Section 8.4

Hydrostatic Pressure and Force

In fluids, pressure increases with depth. To find the total force on a submerged vertical plate (like a dam or submarine window), we integrate the varying pressure across the surface.

Think of a vertical window on a submarine. The water pushes harder at the bottom of the window than at the top because pressure increases with depth.

To find the total force, we slice the surface into thin horizontal strips, calculate the force on each strip, and integrate to sum them all up.

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Pressure and Force

The Key Relationship

Hydrostatic pressure at depth is:

(rho)

Fluid density (for water, ≈ 1000 kg/m³)

Gravitational acceleration (≈ 9.8 m/s²)

Depth below the surface

Common value: For water, N/m³ (or 62.4 lb/ft³ in imperial units).

From Pressure to Force

Force = Pressure × Area. For a thin horizontal strip at depth with width and thickness :

2

The Formula

To find the total hydrostatic force on a submerged vertical plate:

Step-by-Step Process:

1
Set up coordinates: Usually at the surface, increasing downward.
2
Find depth: Express depth as a function of your variable.
3
Find width: Express width as a function of (may require similar triangles for non-rectangular plates).
4
Integrate: Set up and evaluate .
3

Worked Example

Rectangular Plate

A meter plate is submerged vertically with the top edge at the water surface. Find the total hydrostatic force. (Use N/m³)

2×3 meter rectangular plate submerged with pressure arrows
Step 1: Set up coordinates

Let at the surface, positive downward. The plate extends from to .

Step 2: Identify depth and width

At position : depth = , width = (constant for a rectangle).

Step 3: Set up the integral
Step 4: Evaluate
Answer: N

That's about 88 kilonewtons — roughly the weight of 9 metric tons pushing against the plate!

4

Level Up Examples

Example 2: Triangular Gate

An inverted triangular gate (base width 4 m, height 2 m) is submerged with the top edge at the surface. Find the hydrostatic force.

Inverted triangular gate submerged with pressure arrows

Key insight: The width changes with depth! We need to use similar triangles to find .

Step 1: Find width as a function of depth

At the surface (), width = 4. At the bottom (), width = 0.

By similar triangles:

Step 2: Set up the integral
Step 3: Evaluate
Answer: N

Example 3: Deep Submersion

A m square plate is submerged with its top edge 10 m below the surface. Find the hydrostatic force.

1×1 meter plate submerged 10m below the surface

Key insight: The plate is NOT at the surface! Depth ranges from 10 to 11 meters.

Step 1: Identify limits and depth

The plate spans to . At position , depth = , width = .

Step 2: Set up the integral
Step 3: Evaluate
Answer: N

Even though the plate is smaller (1×1 vs 2×3), the force is greater because it's so deep!

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Practice Quiz

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