Mechanical Properties

Mechanical Properties

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Mechanical properties of matter describe how solids and fluids deform and flow under external forces. The chapter splits into Mechanical Properties of Solids and Mechanical Properties of Fluids.

Stress σ = F/A (N/m² or Pa); Strain ε = ΔL/L (dimensionless). Young’s modulus Y = (F/A) ÷ (ΔL/L). Bulk modulus K = −P ÷ (ΔV/V). Shear modulus G relates tangential stress to angular strain. Hooke’s law (F = −kx or σ = Yε) holds only up to the elastic limit.

Pressure in a fluid P = ρgh, isotropic and depth-dependent. Pascal’s law transmits pressure undiminished (hydraulic lift). Archimedes’ principle: buoyant force = weight of displaced fluid (F_b = ρVg). Continuity: A₁v₁ = A₂v₂. Bernoulli: P + ½ρv² + ρgh = constant. Viscosity (Poiseuille): V/t = πPr⁴/(8ηL). Surface tension S = F/l; excess pressure inside a drop = 2S/R, inside a bubble = 4S/R.

For CUET UG, expect 1–2 MCQs (≈3% weightage), usually one from solids (stress–strain graph/Young’s modulus) and one from fluids (Bernoulli/viscosity).

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Stress, Strain, and Moduli

Stress is the restoring force per unit area developed inside a body when deforming forces act on it; it has the same units as pressure (N/m² or pascal). Longitudinal strain is the fractional change in length ΔL/L, shearing strain is the lateral displacement divided by the perpendicular height, and volumetric strain is ΔV/V.

Hooke’s law states that within the elastic limit, stress is proportional to strain. The constant of proportionality is the relevant modulus:

• Young’s modulus Y = (F/A) / (ΔL/L) — for lengthwise stretching of a wire/rod.
• Shear modulus G = (F/A) / θ — for tangential distortion.
• Bulk modulus K = −ΔP / (ΔV/V) — for uniform compression; its reciprocal is compressibility β = 1/K.

Stress–Strain Curve

A ductile material (mild steel) shows a linear elastic region, a yield point beyond which plastic deformation begins, an ultimate tensile strength (maximum stress), and a fracture point. A brittle material (glass, chalk) fractures just past the elastic limit with negligible plastic region. Elastic fatigue occurs when a material fails below its yield point due to repeated cyclic loading.

Fluid Statics

A fluid at rest exerts pressure that depends only on depth: P = P₀ + ρgh. Pascal’s law (a consequence of the isotropy of fluid pressure) is the principle behind hydraulic brakes and lifts: F₂ = F₁ × (A₂/A₁). Archimedes’ principle gives the upward buoyant force F_b = ρ_fluid · V_displaced · g, governing flotation (a body floats when its average density is less than the fluid’s).

Fluid Dynamics

For an ideal, incompressible, non-viscous fluid in streamline flow, the equation of continuity A₁v₁ = A₂v₂ expresses mass conservation. Bernoulli’s equation P + ½ρv² + ρgh = constant along a streamline is the energy-conservation statement for such a flow; it explains the lift on aerofoils, Venturi meters, and the speed of efflux from a tank (Torricelli’s law: v = √(2gh)).

Viscosity and Surface Tension

Viscosity η is internal friction in a real fluid. For laminar flow through a cylindrical pipe, Poiseuille’s formula gives the volume flow rate V/t = πPr⁴/(8ηL), showing the strong r⁴ dependence. Stokes’ law F = 6πηrv gives the viscous drag on a small sphere, used to determine η by terminal-velocity measurements.

Surface tension S = F/l arises from cohesive forces; it produces a spherical shape in liquid drops and a capillary rise h = 2Scosθ/(ρgr) in narrow tubes. Excess pressure across a curved surface is 2S/R for a liquid drop (one surface) and 4S/R for a soap bubble (two surfaces).

Common Exam Patterns in CUET UG

Numerical problems usually involve (i) finding the elongation ΔL = FL/(AY) of a wire, (ii) applying Bernoulli’s equation between two points in a pipe, and (iii) using Poiseuille’s equation to compare flow rates when the radius is changed. Assertion–reason type questions often test whether students can distinguish elastic vs. plastic deformation and stress vs. pressure.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Nuances

Compressibility contrast: solids have K of order 10¹⁰–10¹¹ Pa, liquids ~2 × 10⁹ Pa, and gases ~10⁵ Pa — explaining why liquids and solids are nearly incompressible in everyday situations. The negative sign in K = −ΔP/(ΔV/V) is required because an increase in pressure causes a decrease in volume.

Beyond the elastic limit: Hooke’s law fails. The material enters a plastic region where strain is no longer recoverable. For a polycrystalline metal, the elastic limit lies just below the proportional limit (where σ–ε is strictly linear), and the yield point marks the onset of permanent deformation. Work hardening raises the elastic limit but reduces ductility.

Streamline vs. turbulent flow: the dimensionless Reynolds number R = ρvD/η determines the regime. R < ~1000 is laminar; R > ~2000 is turbulent. Bernoulli’s equation strictly applies only to streamline flow — applying it to turbulent flow is a frequent error trap.

Stokes’ law assumptions: valid for a small rigid sphere moving slowly (low R) in an infinite, otherwise quiescent, Newtonian fluid. At terminal velocity v_t = 2r²(ρ−σ)g/(9η) (where σ is the fluid density), the drag balances the net weight.

Surface tension in drops vs. bubbles: the factor-of-two difference (2S/R vs. 4S/R) comes from the number of liquid surfaces enclosing the gas — a drop has one, a bubble has two. Cleaning actions of soap and detergents depend on reducing S, not on producing more bubbles.

Connections to Other CUET Topics

This chapter is prerequisite for oscillations (spring constant k = YA/L appears in SHM problems), thermal physics (thermal expansion coefficients relate to interatomic spring constants), and wave motion (speed of sound in a solid Y/ρ depends directly on Young’s modulus). Bernoulli’s principle links to kinetic theory through the energy-density interpretation ½ρv².

Common Mistakes

1. Confusing stress (internal restoring force per area, valid in solids) with pressure (force per area in a fluid, always normal). 2. Using compressive strain with the wrong sign, or applying Young’s modulus to a volume change (that needs K). 3. Forgetting the 4S/R factor for bubbles. 4. In Poiseuille’s formula, halving r reduces flow by a factor of 16, not 2. 5. Treating Bernoulli as valid in viscous flow — the equation must be augmented with head-loss terms for real fluids.

Practice Prompts

1. A copper wire of length 2.0 m and cross-section 1.0 mm² stretches by 0.5 mm under a 50 N load. Compute the Young’s modulus and identify the energy stored per unit volume (½ × stress × strain).
2. Water flows through a horizontal pipe of diameter 6 cm at 2 m/s. At a constriction the diameter halves. Using Bernoulli, find the pressure drop between the wide and narrow sections, and verify mass conservation via the continuity equation.

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