Three spring two mass system, compression of the middle spring

[Leb]

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 $x_{1}$ and $x_{2}$ respectively ($x_{1}$ is to the -ve dirrection) ?

The attempt at a solution

$F_{1,1}=k_{1}.x_{1}$
since the spring needs to return to the equilibrium position which is to the right.
$F_{2,1}=k_{2}.(x_{1}+x_{2})$
since the spring needs to return to the equilibrium position which is to the right. (Spring is streched by x1+x2)
$F_{2,2}=-k_{2}.(x_{1}+x_{2})$
since the spring needs to return to the equilibrium position which is to the right.
$F_{3,2}=-k_{3}.x_{2}$
since the spring needs to return to the equilibrium position which is to the left.

Combining all of this I get the following:
$m_{1} \ddot{x_{1}} =F_{1,1}+F_{2,1}=k_{1}.x_{1}+k_{2}(x_{1}+x_{2})$
$m_{2} \ddot{x_{2}} =F_{2,2}+F_{3,2}=-k_{2}(x_{1}+x_{2})-k_{3}.x_{2}$

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 $x_{1}$ 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.

 

[obafgkmrns]

Looks like you have a mixed sign convention with x1 positive right and x2, e.g. Changing the initial displacements won't change the equations of motions as long as the sign convention is the same.

 

[Leb]

Thank you for the reply !

I am confused with one more thing, the concept of tension in a spring. I do not really understand how it works in a spring, when it is not supported by one side.

a) Should I consider the spring not to move if its center is not moving ?

b) Could someone please explain me the following situations ?

1) Compress a spring on both sides, and release both sides simultaneously.

Will FOn End Of Spring by Restoring Force on both sides(force pointing away from the center of the spring) will be 'coupled' with a Tension which would point towards the center of the spring, with a magnitude that increases as the spring expands until some point (to stop the spring from expanding infinitely) ?

2) Stretch a spring on both sides and release both sides simultaneously.

Will the restoring force (pointing towards the center) be 'coupled' with tension pointing from the center (again with increasing magnitude until some point)

3) I think what I really do not understand is how to view forces in a spring.
For instance in the image below, it states that after some perturbation of the masses to the right, the spring in the middle will have tension pointing towards the center...

OK, so for m1 to move to the right it is clear that we need a force acting on the mass to the right to be greater than on the left. However, I do not get, why the tension is pointing to the center in in the middle spring ( since the spring is compressed from the left and expanded from the right, how can we say, that the center of the spring did not move ?)

http://img535.imageshack.us/img535/8139/massspring.jpg

 

[obafgkmrns]

You might be puzzling over what's called a spring's "effective mass." Here's a link: http://depa.fquim.unam.mx/amyd/archivero/Paraosciladorarmonico_1127.pdf

 

[asteriatic]

I would say that your solution seems correct based on the information provided. However, I cannot fully confirm its accuracy without further information about the specific values of the masses and spring constants.

Regarding your question about the middle spring being compressed or expanded, the generalization you mentioned may be misleading. The behavior of the spring depends on the direction and magnitude of the forces applied to it. In this case, the forces applied to the middle spring are in opposite directions, resulting in a net force of zero and no change in its length. It is important to consider all the forces acting on the system to accurately analyze its behavior.