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Fluid Statics: Pressure, Pascal & Archimedes

Source lecture(s): SC133 Lec 18

Intuition

A fluid — liquid or gas — can't hold a shape, so it pushes only one way: perpendicular to every surface, with a squeeze called pressure. From that single fact flow three famous results: pressure grows with depth (dams), pressure applied anywhere appears everywhere (hydraulic lifts — Pascal), and immersed bodies get buoyed up by the weight of displaced fluid (Archimedes and his bathtub).

Pressure and depth

\[P = \frac{F}{A}\ \text{(Pa)} \qquad\qquad \boxed{\,P = P_0 + \rho g h\,}\ \text{(incompressible fluid, depth } h)\]

Each 10 m of water adds about one atmosphere (\(P_\text{atm} \approx 101\,\text{kPa}\)). Pressure at a given depth is the same in every direction and independent of the container's shape — the hydrostatic paradox: a thin tube and a lake at equal depth exert equal pressure.

Gauge vs absolute: gauges read \(P - P_\text{atm}\); your "2.2 bar" tires hold ~3.2 bar absolute.

Pascal's principle

Pressure applied to an enclosed fluid transmits undiminished to every point:

\[\frac{F_1}{A_1} = \frac{F_2}{A_2}\]

A small force on a small piston yields a large force on a large piston — the hydraulic lever behind car lifts, brakes, and excavators. No free lunch: the small piston moves proportionally farther, and work balances.

Archimedes' principle

A body immersed (fully or partly) feels an upward buoyant force equal to the weight of fluid it displaces:

\[\boxed{\,F_B = \rho_\text{fluid}\, V_\text{displaced}\; g\,}\]

Why: the pressure on the body's bottom exceeds that on its top by \(\rho g \Delta h\); summed over the surface, the imbalance equals the displaced fluid's weight — the force that would have held that fluid parcel in equilibrium.

  • Floats if \(\rho_\text{body} < \rho_\text{fluid}\); the floating fraction submerged is \(\rho_\text{body}/\rho_\text{fluid}\) (an iceberg: \(917/1025 \approx 89\%\) under).
  • A steel ship floats because its average density (hull + air) is below water's.

Worked example: crown or fake?

A "gold" crown weighs 9.8 N in air and 9.0 N submerged. Verdict?

Buoyant force \(= 0.8\,\text{N} = \rho_w V g \Rightarrow V = 8.2\times10^{-5}\,\text{m}^3\). Density \(= \frac{m}{V} = \frac{1.0}{8.2\times10^{-5}} \approx 12\,200\,\text{kg/m}^3\) — far from gold's 19 300. Fake (legend says Archimedes reached the same verdict, then ran through Syracuse).

Common mistakes

  • Buoyancy depends on the body's density? No — on the fluid's density and displaced volume. A sunken cannonball still feels full buoyancy; it's just outweighed.
  • Deeper = more buoyancy? No (incompressible fluid): displaced volume is unchanged, so \(F_B\) is depth-independent.
  • Using total volume for floating bodies — only the submerged part displaces.
  • Forgetting atmospheric pressure in absolute-pressure problems (straws, suction, barometers).

Knowledge graph position

Prerequisites: Newton's laws, Equilibrium. Leads to: Bernoulli, the entire PC316 fluids course.

Quiz

Q1 (computational). A hydraulic lift: input piston 2 cm radius, output 20 cm. Force needed to lift a 1200 kg car?

Answer

Area ratio \(= 100\), so \(F = 1200\times9.8/100 \approx 118\,\text{N}\) — one firm push. (You'll pump the small piston 100× the lift height.)

Q2 (conceptual). An ice cube floats in a full glass. When it melts, does the glass overflow?

Answer

No — the level is unchanged. Floating, the cube displaces its weight in water; melted, it becomes exactly that much water. (Sea-level rise comes from land ice and thermal expansion.)

Q3 (multiple choice). A dam must be built thickest: (a) at the top (b) at the bottom (c) uniformly — pressure depends on the reservoir's total volume

Answer

(b). \(P = \rho g h\) grows with depth and doesn't care about the reservoir's extent — a short deep lake pushes as hard as an ocean of equal depth.