Question on bernoulli's equation

[theBEAST]

Homework Statement

This is from the notes given to my professor and I think that it may be incorrect:

Essentially we have a mercury manometer that measures the pressure difference at points 1 and 2. The fluid flowing through the pipe is water. The goal is to find the height h. Now in these notes, my professor seems to be assuming that the pressure does not depend on the height in the water. This is because on the last line he wrote rho_Hg * g * h = P_1. However I think the distance used should be from the center line since the pressure changes?

Also the pressure he solved for, P_1 = 11.3kPa (which I think is wrong), is the pressure at the streamline? This is wrong because we want to use the pressure RIGHT above the mercury.EDIT: I redid the question and found P1 = 69420Pa. Then I found a relationship between P1, P2 and h:

P2 = P1 + rho_w * g * h - rho_Hg * g * h
0 = 69420 + 9.81 * h * (1000 - 13600)

Solving for h I get 56cm.

 

[Basic_Physics]

The 69420 Pa is for the pressure difference:

P₁ - P₂ = 69420 Pa

The pressure at 2 is not 0, the water is still in the pipe. It is only when it emerges that the pressure is 0. The height difference of columns of mercury in the manometer is due to this pressure difference. So one sets this pressure difference equal to

ρ_Hggh = 69420

for which I get 52 cm?

 

[haruspex]

ρ_Hggh = 69420

for which I get 52 cm?

You have not allowed for the fact that the portion of the manometer pipe that is not filled with mercury is filled with water. The density difference is 12600, giving the 53cm in the OP.