Mechanical Properties

Mechanical Properties

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Mechanical properties describe how a body responds to applied forces and deformations, classified into three moduli regimes: elastic (stress–strain linear), plastic (permanent deformation), and fluid response (flow, viscosity, surface tension).

Must-know formulas:

• Longitudinal strain: ε = ΔL/L; stress: σ = F/A
• Young’s modulus: Y = (F/A) / (ΔL/L)
• Bulk modulus: B = -P / (ΔV/V)
• Shear modulus: G = (F/A) / φ
• Stokes’ drag on a sphere: F = 6π η r v
• Poiseuille flow rate: Q = π P r⁴ / (8 η L)
• Capillary rise: h = 2 T cosθ / (ρ g r)

High-yield pointers:

1. Hooke’s law (F = kx) holds only inside the proportional limit, not up to fracture.
2. Bernoulli’s equation, P + ½ρv² + ρgh = constant, applies along a streamline for steady, incompressible, non-viscous flow — venturimeter and Pitot tube are direct applications.
3. JEE Advanced frequently combines a fluid statics/viscosity problem with electrostatics (pressure analogue of potential) or thermodynamics (work done = P ΔV).

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Stress, Strain and Elastic Moduli

A normal stress σ = F/A (units: N/m² or Pa) produces a longitudinal strain ε = ΔL/L (dimensionless). Within the proportional limit, σ ∝ ε, giving Young’s modulus Y = σ/ε. The stress–strain curve marks several points: A (proportional limit), B (elastic limit), C (yield point), D (ultimate stress), E (fracture). Beyond B, deformation is permanent.

For volume change under uniform pressure P, the bulk modulus B = -P / (ΔV/V) — the negative sign ensures B > 0 for compression (ΔV negative). For shear, G = (F/A) / φ, where φ is the angular shear in radians. Poisson’s ratio σ_p = (Δd/d)/(ΔL/L) links lateral and longitudinal strains; σ_p lies between 0 and 0.5 for stable materials.

Fluids at Rest

Pascal’s law: pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion — operating principle of hydraulic lifts. Buoyancy follows Archimedes’ principle: apparent weight loss = weight of displaced fluid = ρ_fluid × V_submerged × g.

Fluid Dynamics

For steady, incompressible, non-viscous flow, two equations govern:

• Continuity: A₁v₁ = A₂v₂ (mass conservation)
• Bernoulli: P + ½ρv² + ρgh = constant along a streamline

A venturimeter measures flow speed using the pressure drop at a constriction; a Pitot tube measures flow speed by stagnation pressure. The Reynolds number R = ρvD/η distinguishes laminar (R < ~2000) from turbulent flow.

Viscosity

Newton’s law of viscosity gives viscous force per unit area τ = η (dv/dx). For laminar flow through a cylindrical tube, Q = π P r⁴ / (8 η L) — the strong r⁴ dependence is why a slight narrowing drastically reduces flow. For a sphere falling through a viscous medium, terminal velocity satisfies 6π η r v_t = (4/3)π r³ (ρ - σ) g, where σ is the fluid density (Stokes’ law).

Surface Tension

Surface tension T is force per unit length acting along a liquid surface. Across a curved interface, excess pressure ΔP = 2T/R (spherical bubble, one surface) or ΔP = 4T/R (soap bubble, two surfaces). Capillary rise h = 2T cosθ / (ρ g r) where θ is the contact angle and r the tube radius.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Common Mistakes and Traps

1. Sign errors in B: a compression (ΔV < 0) under pressure P > 0 must yield positive B. Dropping the minus sign flips the sign convention.
2. Poiseuille’s law vs Bernoulli: Poiseuille applies to viscous laminar flow through pipes; Bernoulli assumes inviscid flow. Mixing them up — e.g. applying Bernoulli where viscous dissipation dominates — gives nonsense answers.
3. Stokes vs kinematic viscosity: use η (Pa·s, dynamic viscosity), not ν = η/ρ (m²/s, kinematic). A common JEE trap is to swap them in Reynolds number or terminal velocity problems.
4. Capillary rise contact angle: cosθ = 1 only for perfectly wetting liquids (water–clean glass). For mercury on glass, θ > 90° and the liquid depresses; substituting cosθ = 1 here gives wrong direction of h.
5. Streamline vs laminar: streamlines are a geometric property of the velocity field; laminar flow describes regime. Both can co-occur, but turbulent flow has no well-defined streamlines.

Connections to Adjacent Topics

• The work done per unit volume in stretching is U/V = ½ × stress × strain, identical in form to the energy density in a capacitor (½ ε₀ E²). JEE Advanced frequently exploits this electrical–mechanical analogy.
• Pressure in fluid statics plays the role of electrostatic potential: both satisfy Laplace’s equation in source-free regions.
• Terminal velocity from Stokes’ law feeds into Millikan’s oil-drop experiment, combining mechanical properties with quantisation of charge.

Worked Example

A steel wire of length 2.0 m and cross-section 2 mm² supports a 20 kg mass. Find elongation. (Y_steel = 2 × 10¹¹ Pa.)

ΔL = (F L) / (A Y) = (20 × 9.8 × 2.0) / (2 × 10⁻⁶ × 2 × 10¹¹) = 392 / (4 × 10⁵) = 9.8 × 10⁻⁴ m ≈ 1 mm.

Practice Prompts

1. A capillary tube of radius 0.5 mm is dipped in water (T = 0.072 N/m, θ = 0°). Calculate the rise. (Answer: h = 2T/(ρgr) ≈ 2.94 cm)
2. A sphere of radius 1 mm falls through glycerine (η = 1.5 Pa·s). Find terminal velocity if density difference = 8000 kg/m³. (v_t = 2r²(Δρ)g/(9η) ≈ 1.16 cm/s)

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