Pressures on fluids of mercury and water

[Cosmossos]

Homework Statement

Can someone please explain to me why P3-P1=Pb-Pa?
[PLAIN]http://img694.imageshack.us/img694/8123/45074508.jpg
thank you

 

[Andrew Mason]

The mercury is not moving. What does that say about the net force acting on the mercury in the horizontal part of the tube? What does that say about the sum of all the pressures at that point?

The force on a section of the mercury in the horizontal part of the tube is due to: gravity and P1 and P3. How is the net gravitational force related to the difference in height of the two columns?

AM

 

[Cosmossos]

The forces are zero.
I know how to claculate Pb-Pa , but I don't know how it is connected with P3-P1 since the heights are different.

 

[Andrew Mason]

In looking at this again, Pa-Pb = P3-P1 only if the heights of water above a and b are the same. It depends on the difference in height of water between the right and left sides. Let h1 be the height of water at b and h2 the height of water at a. Pa and Pb are the downward pressures at a and b.

$$P_b =P_3 + \rho_wgh_1$$$$P_a =P_1 + \rho_wgh_2$$

Subtracting:

$$P_a - P_b = P_1 - P_3 +\rho_wg(h_2-h_1)$$AM

 

[Cosmossos]

In the question I have here , it doesn't say anything about h2 and h1. it says that Pb-Pa=P3-P1 thanks to the Communicating_vessels law.But I don't understand why.

 

[Andrew Mason]

In the question I have here , it doesn't say anything about h2 and h1. it says that Pb-Pa=P3-P1 thanks to the Communicating_vessels law.But I don't understand why.

The law of communicating vessels is based on P3 and P1 being equal. That is not the case here. You will need to know the height of water above a and b. For example if there was no water above a, Pa=P1. But Pb would not be equal to P3. So Pb-Pa is not equal to P3-P1.

AM