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Imagine a pulley with a massless and frictionless rope. On both sides of the pulley (and both sides of the rope), we attach a mass ($m_1$ and $m_2$). These masses are connected with two other masses ($m_3$ an $m_4$) by means of two ideal springs (spring constants $k_1$ and $k_2$). The rope has a mass ofs(kg/m).

We hold the two upper masses at equal height after which we let the system go its way. How do we have to model the system mathematically?

For two single masses at each end the situation is simple. The heavier mass gets an acceleration of $g-a$, the lighter mass $g+a$ and it's easy to calculate $a$, if the masses are known. But what happens in this (non-equilibrium) case? Can it be that it's an unsolvable four-body problem?

For sure there are four bodies (the four masses). But this is not a four body problem because the masses are not moving in each other's force field. Two pairs of masses are connected by a string and both these pairs are subjected to the force of gravity.

After some thought, the problem is easier than I thought. For the acceleration, a, for both sets of masses after letting them go, we solve the following formula (if $(m_1+m_2)>(m_2+m_4)$):

$(m_1+m_2)(g-a)=(m_3+m_4)(g+a)$

After setting $g=10$ and some rearranging we get:

$a=\frac{10(M-m)} {(M+m)}$, where $M=m_1+m_2$, and $m=m_3+m_4$

So the two spring connected masses

When knowing a, it's easy to find how much the springs have lengthened or shortened.

But what if we take into account the (equally) changing mass on either side of the pulley because of the mass of the rope?

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    $\begingroup$ Obtaining the equations of motion is easy enough in Lagrangian mechanics. However, closed form solutions may or may not be available; but they can certainly be integrated numerically. $\endgroup$ – dmckee Feb 22 '17 at 22:41
  • $\begingroup$ This is no homework in the sense that it is a question for school. I made the problem up myself. But I'll try! $\endgroup$ – descheleschilder Feb 23 '17 at 5:13
  • $\begingroup$ Hi descheleschilder. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. $\endgroup$ – Qmechanic Feb 23 '17 at 22:48
  • $\begingroup$ @Qmechanic-Okidokio! $\endgroup$ – descheleschilder Feb 24 '17 at 0:29
  • $\begingroup$ 1. Calculations are exercises and fall under the homework policy whether or not they have been set as homework. 2. Thank you for showing some effort, but now what is your conceptual question? Are you asking if there is an analytic solution? Or are you asking us to write the equations of motion for you? The 2nd is not a conceptual question, and the 1st is a math question. 3. The description of the problem is not clear : can you post a diagram? $\endgroup$ – sammy gerbil Feb 25 '17 at 1:43
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Assuming that your conceptual question is Is the system solvable?, I give dmckee's answer-in-comments :

Obtaining the equations of motion is easy enough in Lagrangian mechanics. However, closed form solutions may or may not be available; but they can certainly be integrated numerically.

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