Springs and masses connected together in a line

In summary, the conversation discusses a problem involving four masses connected by three springs and the need to write down the f=ma equations of motion for all four masses. The person expresses frustration with not being able to remember how to solve the problem and questions the necessity of memorizing it. They ask for help and mention the possibility of using Hamiltonian or Lagrangian mechanics.
  • #1
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Homework Statement



So there are 4 masses in a line. They are connected by 3 springs between them, denoted by k_12 & l_12 (spring coefficient + natural extension length), k_23 & l_23, etc.

Homework Equations



Write down the f = ma equations of motion for all 4 masses.

The Attempt at a Solution



See my image.



This problem really annoys me because I always learn how to do it for the test but never remember. I think I've gone through 3 courses where this problem came up and every time I could never remember. Well now it's just a random exercise in a linear algebra text I'm reading and I wouldn't mind learning it right once and for all. After all I'm supposed to be an engineer.

Thank you for any help.
 

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  • #2
Well do you really need to memorize the problem? Besides, the time you do, someone will give you one with 5 or 3 masses, or springs with varying k :)

It's like this one only with four masses instead of two. (You could use the Hamiltonian or Lagrangian mechanics of course.)
 

1. What is the relationship between the mass and the spring constant in a system of springs and masses connected together in a line?

The mass and the spring constant have an inverse relationship in this system. As the mass increases, the spring constant decreases, and vice versa. This means that a smaller mass will result in a stiffer spring, while a larger mass will result in a more flexible spring.

2. How does the number of springs affect the overall behavior of the system?

The number of springs connected together in a line determines the overall stiffness of the system. The more springs that are added, the stiffer the system will be. This is because each spring contributes to the overall spring constant of the system.

3. What is the role of damping in a system of springs and masses connected together in a line?

Damping is the force that counteracts the motion of the system, reducing its amplitude and slowing down its oscillation. In a system of springs and masses, damping is important in controlling the rate at which the system returns to equilibrium after being disturbed.

4. What is the difference between series and parallel connections of springs and masses?

In a series connection, the springs and masses are arranged one after the other, creating a linear system. In a parallel connection, the springs and masses are connected side by side, creating a more complex system. The way in which the forces are distributed and the overall behavior of the system can differ between series and parallel connections.

5. How does the amplitude of the system's motion change over time?

The amplitude of the system's motion decreases over time due to the effects of damping. As the system oscillates, the damping force dissipates some of the energy, causing the amplitude to gradually decrease until the system reaches equilibrium.

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