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Masses M and m are connected to a system of strings and pulleys as shown in the diagram.The strings are massless and inextensible, and the pulleys are massless and frictionless. The cross hatched horizontal beams are fixed in place.Is there any kind of setting where mass M would move up and mass m decend considering that the system is in static situation (i.e., when the two masses are in equilibrium with neither M nor m moving).

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  • $\begingroup$ just increase the mass of m $\endgroup$
    – user65081
    Commented Sep 14, 2020 at 22:16
  • $\begingroup$ Are you sure this is what you want to ask. If neither M nor m are moving then obviously M is not moving up and m is not descending. $\endgroup$
    – Dale
    Commented Sep 14, 2020 at 22:29
  • $\begingroup$ They are moving, sorry for my wordings $\endgroup$ Commented Sep 14, 2020 at 22:37
  • $\begingroup$ I’m voting to close this question because homework-like questions should include an attempt at a solution. $\endgroup$
    – The Photon
    Commented Sep 14, 2020 at 23:29
  • $\begingroup$ If M equals 2 times m then they are at equilibrium. $\endgroup$ Commented Sep 15, 2020 at 0:26

2 Answers 2

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If M is an elephant and m is a mouse, M will accelerate downward, pulling m up.

If M is a mouse and m is an elephant, m will accelerate downward, pulling M up.

Somewhere in between, the there is a balance where neither mass accelerates up or down. If the system is at rest, it will stay at rest. If you pull on a weight to start it moving, it will keep on moving.

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If the system is in equilibrium then the forces on both masses and on the lower pulley must net to zero. By considering the forces on each of the three objects you can derive three equations which must be met for equilibrium to exist.

But if the values of $m$ and $M$ are known then you only have two unknowns - the tensions in the two ropes. You can eliminate these two unknowns from the three equations and derive a relationship between $m$ and $M$ that must be satisfied in equilibrium.

Note, however, that the system can be in equilibrium even if the masses are not stationary. The equilibrium condition only means that they are not accelerating. The masses could both be moving with a constant velocity and still be in equilibrium - the term mechanical equilibrium is sometimes used to avoid the word static. If the masses are both moving with a constant velocity then by considering the geometry of the strings you can derive a relationship between their velocities.

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