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This problem refers particulary to the model in my attachment.

The blocks are free to move but only along the x-axis. The springs connecting adjacent blocks have spring constant k1, while the two outer springs have stiffness k2. All the springs have a rest lengt of one. Write the matrix equation for the equilibrium position of the blocks and plot the total length of the system as a function of k1/k2

The model of the problem is throwing me off here because I am used to spring systems being fixed to a wall at one end and having something else attached to it on the other end. The way I'm approaching the problem is by finding out the individual force on each block.. F1 + F2 + F3 + F4. However, I keep getting stuck in a loop by trying to focus on an individual block because each block seems dependent on the other ones at all times.

I also can't picture the equation f(k1/k2) at all in my head.

I know it's not much but this is what I have so far.

Springs in a parallel

kp = k1+k2

Springs in a series

1/ks = 1/k1 + 1/k2

The length of the system should be related to the equilibrium length when all springs are at rest.

so:

L_equilibrium = L_1e + L_2e + L_3e

(these are the equilibrium distances between each block)

I'm guessing that the force on each end is something like:

F = -k_springtotal(L_system - L_equilibrium)

I would really appreciate it if someone could give me a push start with this problem. I mainly want to understand what force the individual blocks are experiencing if displaced. Also, I don't really understand the question that well either to be completely honest. Particulary the part about finding the length based on the ratio of constants? How do I get the individual forces on each block if I'm not given specific constants for k1 and k2?

Any help is greatly appreciated.

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# Spring-mass system with nearest neighbor couplings

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