General manometer Equation, Derivation & Rearrangements

Core idea

Overview

A general manometer equation is a pressure walk through connected static fluids. Moving downward through a fluid increases pressure by rho g d, and moving upward decreases it by the same kind of term.

When to use: Use this for manometer problems where both sides may contain process fluids as well as a separate manometer fluid.

Why it matters: Manometers give a mechanical pressure measurement that is still useful for calibration, differential pressure checks, and teaching fluid statics.

Symbols

Variables

$P_{1}$ = Pressure 1, $P_{2}$ = Pressure 2, $ρ_{1}$ = Fluid 1 Density, $d_{1}$ = Fluid 1 Height, $ρ_{2}$ = Fluid 2 Density

P_{1}

Pressure 1

Pa

P_{2}

Pressure 2

Pa

ρ_{1}

Fluid 1 Density

k g / m^{3}

d_{1}

Fluid 1 Height

m

ρ_{2}

Fluid 2 Density

k g / m^{3}

d_{2}

Fluid 2 Height

m

ρ_{m}

Manometer Fluid Density

k g / m^{3}

h

Manometer Height Difference

m

g

Gravitational Acceleration

m / s^{2}

Walkthrough

Derivation

Derivation of General manometer

A general manometer equation is obtained by walking through static fluid columns and adding or subtracting rho g height terms.

All fluids are static.
Each fluid density is constant over its column.
Heights are vertical distances.

1

Use hydrostatic pressure changes

Moving downward through a static fluid increases pressure by rho g times vertical distance.

Δ P = ρ g Δ z

2

Walk from side 1 to side 2

Each fluid column contributes its own density and vertical height along the pressure path.

P_{1} + ρ_{1} g d_{1} = P_{2} + ρ_{2} g d_{2} + ρ_{m} g h

Result

P_{1} + ρ_{1} g d_{1} = P_{2} + ρ_{2} g d_{2} + ρ_{m} g h

Source: Munson, Young, Okiishi, Huebsch, and Rothmayer, Fundamentals of Fluid Mechanics, Wiley, 2013; Engineering LibreTexts, 4.3.2.3: Magnified Pressure Measurement, accessed 2026-04-09

Free formulas

Rearrangements

Solve for $P_{1}$

Make pressure 1 the subject

P_{1} = P_{2} + ρ_{2} g d_{2} + ρ_{m} g h - ρ_{1} g d_{1}

Move every column term except P1 to the other side.

Difficulty: 3/5

Solve for $h$

Make height difference the subject

h = \frac{P _{1} + ρ _{1} g d _{1} - P _{2} - ρ _{2} g d _{2}}{ρ _{m} g}

Collect the endpoint and side-fluid terms, then divide by $r h o_{m}$ g.

Difficulty: 3/5

Solve for $ρ_{m}$

Make manometer density the subject

ρ_{m} = \frac{P _{1} + ρ _{1} g d _{1} - P _{2} - ρ _{2} g d _{2}}{g h}

Isolate $r h o_{m}$ g h, then divide by g h.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph shows Pressure 1 increasing linearly with Pressure 2, forming a straight line with a positive slope. For a student, this means that if Pressure 2 goes up, Pressure 1 will also go up by a corresponding amount. The most important feature is this direct, proportional relationship between P_1 and P_2. This indicates that changes in Pressure 2 directly influence Pressure 1.

Graph type: linear

Why it behaves this way

Intuition

Imagine a U-shaped tube connected between two points, P1 and P2. Inside, three fluid columns act like a liquid balance scale. On the left, a fluid of density ?1 pushes down from height d1. On the right, a second fluid of density ?2 pushes down from height d2, while a heavier 'manometer fluid' (?m) fills the bottom of the U, with its level h higher on the left side than the right. The equation ensures that the total weight-induced pressure from the fluids and the source pressures equalizes at the interface.

P_{1}, P_{2}

Static pressures at the two endpoints

The source pressures being measured or compared, typically from pipes or tanks.

?_{1} g d_{1}

Hydrostatic pressure of the first process fluid leg

The weight of the fluid column on the side of P1; as you move deeper (d1), the pressure increases.

