Potential Energy of Modified Atwood Machine

[McCoy13]

Homework Statement

The two weights on the left have equal masses m and ar connected by a massless spring of force constant k. The weight on the right has mass 2m, and the pulley is massless and frictionless. The coordinate x is the extension of the spring from its equilibrium length; that is, the length of the spring is l+x where l is the equilibrium length with all the weights in position and with the 2m weight held stationary.

Show that the total potential energy is just $U=\frac{1}{2}kx^{2}$

Homework Equations

$$U_{grav}=mgh$$
$$U_{spr}=\frac{1}{2}kx^{2}$$

The Attempt at a Solution

Attempt 1 (ignoring all sorts of constants added onto potential energy, since they'll disappear in Lagrange's/Hamilton's equations):
$$U=2mgy-mgy-mg(y+l+x)+\frac{1}{2}kx^{2}$$
$$U=-mgx+\frac{1}{2}kx^{2}$$
(the l was dropped for the noted reason above)

Attempt 2 (preserving all constants incase I missed something):
let the length of the rope be L;
$$U=2mg(y-L)-mgy-mg(y+l+x)+\frac{1}{2}kx^{2}$$
$$U=\frac{1}{2}kx^{2}-mgx-mg(2L+l)$$

I turned up something on a Dutch forum about changing the quadratic in the bottom from the form Ax^2+Bx+C to something like Ax^2+D and then just letting D=0 (since constant adjustments of potential don't matter), but I have absolutely no idea how to do that and it seems kind of mathematically spurious that you can change a linear term into a constant.

Am I missing something here?

 

[kuruman]

Think center of mass. If you stretch the spring and let go, what kind of motion does the CM undergo?

 

[McCoy13]

Well, I worked through the whole problem working out the equations of motion, and for x it turns out that you get the oscillatory motion plus a constant (mg/k), and that constant is the displacement from equilibrium that occurs when you let the spring hang at rest, which is accounted for in the original problem statement by the definition of l, so it doesn't belong in the equation for x. I argued that you should ignore -mgx in the potential from this fact.

But from your question even, I still can't see how you can argue it from the beginning. The mass on the spring would oscillate about x=0. I'm not sure what CM you're talking about. The CM of the whole system? It would oscillate too.