Class 11 · Chapter 9

Mechanical Properties of Solids

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

Mechanical Properties of Solids Short Notes

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

Stress = F/A — SI unit N/m² (Pa); dimensions [M¹L⁻¹T⁻²].
Longitudinal stress: tensile (stretching) or compressive (squeezing) — normal to the surface.
Volume stress: equal normal forces over the entire surface, changes only volume.
Shear stress: tangential to the surface, changes shape but NOT volume.
Strain = (change in dimension)/(original dimension) — dimensionless, no units.
Longitudinal strain = ΔL/L; volume strain = ΔV/V; shear strain φ = Δx/L (≈ tanφ).
Angle of twist θ and angle of shear φ for a twisted cylinder: rθ = lφ ⇒ φ = rθ/l.
Breaking stress depends on material, temperature, impurities — NOT on area or force. Max load a wire can bear ∝ area.

Proportional limit: linear region, Hooke's law holds, body is elastic.
Elastic limit (yield point): stress–strain no longer linear, but full recovery still occurs if unloaded here.
Plastic region: beyond elastic limit, body keeps a permanent set even after unloading.
Tensile strength (ultimate point): beyond this, strain grows even with reduced force → fracture point follows.
Ductile material: large plastic region between elastic limit and fracture (can be drawn into wire).
Brittle material: small/negligible plastic region — fractures soon after the elastic limit.
Elastomers (e.g., rubber): huge elastic strain (up to ~30%), don't obey Hooke's law over most of the range.

Hooke's law: stress ∝ strain, within elastic limit only.
Y = longitudinal stress/longitudinal strain = FL/(AΔL).
Y depends ONLY on material, temperature, and impurities — never on the magnitude of stress or strain applied.
Elongation under an end-hung load Mg: ΔL = MgL/(AY) = MgL/(πr²Y).
Y = slope of the stress–strain graph in the linear region.

ΔL = MgL/(2AY) = ρgL²/(2Y) — note the extra factor of 2 vs. an end-loaded wire.
Tension (and stress) is maximum at the point of suspension, zero at the free lower end.
The factor of 2 arises because the weight acts effectively as if concentrated at the midpoint of the rope, not at the free end.

K = volume stress/volume strain = −ΔP/(ΔV/V); compressibility C = 1/K.
η = shear stress/shear strain = F/(Aφ).
K is maximum for solids, minimum for gases.
For liquids & gases: Y = 0 and η = 0 (no fixed length or shape to resist) — only K is meaningful.
For an ideal rigid body: Y, K, η are all infinite (zero strain for any stress).
η is a property exclusive to solids — fluids cannot resist a shearing force permanently.

σ = lateral strain/longitudinal strain — dimensionless.
Theoretical range: −1 ≤ σ ≤ 0.5; practical range for real materials: σ ≈ 0.2–0.4.
Y = 3K(1 − 2σ) = 2η(1 + σ).
9/Y = 3/η + 1/K — links all four constants directly.
Given any two of {Y, K, η, σ}, the other two can always be found from these relations.

W = ½FΔl (work done in stretching, exactly like an ideal spring).
W = ½ × stress × strain × volume = ½Y(strain)² × volume.
Energy density (per unit volume) = ½ × stress × strain = area under the stress–strain curve.
This energy is fully recoverable as long as the wire stayed within its elastic limit.

Temperature ↑ ⇒ Y generally ↓ (weaker intermolecular forces); material becomes more plastic.
Invar steel: exception — elastic constants barely change with temperature (used in precision clocks/instruments).
Impurities ⇒ Y slightly ↑ (intermolecular attraction strengthens marginally).
Interatomic force constant: k = Y × r₀ (r₀ = equilibrium interatomic spacing) — directly proportional to Y.

Crane cable: minimum safe cross-section A ≥ mg/S_y (S_y = yield strength of the material).
Girder: a deeper cross-section (larger d) resists sagging far better than a wider/shallower one for the same material & load.
Maximum height of a mountain: H ≈ (breaking stress of rock)/(ρg) ≈ 10 km — beyond this, base rock would simply give way.

Breaking stress is independent of area and applied force — but the maximum LOAD a wire can take before breaking DOES depend on area (Load = Breaking stress × A). Don't mix the two up.
Y is a fixed ratio for a given material — it does NOT increase just because more load/stress is applied; it stays constant within the elastic limit regardless of the values of stress or strain.
Own-weight elongation has an extra factor of 2 in the denominator (ΔL = MgL/2AY) compared to end-loaded elongation (ΔL = MgL/AY) — a very common one-mark slip.
Shear strain changes shape only, never volume — don't confuse it with volume strain (ΔV/V), which is the only strain that changes volume.
For liquids and gases, only K exists; Y = η = 0 — questions sometimes wrongly ask for the 'Young's modulus of water', which is meaningless.
Watch the sign in K = −ΔP/(ΔV/V): pressure increase causes volume decrease, but K itself is always quoted as a positive number.
Y = 2η(1+σ) and Y = 3K(1−2σ) — easy to swap which modulus pairs with which factor; re-derive from 9/Y = 3/η + 1/K if unsure during the exam.

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