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Homework Statement
I had to answer which of the pulley arrangements illustrated (see picture) is in equilibrium (the drawing went a bit difficult, please see all outer boxes as equal in size and the middle ones as well) I managed to answer that it is arrangement C, by finding out that equilibrium seems to be the case when the angle between the middle pulley and the rope on each side of it is symmetrical. However, now I want to know the exact physics behind it. Below, I already described my assumptions about it and am wondering if I have concluded it correctly. Plus, I get stuck at how slightly different pulley arrangements will effect equilibrium and movement of the weights. Please, see below, for more details.
Homework Equations
Possibly: Torque (F*R)
The Attempt at a Solution
I assume that C is the one that is in equilibrium, because there the outer weights must be equal in size since the angle that the rope makes with the middle pulley is equally at each side; in other words symmetrical. Looking at the other arrangements, the angle that the rope makes with the middle pulley is asymmetrical. In order to get the middle pulley hanging asymmetrically, the outer boxes must be different in weight. Do I get it right that if the weight hangs more toward the left side, the left weight must be heavier (picture B), and that if the that same weight is hanging even higher, the left weight must be much heavier (picture D)? In the case of arrangement A, the box at the right must be heavier. Am I right in saying that I should not get confused and think there is equilibrium just because the rope won’t move in any of the arrangements. It seems to me that equilibrium is just the case when the weights are equal in size and the weight in the middle is hanging at symmetrical angles in respect to the rope when the pivot points of the outermost pulleys are at the same height.
Now, there are four other scenarios at which I get stuck understanding what the arrangement will look like, and in which direction the weight will possibly move:
1) a height difference between the outer pulleys with the same amount of weight hanging at the outermost pulleys
2) a height difference between the outer pulleys with a different amount of weight hanging at the outermost pulleys
3) much radius difference between the outermost pulleys while having the same amount of weight hanging at the outermost pulleys
4) much radius difference between the outermost pulleys while having a different amount of weight hanging at the outermost pulleys
Please, see picture two for the illustration of those arrangements.
It appears to me that the theory behind how those arrangements will look and in which direction possible movement will be, must be all about Torque and that pulley radius and difference in height between the outer pulleys must play a role. However, I do not understand in which way the rules of torque can be applied here, and thus would like to get assistance in figuring out how.
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