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.

Frequently Asked Questions (FAQ) on Pressure in Fluids

Q1. Why isn't pressure considered a vector in physics?
A: Pressure is a scalar because it has magnitude but no specific direction. It acts equally in all directions at a point in a fluid and always exerts force perpendicular to a surface. While the resulting force is a vector, pressure itself is not.

Q2. Why doesn’t the shape of a container affect the hydrostatic pressure at a certain depth?
A: Hydrostatic pressure only depends on the fluid’s density, gravity, and the depth below the surface. The shape of the container doesn’t matter. That’s why pressure at 1 meter deep is the same whether you're in a wide tank or a narrow pipe.

Q3. Why can't water be compressed like air or gas?
A: Water molecules are packed much closer together than gas molecules, leaving little space to compress. That’s why liquids like water are nearly incompressible, while gases are easily compressed due to the large gaps between their molecules.

Q4. Why does water pressure increase the deeper you go?
A: As you go deeper, more water is stacked above you, and its weight adds to the pressure. The deeper the fluid, the more weight it exerts downward, increasing the pressure with depth.

Q5. Can any gases be denser than liquids and sink in them?
A: Almost all gases are less dense than liquids under normal conditions. However, in special cases, like very dense gases (e.g., xenon) and very light liquids (like liquid hydrogen), a gas can be denser and sink—though this is extremely rare and requires specific conditions.

Q6. How can the force on the bottom of a slanted tube be the same as in a straight cylinder? (Hydrostatic paradox)
A: Even if the container is slanted, pressure at a given depth depends only on vertical height—not shape. So the pressure at the bottom is the same, and since force equals pressure times area, the total force on the base remains unchanged.

Q7. What’s the difference between gauge pressure and absolute pressure?
A: Absolute pressure includes atmospheric pressure, while gauge pressure measures pressure above atmospheric pressure. For example, if atmospheric pressure is 1 atm and the gauge reads 2 atm, the absolute pressure is 3 atm. There's also vacuum pressure, which is below atmospheric.

Q8. Why subtract (8 atm - 3 atm) - 1 atm = 4 atm in one case but not subtract 1 atm in the other?
A: Gauge pressure ignores atmospheric pressure by default, while absolute pressure includes it. If you're comparing fluids and one side uses absolute and the other uses gauge, it’s important to keep the reference consistent—subtracting 1 atm only when switching between the two.

Q9. Why is fluid pressure always perpendicular to a surface, even if the surface is angled?
A: Fluid pressure acts in all directions at a point, but it always exerts force normal (perpendicular) to any surface. This happens because fluids can’t resist shear forces and can only push, not pull or slide, against a surface.

Q10. What's the difference between mass and density?
A: Mass is how much matter something has. Density is how tightly that matter is packed—it's the mass per unit volume. So two objects can have the same mass, but if one is more compact, it has a higher density.