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Pressure in Fluids

Discover how pressure at a depth depends only on density, gravity, and depth — not the container width or shape.

Pressure at 0.50m depth: 4.9 kPa

Simulation Controls

Adjust side length of container

Position of horizontal plane (max: 0.9m)

3D Container

Horizontal plane at depth h
Atmospheric force
Fluid weight
Pressure at depth

Real-Time Calculations

Top surface area (A)1.00 m²

A = L² = 1.0²

Weight of fluid above plane4.91 kN

ρ × A × h × g = 1000 × 1.00 × 0.5 × 9.81

Force due to atmosphere101.33 kN

P₀ × A = 101.3 kPa × 1.00 m²

Total force on plane106.23 kN

Weight of fluid above plane + Atmospheric force

Pressure at depth h4.91 kPa

P = ρgh = 1000 × 9.81 × 0.5

Key insight: Independent of container size L!

Pressure Formula

P = ρgh
ρ (density) = 1000 kg/m³
g (gravity) = 9.81 m/s²
h (depth) = 0.5 m

When L increases:

  • • Force ↑ (proportional to L²)
  • • Area ↑ (proportional to L²)
  • • Pressure = Force/Area = constant
!

Key Insight

At any fixed depth h:

  • • Fluid above that plane gets heavier if the tank gets wider ✅
  • • But the area it presses on also gets bigger ✅
  • • So the pressure = force ÷ area = ρgh stays the same 🎯

Understanding the Pressure Equation

The pressure equation P = ρgh is fundamental to understanding fluid statics. This equation tells us that the pressure at any point in a fluid depends on three factors:

  • ρ (rho) - The density of the fluid
  • g - The acceleration due to gravity
  • h - The depth below the surface

Key Insight: Independence from Container Shape

The remarkable property of this equation is what it doesn't include - the shape or width of the container. As you experiment with the simulation above, you'll notice that changing the container size (L) affects the total force on any horizontal plane, but the pressure remains constant at the same depth.

Why Does This Happen?

When you increase the container width:

  • The weight of fluid above any plane increases (more fluid volume)
  • The area of the plane also increases proportionally
  • Since pressure = force ÷ area, these effects cancel out
  • The pressure at depth h remains ρgh regardless of container width

Derivation

Consider a horizontal plane at depth h in a fluid:

  • Volume of fluid above the plane = A × h (where A is the area of the plane)
  • Weight of this fluid = ρ × A × h × g
  • Adding atmospheric pressure: Total force = P₀A + ρAhg
  • Pressure at depth h = Total force ÷ Area = P₀ + ρgh

In most cases, we consider gauge pressure (pressure above atmospheric), so P = ρgh.

Remember

At any fixed depth, the pressure is the same regardless of the container's shape or size. This principle applies to everything from water glasses to ocean depths!