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**Homework Statement**

I decided to revise my mechanics course and came across a problem involving a system with three springs (say, different k's) and two masses (say different).

The s1-m1-s2-m2-s3 system has the outer springs connected to walls of infinite mass.

I was interested in what happens to the system (i.e. in terms of forces) if we were to pull the masses in

*opposite*directions by displacement [itex]x_{1}[/itex] and [itex]x_{2}[/itex] respectively ([itex]x_{1}[/itex] is to the -ve dirrection) ?

**The attempt at a solution**

[itex] F_{1,1}=k_{1}.x_{1}[/itex]

since the spring needs to return to the equilibrium position which is to the right.

[itex]F_{2,1}=k_{2}.(x_{1}+x_{2})[/itex]

since the spring needs to return to the equilibrium position which is to the right. (Spring is streched by x1+x2)

[itex]F_{2,2}=-k_{2}.(x_{1}+x_{2})[/itex]

since the spring needs to return to the equilibrium position which is to the right.

[itex]F_{3,2}=-k_{3}.x_{2}[/itex]

since the spring needs to return to the equilibrium position which is to the left.

Combining all of this I get the following:

[itex]m_{1} \ddot{x_{1}} =F_{1,1}+F_{2,1}=k_{1}.x_{1}+k_{2}(x_{1}+x_{2}) [/itex]

[itex]m_{2} \ddot{x_{2}} =F_{2,2}+F_{3,2}=-k_{2}(x_{1}+x_{2})-k_{3}.x_{2}[/itex]

Does this seem correct ?

I was trying to compare this with the usual example, where masses are moved to one direction (which stated, that the middle spring can be either expanded

*OR*compressed, and I would get the same, if I were to replace the value of [itex]x_{1}[/itex] to its negative. However, this would imply that the length of spring 2 actually decreases (instead of increasing), so, is their generalization (i.e. works for both compressed and expanded spring) wrong or is there something I do not get ?

Thank you for your time.

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