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I am following the advice given in community wiki answer and making a request to the Physic.SE community to reopen this question.

The reason I believe the question should be reopened is because it is not a homework question but more of a conceptual question. This question falls in the type 4 category as per ACuriousMind's post here, which most members accept are worthy of being kept open. The OP had provided proper justification for both the type of derivation and is not a solve-my-problem type of question.

As such I believe that the question should be reopened.

P.S: I have also edited the question and improved the LaTeX.

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Honestly I don’t think the item 4) you suggest applies. This item states that check-my-work question could be allowed when they are of the type

It seems as if step X is wrong? But it should be right because of Y, so why is this not the case?

Here, we have two competing derivations. Nowhere in either exposition does the OP identify a step X which is wrong, and justify it should be right because of Y. Rather OP is plainly asking for other users to decide which of the two methods is right.

I was one of the original closers, and I don’t remember why I voted to close. Looking at it now the question suffers from other problems. For starters, the variable $x$ of method 1 is defined as the extension of the spring whereas in method 2 the variables $x_1$ and $x_2$ are defined from the center of mass.

The extension of the spring should be (if I understand well) $x_1-x_2$, not $x_1+x_2$. This way, if both $x_1$ and $x_2$ are displaced by the same amount, the spring is not stretched or compressed. Because so little details are provided, it is impossible to verify one way or the other.

I would vote to close again as check-my-work or as unclear.

Clarifications could include the actual FBD illustrating the forces, a coordinate system from which the definition of $x_1$, $x_2$ and $x$ can easily be grasped. This way, the OP could identify steps where the parallel development of the two solutions differ in an essential way, and could then proceed to identify a principle which seems to apply in one case but not another - at least bring it closer to the spirit of the format you quote.

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