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Class XI · Chapter 09

Mechanical Properties of Fluids

Why a hydraulic jack can lift a car with one hand, why airplane wings generate lift, and why a water strider can walk on a pond without sinking.

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1Pressure and Pascal's Law

Pressure
P = F / A
Unlike force, pressure has no direction of its own — a fluid pushes equally in every direction at a given point, which is precisely what makes fluids behave so differently from solids.
Pascal's Law Pressure applied anywhere to an enclosed, incompressible fluid is transmitted undiminished to every point of the fluid and to the walls of its container.

2Variation of Pressure with Depth

Pressure at Depth h
P = P₀ + ρgh
P₀ = pressure at the surface (often atmospheric pressure), ρ = fluid density. Pressure grows linearly with depth — exactly why ears "pop" diving into a deep pool, and why deep-sea vessels need to be built to withstand enormous pressure.

Gauge pressure is pressure measured relative to atmospheric pressure (gauge pressure = absolute pressure − atmospheric pressure) — what a typical tyre pressure gauge actually reads.

3Hydraulic Machines

Pascal's Law makes force multiplication possible: apply a small force to a narrow piston, and the same pressure — transmitted undiminished through the enclosed fluid — acts on a much wider piston, producing a much larger force.

Hydraulic Lift
F₂ = F₁ × (A₂ / A₁)
A₂/A₁ is the mechanical advantage. This is exactly how a hydraulic car jack lets one person lift a vehicle, and how hydraulic brakes multiply a driver's foot pressure into strong braking force at each wheel.

4Archimedes' Principle

Archimedes' Principle
Buoyant force = weight of fluid displaced = ρfluid Vdisplaced g
An object floats when its buoyant force can equal its own weight — meaning it sinks only as far as needed to displace a volume of fluid equal in weight to itself. A steel ship floats not because steel is light, but because its hollow shape displaces far more water than a solid steel block of the same mass would.

5Streamline Flow and the Equation of Continuity

In smooth, streamline (laminar) flow, fluid particles follow well-defined, non-crossing paths. For an incompressible fluid flowing through a pipe of varying cross-section, mass conservation gives a simple, powerful relation:

Equation of Continuity
A₁v₁ = A₂v₂
Fluid speeds up where the pipe narrows, and slows down where it widens — the same reason a garden hose nozzle makes water shoot out faster by narrowing its exit.

6Bernoulli's Principle

Bernoulli's equation is essentially energy conservation applied to a flowing fluid — combining pressure, kinetic, and potential energy per unit volume along a streamline.

Bernoulli's Equation
P + ½ρv² + ρgh = constant
Valid for an ideal fluid (incompressible, non-viscous) in steady flow. The key consequence: where speed is higher, pressure must be lower, to keep the sum constant — the single most important takeaway from this whole chapter.

7Applications of Bernoulli's Principle

  • Venturi meter: measures flow speed by measuring the pressure drop as fluid passes through a constriction.
  • Atomizers and perfume sprayers: fast airflow across a tube's opening lowers pressure there, drawing liquid up and out.
  • Aerofoil lift: a wing's shape makes air travel faster over the curved top than the flatter bottom, creating lower pressure above and higher pressure below — the pressure difference produces net upward lift.
  • Blood flow: narrowed arteries (from plaque buildup) force blood to speed up through the constriction, dropping local pressure — a genuinely medically relevant consequence of the same principle.

8Viscosity

Real fluids resist relative motion between their own layers — an internal friction called viscosity, which is exactly what Bernoulli's "ideal fluid" assumption ignores.

Newton's Law of Viscosity
F = η A (dv/dx)
η = coefficient of viscosity, dv/dx = velocity gradient between fluid layers. Honey (high η) resists layer-sliding far more than water (low η) — that's the whole intuition behind viscosity.

9Stokes' Law and Terminal Velocity

Stokes' Law (Drag on a Sphere)
F = 6π η r v
Terminal Velocity
vt = 2r²(ρ − σ)g / (9η)
ρ = density of the falling sphere, σ = density of the fluid. A falling object speeds up only until viscous drag exactly balances the net downward force (gravity minus buoyancy) — after that, it falls at this constant terminal velocity, which is exactly why raindrops don't accelerate indefinitely on the way down.

10Reynolds Number

Reynolds Number
Re = ρvd / η
A dimensionless number comparing inertial forces to viscous forces. Low Re (roughly below 2000) means smooth, laminar flow; high Re (roughly above 3000) means chaotic, turbulent flow — this single number predicts which regime a given flow falls into, from blood in capillaries to water in a river.

11Surface Tension and Capillary Rise

A liquid's surface behaves like a stretched elastic membrane, since molecules at the surface are pulled inward by neighbours (with nothing pulling outward), minimising surface area — this is surface tension.

Surface Tension
T = F / L
Force per unit length along the surface. This is why a water strider can stand on a pond, and why small water drops pull themselves into spheres — the shape that minimises surface area for a given volume.
Capillary Rise
h = 2T cos θ / (ρgr)
θ = angle of contact, r = tube radius. Narrower tubes pull liquid higher — the mechanism by which water rises through the narrow vessels of plants, against gravity, without any pump.

Formula Summary

Pressure
P = F/A
Pressure at Depth
P = P₀ + ρgh
Hydraulic Lift
F₂ = F₁(A₂/A₁)
Buoyant Force
= ρ_fluid V g
Continuity
A₁v₁ = A₂v₂
Bernoulli's Equation
P + ½ρv² + ρgh = const
Stokes' Law
F = 6πηrv
Terminal Velocity
v_t = 2r²(ρ−σ)g/9η
Reynolds Number
Re = ρvd/η
Capillary Rise
h = 2Tcosθ/ρgr

Solved Examples

Example 1 · Hydraulic Lift

In a hydraulic lift, the small piston has area 5 cm² and the large piston has area 500 cm². A force of 100 N is applied on the small piston. Find the force produced on the large piston.

Solution: F₂ = F₁ × (A₂/A₁) = 100 × (500/5) = 100 × 100 = 10,000 N = 10 kN.

Example 2 · Capillary Rise

Find the height to which water rises in a capillary tube of radius 0.3 mm. (T = 0.072 N/m, ρ = 1000 kg/m³, g = 9.8 m/s², angle of contact ≈ 0°)

Solution: h = 2T cosθ / (ρgr) = (2 × 0.072 × 1) / (1000 × 9.8 × 0.0003) = 0.144 / 2.94 ≈ 0.049 m ≈ 4.9 cm.

Quick Check

1. According to Bernoulli's principle, where fluid speed is higher, pressure is:
Also higher
✓ Lower (correct)
Unchanged
Zero
2. Reynolds number is used to predict:
Buoyant force
Surface tension
✓ Whether flow is laminar or turbulent (correct)
Capillary rise
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Mechanical Properties of Solids

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Thermal Properties of Matter

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