Tension in Fixed Pulleys: Understanding Forces and Equilibrium

[Bipolarity]

Homework Statement

http://learn.uci.edu/media/OC08/11004/OC0811004_L6Pulley.gif

Homework Equations

The Attempt at a Solution

OK so I am a bit confused about how pulleys work. Assume the pulley in the diagram is massless and frictionless and that ropes/hinges are massless.
Now in the diagram, there are three tensions in consideration. The tension in the metal hinge connecting the ceiling to the pulley, the tension in the rope due to the mass on the left, and the tension in the rope due to the mass on the right.

First off, what is the tension in the metal hinge? I am sure it is equal to either the sum of the weights of the two masses, or equal to the difference between their weights. Which of these is it? I am confused about this.

Second off, the tension in the rope due to the mass on the left equals the tension in the rope due to the mass on the right, provided the rope is taut. Why is it that these two tensions are equal? What would happen were they not equal?

Thanks! Not a textbook problem, just my personal confusion over pulleys.

BiP

 

[tiny-tim]

Hi Bipolarity!

First off, what is the tension in the metal hinge? I am sure it is equal to either the sum of the weights of the two masses, or equal to the difference between their weights. Which of these is it? I am confused about this.

Neither.

Since the pulley is not moving vertically, the force holding the pulley up must be 2T …

you'll need to use F = ma on each weight separately to find T.

Second off, the tension in the rope due to the mass on the left equals the tension in the rope due to the mass on the right, provided the rope is taut. Why is it that these two tensions are equal? What would happen were they not equal?

If they weren't equal, the net force would cause infinite acceleration of a stretch of rope, since the mass of the rope is zero!