Class 11 Physics - Chapter 9: Mechanical Properties of Fluids

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CLASS 11 PHYSICS

CHAPTER 9 — MECHANICAL PROPERTIES OF FLUIDS

Conceptual • NCERT-Focused • NEET/JEE-Oriented

WHAT IS THIS CHAPTER REALLY ABOUT?

Liquids and gases are grouped as fluids because they share one defining behaviour: they offer almost no resistance to shear stress and therefore flow. The chapter splits neatly into two halves. FLUIDS AT REST give us pressure, Pascal’s law and buoyancy. FLUIDS IN MOTION give us the continuity equation and Bernoulli’s principle. Two ‘extra’ forces then refine the ideal picture — viscosity (internal friction) and surface tension (the skin-like energy of a free surface).

FLUID AT REST (pressure, Pascal) → FLUID IN MOTION (continuity, Bernoulli) → VISCOSITY (real friction) → SURFACE TENSION (free surface)

PRESSURE IN A FLUID

P = F / A (SI unit: pascal, Pa = N m⁻²)

The force a static fluid exerts on any surface is always normal (perpendicular) to that surface — if there were a parallel component, the fluid would flow (Newton’s third law), contradicting ‘at rest’. Pressure is a scalar: at a point it acts equally in all directions and has no assigned direction.

⚠️ NEET/JEE TRAP Pressure is a SCALAR even though it equals force/area. The ‘force’ in the definition is only the COMPONENT normal to the surface — direction belongs to that force, not to pressure itself.

Pascal’s Law

Two linked statements: (1) in a fluid at rest, pressure is the same at all points at the same height; (2) a pressure change applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the container walls.

🔥 HIGH-YIELD The second statement powers every HYDRAULIC MACHINE. In a hydraulic lift F₂ = F₁ × (A₂/A₁): a small force on a small piston lifts a large load on a large piston. Mechanical advantage = A₂/A₁. Hydraulic brakes transmit one pedal force equally to all four wheels.

Variation of pressure with depth

P = Pₐ + ρgh

The pressure at depth h below an open surface exceeds atmospheric by ρgh. The excess, P − Pₐ = ρgh, is the gauge pressure; P itself is the absolute pressure. Notice A does not appear — only vertical depth matters.

📖 NCERT FOCUS Hydrostatic paradox: three differently-shaped vessels connected at the base fill to the SAME level despite holding different amounts of liquid — because pressure depends only on depth (ρgh), not on the shape or the amount of liquid above.

Measuring pressure

Mercury barometer (Torricelli): Pₐ = ρgh; ~76 cm of Hg at sea level = 1 atm = 1.013×10⁵ Pa. 1 torr = 1 mm Hg = 133 Pa.

Open-tube manometer: measures gauge pressure P − Pₐ, proportional to the height difference h.

⚠️ NEET/JEE TRAP Absolute vs gauge: absolute P = Pₐ + ρgh; gauge = P − Pₐ = ρgh. Tyre gauges and the blood-pressure cuff read GAUGE pressure (the excess over atmospheric), not absolute.

STREAMLINE FLOW & EQUATION OF CONTINUITY

A flow is steady (streamline) if the velocity at any fixed point stays constant in time. A streamline is a curve whose tangent gives the fluid velocity there; two streamlines never cross (a particle can’t have two velocities). Above a critical speed the flow becomes turbulent (whirlpools, white-water).

A·v = constant (equation of continuity)

This is conservation of mass for an incompressible fluid: where the pipe is narrow, streamlines crowd and the fluid speeds up; where it is wide, it slows down. Av is the volume flux (flow rate).

BERNOULLI’S PRINCIPLE

Energy conservation for a non-viscous, incompressible, steady flow. Along a streamline:

P + ½ρv² + ρgh = constant

The three terms are pressure energy, kinetic energy per unit volume, and potential energy per unit volume. Key consequence: where speed is high, pressure is low.

⚠️ NEET/JEE TRAP Bernoulli’s equation assumes NO viscosity and NO turbulence and an INCOMPRESSIBLE fluid. It fails for a river rapid (turbulent) and for real viscous flow — friction converts kinetic energy into heat, so the downstream pressure is lower than the ideal value predicts.

Speed of efflux — Torricelli’s law

v = √(2gh) (tank open to atmosphere)

The jet from a hole a depth h below the surface leaves with the same speed a body would reach falling freely through h. If the tank is pressurised (P ≫ Pₐ), the efflux speed is set instead by the container pressure — the basis of rocket propulsion.

Dynamic lift

Magnus effect: a spinning ball drags air around it, crowding streamlines on one side (faster, lower pressure) — the net sideways force curves the ball off its parabola.

Aerofoil (wing): shaped so air moves faster over the top than the bottom ⇒ lower pressure above ⇒ upward lift that balances the plane’s weight.

🔥 HIGH-YIELD Both the Magnus effect and wing lift are direct applications of Bernoulli: FASTER flow ⇒ LOWER pressure. The pressure difference × wing area supports the aircraft’s weight.

VISCOSITY

Real fluids resist relative motion between layers — an internal friction called viscosity. The coefficient of viscosity is the ratio of shearing stress to the rate of shear strain (not the strain itself, as in a solid).

