Here is a description of the motion of two springs in series. The premise is that the force on the two springs is the same.
This is derived from the following reasoning: when I pull the mass with a certain force $F$ at some point I reach the equilibrium position.
So every piece is stationary and in particular the point between the springs is stationary: we conclude that the forces applied in that point (the two elastic forces) sum up to zero. We derive that the elastic forces are equal and they're equal to $F$.
The problem is: this is valid in the equilibrium position. I can't understand why this conclusion is extended for every position in order to derive the equation of motion:
Answer
Because the springs are considered massless. So if there were a force difference between the ends, you would get infinite acceleration. And this is valid not only for the ends, but for any two points, if there is no mass between them.
The issue is somehow explained in this post:
Is the tension in both ends the same (on a massed string)?
And indeed, if the springs are massive (easier: if you have one massive spring), you get another value for the period. That's the same issue - why should we take the tension inside one string to be equal?
So, to sum it up: if the part between two points is massless, then the force on both sides (i.e. the strain in the end-points) has to be exactly equal. Thus, the strain inside the spring is equal everywhere. Only this fact allows you to talk about the elastic force of the spring. Otherwise is would change througout the spring.
The reasonig is the same as for a static situation; equal forces are implied by a finite mass and zero acceleration -- as well as by zero mass and finite acceleration.
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