Bernoulli's Equation and pressure differences

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SUMMARY

The discussion focuses on Bernoulli's Equation and the pressure differences in a U-tube scenario. The equation presented is $$P_1+\frac{\rho v_1^2}{2}=P_2+\frac{\rho v_2^2}{2}$$, which relates pressure and velocity in fluid dynamics. A key point of confusion is the derivation of the pressure difference $$P_A-P_B=h(\rho_{Hg}-\rho)g$$, specifically why the density of water is subtracted from the density of mercury. The explanation involves analyzing pressure changes from point B to point A through the U-tube.

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  • Understanding of Bernoulli's Equation
  • Knowledge of fluid dynamics principles
  • Familiarity with pressure concepts in U-tube manometers
  • Basic calculus for fluid velocity and density calculations
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  • Study the derivation of Bernoulli's Equation in fluid dynamics
  • Learn about U-tube manometer applications and calculations
  • Explore the relationship between pressure, density, and height in fluids
  • Investigate the effects of fluid density on pressure measurements
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Homework Statement


Problem in attached image

Homework Equations



$$P_1+\frac{\rho v_1^2}{2}=P_2+\frac{\rho v_2^2}{2}$$
bernoulii.PNG

The Attempt at a Solution


I understand everything in the solution except why $$P_A-P_B=h(\rho_{Hg}-\rho)g$$ Why do we have to subtract the density of water from that of mercury instead of just taking the density of mercury??

Any help is appreciated.
 
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Consider starting at point B and moving down through the U-tube all the way around until you come up the left side of the U-tube to point A. Keep track of all the pressure changes.
 

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