How to Calculate Pressure in a Fluid

This lesson introduces students to the fundamental fluid properties of density and pressure, explaining how these concepts govern fluid behavior in static situations.

What You’ll Learn

1. Define and calculate density and pressure in fluids
2. Distinguish between absolute pressure and gauge pressure
3. Explain why pressure is a scalar quantity despite being derived from force
4. Use the hydrostatic pressure equation to determine fluid pressure at a given depth
5. Understand how atmospheric pressure changes with altitude
6. Solve fluid statics problems using pressure equilibrium in a U-tube

Key Concepts Covered

• Fluid mechanics
• Density and pressure in fluids
• Absolute pressure vs. gauge pressure
• Hydrostatic pressure formula: p = p₀ + ρgh
• Atmospheric pressure and altitude
• Incompressible fluids vs. compressible gases
• Scalar vs. vector quantities
• Static equilibrium in fluid columns

Understanding how fluids behave at rest is essential for real-world applications like engineering, meteorology, and physiology. It also forms the foundation for topics in AP Physics, IB Physics, and competitive exams such as JEE and NEET. Mastery of fluid statics is crucial for solving problems involving submerged objects, barometers, and fluid-containing systems.

Prerequisite or Follow-Up Lessons

• Introduction to Forces and Free-Body Diagrams
• Buoyant Force and Archimedes' Principle

Full Lesson: Pressure and Density in Fluids

What is a Fluid?

Fluids are substances that flow and take the shape of their containers. This includes both liquids and gases. Unlike solids, fluids do not have a fixed shape, which makes their mechanical behavior distinct.

Understanding Density

Density (ρ) is the mass per unit volume of a substance:

ρ = m / V

Where:

• ρ is density (kg/m³)
• m is mass (kg)
• V is volume (m³)

Example: If 4 kg of fluid occupies 2 m³, then ρ = 2 kg/m³.

Gases vary in density with pressure, but liquids are largely incompressible, so their density remains constant.

Defining Pressure

Pressure (p) in a fluid is the force per unit area:

p = F / A

Where:

• p is pressure (Pa)
• F is the normal force (N)
• A is the area (m²)

Despite being derived from force (a vector), pressure is a scalar because it only uses the magnitude of the force perpendicular to a surface. In a fluid, pressure acts equally in all directions.

Units and Atmospheric Pressure

In SI units, pressure is measured in Pascals (Pa), where:

1 Pa = 1 N/m²

At sea level, atmospheric pressure is approximately:

1 atm = 1.01 × 10⁵ Pa

At higher altitudes, atmospheric pressure decreases due to fewer air layers above.

Pressure Variation with Depth

In a liquid, pressure increases with depth due to the weight of the fluid above. For a fluid at rest, the pressure at depth h is given by:

p = p₀ + ρgh

Where:

• p is the absolute pressure at depth h
• p₀ is the atmospheric pressure at the surface
• ρ is the fluid density
• g is gravitational acceleration
• h is the depth below the surface

This equation shows that pressure depends only on depth, not on container shape.

Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure:

gauge pressure = p - p₀ = ρgh

Gauge pressure ignores atmospheric pressure, which acts equally at all points.

Pressure Above a Fluid Surface

If a point is located d meters above the fluid surface, the pressure there is:

p = p₀ - ρ_air g d

This explains why atmospheric pressure drops at high altitudes.

U-Tube with Two Fluids: Solving for Unknown Density

Given:

• Water density ρₓ = 998 kg/m³
• Oil density ρₒ = ?
• Water column height l = 135 mm
• Oil depth d = 12.3 mm

At equilibrium, pressures at the same level in the two arms are equal:

p_i (water side) = p₀ + ρₓ g l

p_i (oil side) = p₀ + ρₒ g (l + d)

Equating:

ρₒ = ρₓ × [l / (l + d)] = 915 kg/m³

This shows that pressure equilibrium allows us to compute unknown fluid densities, and results are independent of g or p₀.