Statics: Pulley System In Equilibrium?

Diagram: http://i.imgur.com/LYv1bYv.jpg
Frictionless pulley system. tension in each rope and the force P, such that the system is in equilibrium? (Side question. Why is the tension in the rope around the pulley constant?)
From an assignment a while ago. Always sucked at pulley questions, but I think I should probably know how to solve them. Help, hints, directions to some helpful resources - all welcome. Thanks a bunch!
Posted: May 19, 2013

Best Answer

I always sucked at pulley problems too, Maybe I still do. But I think I can muddle through this one. First, your side question: good question! It isn't intuitive to me either. But think about it: at what point would the tension dissipate from the rope? I think you can generalize that as long as there are no "corners" in the rope (that is, nothing other than smooth curves, as around a pulley) that the tension is constant. Maybe you just have to put it in the "faith" column.
Let's call the pulleys A, B and C from left to right. Given a force of P, the tension in the rope supporting pulley C must be 2P. (Right? P down on the right, therefore P down on the left, therefore 2P up on the pulley.) Therefore the tension in the rope looping over pulley B and supporting pulley A is 2P. The rope on either side of pulley A still has tension P, meaning that there is a total of 4P holding up pulley A, which in turn holds up 400 N.
Then P + P + 2P = 400 N,
or P = 100 N.
To feel really good about it, consider that the force on pulley B is
2P + 2P = 4P
and the tension at the end of the rope tied to the ceiling is P,
for a total upward force of 5P = 500 N.
Downward we've got 400N + P = 500 N.
Sweet!

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