Solving Pressure Puzzle: Find P1 in Diagram

[goldfish9776]

Homework Statement

i couldn't understand why the pressure P1 is given by P atm - ρ water (gh2) - ρ(oil)(gh1) + ρ(mercury)(gh3) .Where is the point P1 ? It's not indicated in the diagram ...

Homework Equations

The Attempt at a Solution

 

[gneill]

It looks like P1 is meant to be the pressure at the water/air interface in the tank. So, it's the same as the air pressure in the tank.

 

[goldfish9776]

It looks like P1 is meant to be the pressure at the water/air interface in the tank. So, it's the same as the air pressure in the tank.

isn't the air pressure at water surface = atmospheric pressure , which is 85.6Pa ?

 

[gneill]

isn't the air pressure at water surface = atmospheric pressure , which is 85.6Pa ?

Nope. That would be the pressure at the open end of the "S" tube.

The tank is a sealed enclosure except for the tube, and the enclosed air can have a different pressure than atmospheric.

 

[goldfish9776]

Nope. That would be the pressure at the open end of the "S" tube.

The tank is a sealed enclosure except for the tube, and the enclosed air can have a different pressure than atmospheric.

ok , can you explain why the pressure at P1 is found by subtracting the ρ water (gh2) and ρ(oil)(gh1) and plus ρ(mercury)(gh3) ??

 

[gneill]

Sure. The calculation follows a path from the open air at the open end of the S-tube to a point where the pressure is that of the water/air interface. In the system, horizontal lines through the fluids are lines of equal pressure. Look at the path described in orange:

Edit: Modified picture to describe a pressure change path from atmosphere to tank water surface with labels so points with equal pressures can be seen.

 

[Chestermiller]

If looks to me like they have the h2 and the h1 incorrectly switched in the equation.

Chet

 

[gneill]

If looks to me like they have the h2 and the h1 incorrectly switched in the equation.
Chet

I think you are right. I hadn't written out and simplified the full path equation before, now I have and I agree with you. Thanks for catching that.

I've modified the diagram in my previous post to reflect the full path. Points with the same letter labels are at the same pressure. The OP should be able to follow the path summing the pressure changes, then simplify the result.

 

[Chestermiller]

He also needs to recognize that moving upward lowers the pressure and moving downward increases the pressure. I'm wondering if he is aware of this.

 

[goldfish9776]

I think you are right. I hadn't written out and simplified the full path equation before, now I have and I agree with you. Thanks for catching that.

I've modified the diagram in my previous post to reflect the full path. Points with the same letter labels are at the same pressure. The OP should be able to follow the path summing the pressure changes, then simplify the result.

so P1 = Patm +ρ (water)gh2 - ρ(oil)g(h1+h2) + ρ(oil)gh1 - ρ(water)g(h1+h2) + ρ(mercury)gh3
P1 = Patm - ρ(water)gh1 - ρ(oil)gh2 + ρ(mercury)gh3 ?

 

[Chestermiller]

so P1 = Patm +ρ (water)gh2 - ρ(oil)g(h1+h2) + ρ(oil)gh1 - ρ(water)g(h1+h2) + ρ(mercury)gh3
P1 = Patm - ρ(water)gh1 - ρ(oil)gh2 + ρ(mercury)gh3 ?

Yes. That's correct.