Fluids
Pressure in a fluid increases with depth, which means the force on any submerged surface isn't simply "pressure times area" — that only works if pressure is uniform over the surface, and here it isn't. Finding the actual total force on a submerged wall means integrating pressure over the surface, the exact same variable-quantity-times-integration pattern that showed up for work (force varying with position) and impulse (force varying with time).
What you'll learn
- Derive the hydrostatic pressure formula P(h)=P₀+ρgh by integrating dP/dh=ρg
- Set up and evaluate an integral for the total force on a submerged vertical surface, where pressure varies with depth
- Apply continuity (A₁v₁=A₂v₂) and Bernoulli's equation to flowing fluids
- Connect buoyant force to the pressure difference between the top and bottom of a submerged object
1. Deriving Hydrostatic Pressure by Integration
Pressure in a fluid increases with depth because each additional layer of fluid adds its own weight to everything below it: \(\dfrac{dP}{dh}=\rho g\). Integrating this directly gives \(P(h)=P_0+\rho g h\) — the "hydrostatic pressure formula" isn't a separate fact, it's this rate integrated from the surface down to depth \(h\).
2. Total Force on a Submerged Surface
Since pressure varies with depth, the force on a submerged vertical surface isn't simply pressure times area — that shortcut only works when pressure is the same everywhere on the surface, which it isn't here. The correct approach: divide the surface into thin horizontal strips, each at its own depth \(h\) with its own pressure \(P(h)\), and integrate \(F=\int P(h)\,dA\) over the whole surface.
3. Buoyant Force from Pressure Difference
Buoyant force is a direct consequence of pressure increasing with depth: a submerged object's bottom sits at greater depth (higher pressure) than its top, so the upward push on the bottom exceeds the downward push on the top. The net result, after accounting for the object's whole submerged shape, is Archimedes' principle: \(F_{buoyant}=\rho_{fluid}\,g\,V_{submerged}\) — not a separate law, but what pressure varying with depth produces once integrated over a closed surface.
4. Continuity and Bernoulli's Equation
For fluid flowing through a pipe of changing cross-section, conservation of volume gives continuity, \(A_1v_1=A_2v_2\) — a narrower section must carry faster-moving fluid. Bernoulli's equation extends energy conservation to flowing fluids, relating pressure, speed, and height at any two points along the same flow.
Key equations
- dP/dh = ρg → P(h) = P₀ + ρgh — Hydrostatic pressure — derived by integrating the rate pressure increases with depth, not assumed as a standalone fact.
- F = ∫P(h)dA — Total force on a submerged surface where pressure varies with depth — reduces to PA only when pressure is uniform over the whole surface.
- F_buoyant = ρ_fluid·g·V_submerged — Archimedes' principle — the net upward force from higher pressure on an object's bottom than its top, following from the same pressure-varies-with-depth relationship.
- A₁v₁ = A₂v₂ — Continuity — conservation of fluid volume flowing through a pipe of changing cross-section.
- P₁+½ρv₁²+ρgh₁ = P₂+½ρv₂²+ρgh₂ — Bernoulli's equation — energy conservation along a streamline, connecting pressure, speed, and height at two points in a flow.