PN02 — Properties of Matter

Bulk modulus K = -V(dP/dV) = volumetric stress / volumetric strain. The negative sign ensures K is positive (volume decreases with increasing pressure). K of steel ≈ 160 GPa; water ≈ 2.2 GPa; gases have very low K. Incompressible materials have infinite bulk modulus.

A soap bubble has TWO surfaces (inner and outer). Excess pressure across each surface = 2T/r. Total excess pressure = 4T/r. For a liquid drop or one surface: ΔP = 2T/r. For a cylindrical soap film: ΔP = T/r. The 4T/r formula for soap bubbles is a standard NDA question.

Q ∝ r⁴ (Poiseuille). If r → 2r: Q_new = π P (2r)⁴ / (8ηL) = 16 × Q. Flow rate increases by 16 times. This r⁴ dependence explains why small blockages in blood vessels (slightly reducing r) dramatically reduce blood flow. Used in oil pipeline design and medical blood flow analysis.

Stokes Law: drag force F = 6πηrv. At terminal velocity: 6πηrv = (4/3)πr³(ρ-σ)g → v_t = 2r²(ρ-σ)g/(9η). Larger radius and greater density difference → faster terminal velocity. Smaller droplets fall slowly (cloud droplets). Negative (ρ < σ) means the object rises (bubble in water).

Contact angle θ: if θ < 90° (water-glass: ~20°) — liquid wets the solid, capillary rise occurs. If θ > 90° (mercury-glass: ~135°) — liquid does not wet, capillary depression occurs. At θ = 0°: perfect wetting. Surface tension, adhesion-cohesion balance, and contact angle are interconnected.

Work done = increase in surface energy = surface tension × increase in surface area. Soap bubble has 2 surfaces: total surface area = 2 × 4πR² = 8πR². Work = T × 8πR² = 8πR²T. For a single liquid surface (drop): W = 4πR²T. This NDA-level question requires knowing the bubble has two surfaces.

Bernoulli's equation holds for: steady (time-independent), irrotational (no vortices), incompressible (constant ρ), non-viscous (zero viscosity) flow along a streamline. Real fluids deviate from these ideal conditions. For viscous flow: Poiseuille equation applies. Bernoulli is conservation of energy per unit volume.

From F = ηA(dv/dy): η = F/(A × dv/dy) = [MLT⁻²]/([L²] × [LT⁻¹/L]) = [MLT⁻²]/[L²T⁻¹] = [ML⁻¹T⁻¹]. SI unit = Pa·s = kg/(m·s). Also called dynamic viscosity. Kinematic viscosity ν = η/ρ has dimensions [L²T⁻¹] (m²/s).

Stokes Law: F = 6πηrv. This formula was derived by George Gabriel Stokes for flow at low Reynolds number (Re < 1). It is the basis for calculating terminal velocity and is used in Millikan's oil drop experiment to find electron charge. The coefficient 6π arises from the exact solution of the Navier-Stokes equations.

Surface tension T = Force/Length = N/m. Equivalently: T = Energy/Area = J/m². Both are the same (1 N/m = 1 J/m²). This dual interpretation (force or energy per area) is important: T acts like a stretched elastic membrane. Surface tension decreases with temperature and is destroyed by surfactants.

Reynolds number predicts flow regime: Re < 2000 = laminar (streamline); 2000 < Re < 4000 = transitional; Re > 4000 = turbulent. Re is dimensionless. High Re means inertial forces dominate over viscous forces → turbulence. Low Re means viscous forces dominate → laminar flow. Used in aerodynamics, pipe design, and blood flow analysis.

In a rotating fluid (rigid body rotation), the centrifugal acceleration ω²r acts radially outward. Combined with downward gravity g, the effective gravity is g_eff(r) = √(g² + ω²r²) at radius r. The free surface of the rotating liquid becomes a paraboloid: z = ω²r²/(2g) + constant.

h ∝ 1/r (from h = 2T cosθ/ρgr). If r doubles → h halves. The product h × r = 2T cosθ/ρg = constant. Narrow tubes → greater capillary rise; wide tubes → less rise. This explains why narrow wick in an oil lamp draws oil higher than a wide tube.

Elastic PE = ½ × stress × strain × volume = ½ × (F/A) × (FL/AY) × AL = F²L/(2AY). Alternatively: PE = ½ × F × extension = ½F × (FL/AY) = F²L/(2AY). This is equivalent to ½kx² for a spring where k = AY/L (spring constant of the wire). NDA requires this derivation.

Compressibility = 1/K = volumetric strain / volumetric stress = (-dV/V)/dP. Highly compressible materials (gases) have low bulk modulus. Steel is nearly incompressible (very high K). Compressibility of water: ~4.5 × 10⁻¹⁰ Pa⁻¹. Used in hydraulic system design and acoustic wave propagation calculations.

Venturi meter uses Bernoulli: at the narrow throat, velocity increases and pressure decreases. By measuring the pressure difference ΔP between the wide pipe (area A₁) and the throat (area A₂): flow rate Q = A₁A₂√(2ΔP/ρ(A₁²-A₂²)). Used in carburettors, aircraft airspeed indicators, and industrial flow measurement.

For an isotropic material: Y = 2G(1 + ν) where ν = Poisson ratio. Also: Y = 3K(1 - 2ν) where K = bulk modulus. For most metals: ν ≈ 0.3 → Y/G ≈ 2.6. For rubber: ν ≈ 0.5 (incompressible) → Y = 3G. These interrelations between elastic constants are tested in NDA.

The vertical height of liquid in an inclined capillary is still h (same capillary pressure balances the hydrostatic pressure ρgh). But the LENGTH of the liquid column along the inclined tube = h/sinθ (longer column, smaller vertical component = h). At θ = 90° (horizontal tube), liquid fills the tube completely (h/sin90° → ∞).

Elasticity relations: Y = 2G(1+ν); Y = 3K(1-2ν); Y = 9KG/(3K+G). For an incompressible material (ν = 0.5): K → ∞ (cannot be compressed). For ν = 0 (no lateral contraction): Y = 3K. These relations are important for NDA-level elasticity problems.

Pascal law: pressure applied to any point of an enclosed incompressible fluid is transmitted undiminished to every other point and to the walls of the container. This is the principle of hydraulic machines. The force multiplication factor = ratio of areas (F₂/F₁ = A₂/A₁).