?_{2} g d_{2}

Hydrostatic pressure of the second process fluid leg

The weight of the fluid column on the side of P2.

?_{m} g h

Pressure difference due to the manometer fluid displacement

The 'imbalance' in the heavy fluid levels; this column usually does the 'heavy lifting' in counteracting a large pressure difference between P1 and P2.

Signs and relationships

? g d (Positive addition): In fluid statics, moving vertically downward through a continuous fluid adds to the pressure because of the weight of the fluid above.
P_1 + ?_1 g d_1: This represents the total pressure at the bottom interface on the left side, summing the source pressure and the weight of the process fluid.
P_2 + ?_2 g d_2 + ?_m g h: This represents the total pressure at the equivalent horizontal plane on the right side, summing the source pressure, the process fluid weight, and the weight of the manometer fluid column.

Free study cues

Insight

Canonical usage

This equation is used to relate pressure differences across a fluid column by balancing the hydrostatic pressures of different fluids in a manometer system.

Common confusion

Students may incorrectly assume all fluids in the manometer have the same density or fail to convert all pressure units to a consistent system (e.g., SI Pascals) before calculation.

Dimension note

This equation involves quantities with physical units; the result is a pressure difference or absolute pressure, not a dimensionless quantity.

One free problem

Practice Problem

Given P2 = 100000 Pa, rho1 = 1000 kg/ $m^{3}$ , d1 = 0.20 m, rho2 = 850 kg/ $m^{3}$ , d2 = 0.10 m, $r h o_{m}$ = 13600 kg/ $m^{3}$ , h = 0.050 m, and g = 9.81 m/ $s^{2}$ , find P1.

Pressure 2100000 Pa

Fluid 1 Density1000 kg/m^3

Fluid 1 Height0.2 m

Fluid 2 Density850 kg/m^3

Fluid 2 Height0.1 m

Manometer Fluid Density13600 kg/m^3

Manometer Height Difference0.05 m

Gravitational Acceleration9.81 m/s^2

Solve for: pressure1

Hint: Move all terms except P1 to the right-hand side.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When Finding the pressure difference between two pipe taps connected to a U-tube manometer, General manometer is used to calculate Pressure 1 from Pressure 2, Fluid 1 Density, and Fluid 1 Height. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

Study smarter

Tips

Trace from one known pressure to the other, adding rho g height when moving downward.
Use vertical height differences, not tube length.
Keep pressure units and density units consistent throughout.

Avoid these traps

Common Mistakes

Assigning the wrong sign to a fluid-column term.
Using the manometer-fluid density for every leg of the pressure path.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

A general manometer equation is obtained by walking through static fluid columns and adding or subtracting rho g height terms.

Use this for manometer problems where both sides may contain process fluids as well as a separate manometer fluid.

Manometers give a mechanical pressure measurement that is still useful for calibration, differential pressure checks, and teaching fluid statics.

Assigning the wrong sign to a fluid-column term. Using the manometer-fluid density for every leg of the pressure path.

When Finding the pressure difference between two pipe taps connected to a U-tube manometer, General manometer is used to calculate Pressure 1 from Pressure 2, Fluid 1 Density, and Fluid 1 Height. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

Trace from one known pressure to the other, adding rho g height when moving downward. Use vertical height differences, not tube length. Keep pressure units and density units consistent throughout.

References

Sources

Munson, Young, Okiishi, Huebsch, and Rothmayer, Fundamentals of Fluid Mechanics, Wiley, 2013
Engineering LibreTexts, 4.3.2.3: Magnified Pressure Measurement, accessed 2026-04-09
OpenStax University Physics Volume 1, Pressure Gauges and Manometers, accessed 2026-04-09
NIST CODATA
IUPAC Gold Book
Fluid statics (Wikipedia)
NIST Chemistry WebBook
University Physics (e.g., Sears and Zemansky's)

General manometer

Overview

Variables

Derivation

Use hydrostatic pressure changes

Walk from side 1 to side 2

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Hydrostatic pressure

Hydrostatic head

Differential manometer

Manometer for gas

Frequently Asked Questions

Sources