Class 11 · Chapter 10

Mechanical Properties of Fluids

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Mechanical Properties of Fluids Short Notes

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Condensed revision points for Mechanical Properties of Fluids — for quick recall before exams, not a substitute for the full notes.

P = ΔF/ΔA — SI unit Pa; dimensions [M¹L⁻¹T⁻²]. Pressure is a SCALAR.
1 atm = 1.013 × 10⁵ Pa = 760 mm of Hg.
Pressure at depth h: P = P₀ + hρg.
Gauge pressure = hρg (excess over atmospheric); Absolute pressure = P₀ + hρg.
Average pressure on a vertical wall of height h = hρg/2 (half the bottom value, since P rises linearly with depth).

Pascal's law: pressure change in an enclosed fluid is transmitted equally & undiminished to every point.
Hydraulic lift: F₁/A₁ = F₂/A₂ ⇒ F₂ = (A₂/A₁)F₁.
Applications: hydraulic lift, press, jack, brakes — force is gained only by sacrificing displacement (no free energy).

Buoyant force Th = weight of fluid displaced = V_in·σ·g.
Th depends ONLY on submerged volume and fluid density — NEVER on the object's own mass, density, or shape.
Acts upward through the centre of buoyancy (C.G. of displaced fluid, not of the object).
Apparent weight = W − Th. This is why things feel lighter in water.

ρ > σ ⇒ sinks. ρ = σ ⇒ floats fully submerged (neutral equilibrium). ρ < σ ⇒ floats partly submerged.
R.D. of solid = W_A/(W_A − W_W) (Archimedes' method).
Specific gravity of unknown liquid = (W_A − W_L)/(W_A − W_W).

A₁v₁ = A₂v₂ = constant for steady, incompressible flow; Q = Av (volume flow rate).
Steady/streamline flow: velocity at a fixed point doesn't change with time; streamlines never cross.
Laminar: smooth parallel layers. Turbulent: chaotic, eddies, high energy loss — occurs at high speed.
Narrower cross-section ⇒ faster flow (continuity); this is the basis for every Bernoulli application below.

P + ½ρv² + ρgh = constant (per unit volume) — valid for ideal fluid: incompressible, non-viscous, steady, irrotational.
Per unit mass: P/ρ + v²/2 + gh = constant. Per unit weight: P/ρg + v²/2g + h = constant (pressure head + velocity head + gravitational head).
Horizontal flow (h const): P + ½ρv² = constant — faster flow ⇒ lower pressure, and vice versa.
Real (viscous) fluids lose energy as heat — not accounted for in the ideal Bernoulli equation.

Torricelli's law (speed of efflux): v = √(2gh) — same as free-fall speed through height h.
Venturimeter: v₁ = A₂√[2gh/(A₁²−A₂²)] — measures flow speed from a height difference h in side-tubes.
Aerofoil/wing lift: faster flow over the top ⇒ lower pressure there ⇒ net upward lift.
Magnus effect: spin drags air around the ball, speeding flow (lowering pressure) on one side ⇒ curved path (swing/spin).
Atomizer/sprayer: fast air jet ⇒ low pressure ⇒ draws liquid up and disperses it as spray.
Fast trains / storm winds: lower pressure near the fast-moving object pulls nearby objects toward it.

F = ηA(dv/dy); η = (F/A)/(dv/dy) — viscous force ∝ area & velocity gradient.
SI unit: Pa·s (poiseuille); CGS: poise (1 Pa·s = 10 poise). Dimensions [M¹L⁻¹T⁻¹].
η depends only on the fluid's nature & temperature — NOT on area or velocity gradient.
Liquids: η ↓ as T ↑ (weaker cohesion). Gases: η ↑ as T ↑ (faster molecular momentum exchange) — opposite trends!

Stokes' law: viscous drag on a falling sphere F_v = 6πηrv.
Terminal velocity: v_T = 2r²(ρ−σ)g/9η — grows with r² (radius squared).
At terminal velocity: weight = buoyant force + viscous force (zero net acceleration).
Applications: raindrop terminal speed, cloud droplets 'floating', parachutes, Millikan's oil-drop experiment.

Re = ρvd/η — ratio of inertial to viscous forces.
Re < 1000: laminar. Re > 2000: turbulent. 1000–2000: unstable transition.
Poiseuille's formula: Q = πPr⁴/(8ηL) — flow rate ∝ r⁴, extremely sensitive to radius.
Viscosity is largely independent of pressure for both liquids and gases.

T = F/L — SI unit N/m (= J/m²); dimensions [M¹L⁰T⁻²].
T = dW/dA — surface tension = work done per unit increase in surface area.
Cause: surface molecules have fewer neighbours ⇒ net inward pull ⇒ surface minimizes its area (like a stretched membrane).
T decreases as temperature rises; becomes zero at the critical temperature.
Soluble impurities (salt) ⇒ T increases. Partially-soluble/surfactants (detergent, soap) ⇒ T decreases.
Drop (1 surface): W = T × 4πr². Soap bubble (2 surfaces): W = T × 8πr².

Drop or air bubble IN liquid (1 surface): P_excess = 2T/r.
Soap bubble in air (2 surfaces): P_excess = 4T/r — double that of a single-surface drop of the same radius.
Excess pressure ∝ 1/r — smaller bubble/drop has HIGHER internal pressure than a larger one.
Two connected bubbles of different sizes: air flows from the smaller (higher-pressure) into the larger one.

h = 2Tcosθ/(rρg) — capillary rise (or depression).
θ < 90° (acute): liquid wets solid, concave meniscus, liquid RISES (water–glass, θ≈0°).
θ > 90° (obtuse): liquid doesn't wet solid, convex meniscus, liquid is DEPRESSED (mercury–glass).
Jurin's law: h ∝ 1/r — narrower capillary ⇒ greater rise.
Real-world capillarity: lamp wicks, fountain pen nibs, water rising through soil/plant tissue.

Don't confuse the bubble-in-liquid case (1 surface, 2T/r) with the soap-bubble-in-air case (2 surfaces, 4T/r) — a very common one-mark slip.
Buoyant force never depends on the object's own density or mass — only on submerged volume and fluid density. Students often wrongly drag the object's density into the Th formula.
Viscosity of liquids and gases respond OPPOSITELY to rising temperature (liquids: ↓, gases: ↑) — easy to mix up under exam pressure.
Terminal velocity ∝ r² (radius squared, from Stokes' law), NOT ∝ r — a frequent algebra slip when scaling between two drops of different sizes.
Bernoulli's equation requires an IDEAL fluid (non-viscous, incompressible, steady, irrotational) — it does not directly apply to real, viscous flow without modification.
Capillary rise h ∝ 1/r (Jurin's law) — inverse, not direct proportionality; doubling the radius HALVES the rise, not doubles it.
Pressure is a scalar, but the force due to pressure is a vector acting perpendicular to the surface — don't describe 'pressure' itself as having a direction.

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