Force on Each Mass in a Two-Mass Spring System

[Hereformore]

Homework Statement

Two masses attached to opposite ends of a spring. What is the Force felt by each mass if each mass is stretched by Xmeters in opposite directions away from each other.

Homework Equations

F=-kx

The Attempt at a Solution

So my book says the answer is F=k*2x.

Each mass has the same mass and is stretched from their equilibrium by the same amount X. so the spring is stretched a total amount of 2x.

So since the spring force is proportional to the displacement of the spring/stretch of the spring, it would make sense that the spring force would be F=k*2x since 2x is the total displacement.

But what I'm havin gtrouble understanding is how to distinguish between situations like this when the spring is stretched by the same amount on both sides and problems where the same force is applied to both sides.

When the same force is applied to both sides, you use the same formula Fspring = kx , but for the force applied you only consider one of the applied forces rather than both combined, on the notion that the other side is what's keeping the spring from accelerating, Newtons third law.

But when you actaully compress the spring or stretch it by a certain amount, even if you stretch it from both ends, you consider the total stretch added from both ends to consider the recoil force, am i right?

And lastly, (Thanks guys haha) is the reason why you don't consider both forces when you apply the same force to both sides that compression could only occur if there was a counter force? Because the lack of movement allows for the compression to occur? Does that make sense? its weird because it's like we're using the applied force twice,
once to cancel out the counter force from the other end of the spring (i.e. from a wall or from someone on the other end of the spring)and then again when determining the compression caused by said force.

 

[Chestermiller]

I'm having a little trouble understanding your question. Can you give a specific example of the two situations you are referring to?

Chet

 

[haruspex]

A lot of people make the mistake of thinking that if there's force T pulling at each end then the tension is 2T. But in a static set-up, it is not possible to pull at one end only.
In the OP set-up, it might help to think of the spring as two (half-length) springs fixed at the centre. If you chop a spring of constant k in half then each half will have a constant 2k.