JEE/NEET Physics · Mechanical Properties of Solids series · Part 2 of 4 · All parts →
- Hooke’s law: within the elastic limit, stress ∝ strain — stretch is proportional to pull
- Young’s modulus Y = stress/strain — the stiffness rating of the material itself
- Big Y = stiff (steel ~2×10¹¹ Pa); small Y = flexible (rubber ~10⁶ Pa)
- Y depends only on the MATERIAL, not the shape — wire or girder of steel share Y
- Beyond the elastic limit: permanent deformation, then fracture
Stretch a spring twice as far and it pulls twice as hard — Robert Hooke’s 1660s discovery, ‘ut tensio, sic vis’: as the extension, so the force. Every material is a spring; Young’s modulus is its stiffness score. Part 2 of the Mechanical Properties of Solids series.
- Hooke’s law: the proportional promise
- Young’s modulus: the stiffness score
- The stress-strain graph’s five zones
- Elastic limit and breaking
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
Hooke’s Law: The Proportional Promise
Within each material’s comfort zone, doubling stress doubles strain: a straight-line relationship. The spring version you know (F = kx) is the same law; Young’s modulus is the material version, stripped of shape.
Young’s Modulus: The Stiffness Score
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| Y | Young’s modulus — stiffness score | Pa; steel ≈ 2×10¹¹, rubber ≈ 10⁶ |
| k | spring constant of the whole object | Y in disguise: k = YA/L |
| elastic limit | stress beyond which stretch becomes permanent | material-specific |
The Stress-Strain Graph’s Five Zones
| Zone | Behaviour | Meaning |
|---|---|---|
| 1 Proportional | straight line | Hooke’s law lives here |
| 2 Elastic | slightly curved | still springs back fully |
| 3 Yield | long flat region | metal flows with little extra stress |
| 4 Strain-hardening | curves up again | material toughens as it deforms |
| 5 Fracture | ends at breaking point | the wire snaps |
Elastic Limit and Breaking
Below the elastic limit, remove the load and the material returns exactly — a steel wire is a perfect trampoline. Beyond it, atoms have slid into new arrangements: permanent set. Further still, micro-cracks win: fracture. Engineering keeps every working stress well inside zone 1.
Solved Examples
ΔL = FL/(AY) = 100 × 2/(10⁻⁶ × 2×10¹¹) = 1 mm.
✔
Answer: 1 mm
ΔL ∝ 1/Y → aluminium stretches ≈2.9× more.
Same shape, same pull: stiffness is purely the material’s vote. ✔
Answer: Al stretches ~2.9× more
F = (YA/L)ΔL — force proportional to extension.
k = YA/L: long thin wires are soft springs, short fat ones stiff.
Every object is a spring; geometry just tunes the constant. ✔
Answer: k = YA/L
- Using Hooke’s law beyond the elastic limit. Proportionality dies at the end of zone 1 — extrapolating to fracture overestimates strength absurdly.
- Treating Y as shape-dependent. Y is a MATERIAL property; the object’s stiffness (k) is Y dressed in geometry.
- Inverting the ratio. Y = stress/strain (big for stiff); strain/stress would make rubber the ‘stiff’ one.
- Skipping unit conversion. Mix mm² with pascals and the answer is wrong by 10⁶+; convert area and length first.
This Physics in Your Daily Life
- Mattresses and shoe soles — foam’s tiny Y gives cushioning; the steel springs inside carry the big loads: comfort is a modulus sandwich.
- Crane cables and bridge cables — steel’s enormous Y keeps extension millimetric under tonnes: bridges don’t sag visibly for the same reason.
- Dental braces — apply steady sub-yield stress: teeth move through bone while the wire stays in its elastic zone for months.
- Gymnastics floors and diving boards — layered wood/fibreglass with tuned Y store and return jump energy: sports surfaces as engineered springs.
- Smartphone screens — glass with engineered high modulus and pre-stressed surfaces resist the pocket stresses of daily life.
Two rods, identical size — one steel, one rubber. Hang the same weight: steel doesn’t visibly budge, rubber droops. Both felt the same stress; their answers differ a thousandfold. Young’s modulus is the number that grades that answer: the material’s ‘firmness of no’.
Y_steel = 2×10¹¹ Pa: a strain of 10⁻³ (0.1%) needs 2×10⁸ Pa of stress. Y_rubber = 10⁶: the same strain needs only a thousandth of that. Same question, thousandfold different demands — that’s the whole table of materials in two numbers.
The stress-strain graph: a steep straight line for steel (big stress, tiny strain), a lazy shallow slope for rubber (big strains for small stress). Y is the slope. Engineering materials are picked by slope the way musicians pick strings by tension.
Practice set (answers hidden — try first)
(NEET-level) Y’s unit:
(JEE Main-level) ΔL for F = 200 N, L = 1 m, A = 1 mm², Y = 2×10¹¹:
(NEET-level) Same Y and load: doubling length does what to extension?
(Concept) Rubber vs steel Young’s modulus:
(JEE Main-level) Wire as spring: k = YA/L. Y=2×10¹¹, A=1 mm², L=2 m: k =
- Hooke: stress ∝ strain within the elastic limit
- Y = stress/strain = FL/(AΔL)
- big Y stiff (steel), small Y soft (rubber)
- k = YA/L — every object is a spring
- beyond limit: permanent set, then fracture
- 🔁 Hooke’s law and its zone
- 🔁 Y = FL/(AΔL)
- 🔁 Y material-only; k = YA/L shape-tuned
- 🧠 Chant: ‘stiff is big-Y, sloppy is small’.
- 🧠 k = YA/L — the wire is a spring in disguise.
- 🏠 Daily: mattress = modulus sandwich (foam + steel).
- 🏠 Daily: bridges don’t sag because steel’s Y is huge.
Quick revision
- Hooke’s law: within the elastic limit, stress ∝ strain — stretch is proportional to pull
- Young’s modulus Y = stress/strain — the stiffness rating of the material itself
- Big Y = stiff (steel ~2×10¹¹ Pa); small Y = flexible (rubber ~10⁶ Pa)
- Y depends only on the MATERIAL, not the shape — wire or girder of steel share Y
- Beyond the elastic limit: permanent deformation, then fracture
- Hooke’s law: the proportional promise
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