to help :)
 

FAQ: Exploring Coupled Masses with Hooke's Law: Homework Equations and Solutions

What is the concept of 3 coupled masses?

The concept of 3 coupled masses refers to a system of three masses that are connected to each other by springs, ropes, or other means. This system is often used in physics and engineering to study the behavior of interconnected objects.

How do you calculate the forces in a 3 coupled mass system?

The forces in a 3 coupled mass system can be calculated using Newton's Second Law of Motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In this system, the forces can be determined by considering the masses, the spring constants, and the displacements of each mass.

What are the different types of motion that can occur in a 3 coupled mass system?

There are three main types of motion that can occur in a 3 coupled mass system: oscillatory motion, where the masses move back and forth in a periodic manner due to the springs; coupled motion, where the masses move together as a unit due to the connections between them; and chaotic motion, where the system behaves in an unpredictable manner due to the complex interplay of forces.

How does the behavior of a 3 coupled mass system change with different masses and spring constants?

The behavior of a 3 coupled mass system can vary greatly depending on the masses and spring constants used. For example, increasing the mass of one of the masses will result in a slower oscillation period, while increasing the spring constant will result in a faster oscillation period. Changes in either of these parameters can also affect the amplitude and frequency of the motion.

What real-world applications can 3 coupled mass systems be used for?

3 coupled mass systems have many practical applications, such as in shock absorbers for vehicles, earthquake-resistant building designs, and in the study of molecular vibrations in chemistry. They also have applications in the fields of robotics, biomechanics, and aerospace engineering, among others.

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