η = shearing stress / (v/l) (SI unit: poiseiulle, Pl = Pa·s)

⚠️ NEET/JEE TRAP Temperature dependence is OPPOSITE for the two states: viscosity of a LIQUID DECREASES with temperature, while viscosity of a GAS INCREASES with temperature. Frequently tested fill-in-the-blank.

Stokes’ law & terminal velocity

F = 6πηav

A sphere of radius a moving at v through a fluid feels a viscous drag 6πηav, opposing motion. A falling drop accelerates until drag + buoyancy balance gravity; it then falls at a constant terminal velocity:

vₜ = 2a²(ρ − σ)g / 9η

🔥 HIGH-YIELD Terminal velocity ∝ a² (square of radius) and ∝ 1/η. So a bigger drop falls much faster, and a more viscous medium slows the fall. This is why fine mist droplets seem to hang in the air.

SURFACE TENSION

A molecule deep inside a liquid is pulled equally in all directions; a molecule at the surface has liquid only below it, so it carries extra potential energy (~half that of an interior molecule). A liquid therefore tends to MINIMISE its surface area.

S = F / 2l (force per unit length) = surface energy per unit area

Surface tension is both a force per unit length in the plane of the surface AND the extra energy per unit area of the interface. It is a property of the interface between two materials, not of one fluid alone, and it usually FALLS with temperature.

🧠 MEMORY HOOK For a soap-film experiment the film has TWO surfaces, so the extra energy is S(2dl) and the force is S×(2l). A single free liquid surface has just one. Count your surfaces before applying a formula.

Angle of contact

The angle between the liquid surface and the solid, measured inside the liquid. It decides wetting via the balance Sla cosθ + Ssl = Ssa.

Angle of contact	Behaviour	Example
Acute (θ < 90°)	Liquid WETS the solid, spreads, meniscus concave	Water on clean glass; kerosene on almost anything
Obtuse (θ > 90°)	Liquid does NOT wet, forms drops, meniscus convex	Water on a waxy leaf; mercury on glass

Wetting agents (soaps, detergents) LOWER the angle of contact so liquid penetrates; water-proofing agents RAISE it.

Excess pressure — drops & bubbles

System	Number of surfaces	Excess pressure (Pᵢ − Pₒ)
Liquid drop	1	2S / r
Air bubble inside a liquid	1	2S / r
Soap bubble (in air)	2	4S / r

🔥 HIGH-YIELD The concave (inner) side of a curved liquid surface always has HIGHER pressure. Excess pressure ∝ 1/r, so SMALLER bubbles have larger internal pressure. A soap bubble’s two surfaces double the excess pressure to 4S/r — that’s why you must blow a little harder to start a soap bubble.

Capillary rise

h = 2S cosθ / (ρga)

A narrow tube in water: the concave meniscus lowers the pressure just under it, and atmospheric pressure pushes the column up until ρgh balances the surface-tension pull. The rise is LARGER for a smaller radius a. For mercury (obtuse θ, cosθ negative) the level is DEPRESSED instead.

⚠️ NEET/JEE TRAP Capillary rise ∝ 1/a — thinner tube, greater rise (‘capilla’ = hair). And because cosθ flips sign for a non-wetting liquid, mercury goes DOWN in a capillary, not up.

MUST-KNOW DIAGRAMS

⭐ MUST KNOW

Hydraulic lift: small piston (A₁, F₁) and large piston (A₂, F₂) sharing a fluid — shows F₂ = F₁(A₂/A₁) and Pascal transmission.

Streamlines in a varying pipe: crowded (fast, low P) at the constriction, spread (slow, high P) at the wide part — links continuity + Bernoulli.

Aerofoil streamlines: faster flow over the top ⇒ lower pressure ⇒ upward lift.

Understand the structure

Meniscus shapes (concave water / convex mercury) and angle of contact — understand wetting rather than memorising the force balance.

Capillary rise set-up — see why the pressure jump across the curved surface lifts the column.

LAST-MINUTE CONFUSION BUSTERS

Easily confused pair	Correct distinction
Absolute vs gauge pressure	Absolute = Pₐ + ρgh. Gauge = ρgh (excess over atmospheric). Tyre/BP gauges read gauge.
Viscosity of liquid vs gas with T	Liquid: η DECREASES as T rises. Gas: η INCREASES as T rises.
Continuity vs Bernoulli	Continuity = mass conservation (Av = const). Bernoulli = energy conservation (P + ½ρv² + ρgh = const).
Drop / air bubble vs soap bubble	One surface → 2S/r. Two surfaces (soap bubble) → 4S/r.
Water vs mercury in a capillary	Water (acute θ): rises, concave. Mercury (obtuse θ): depressed, convex.
Surface tension with area	Independent of surface area; it is force per length / energy per area, not per surface.

ACTIVE RECALL

Why is the force exerted by a static fluid always normal to a surface, and why is pressure a scalar?

State Pascal’s law and use it to explain the mechanical advantage of a hydraulic lift.

Explain the hydrostatic paradox in one sentence.

Derive the sense of Torricelli’s law: why v = √(2gh)?

How does viscosity of a liquid and of a gas change with temperature?

Write the excess pressure for a liquid drop, an air bubble in liquid, and a soap bubble — and explain the factor difference.

Why does water rise but mercury fall in a fine capillary tube?