JEE/NEET Physics · Mechanical Properties of Solids series · Part 3 of 4 · All parts →
- Shear modulus G = shear stress/strain — resistance to sliding layers
- Bulk modulus B = pressure/(ΔV/V) — resistance to volume squeeze; its reciprocal is compressibility
- Poisson’s ratio: stretch a wire, it thins — σ links the two
- Elastic PE stored: U = ½ × stress × strain × volume = ½FΔL (the spring’s ½kx² in disguise)
- Thermal stress: a clamped rod that can’t expand develops stress YαΔT
Steel can be stretched, slid, squeezed and heated — and every one of those has its own stiffness number. Plus: deformed solids store energy like batteries, which is what makes springs springs. Part 3 of the Mechanical Properties of Solids series.
- Shear modulus: the slide rating
- Bulk modulus: the squeeze rating
- Poisson’s ratio: the thinning effect
- Elastic potential energy
- Thermal stress
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
Shear Modulus: The Slide Rating
Push the top of a book sideways while the bottom stays: layers slide. The shear modulus G grades resistance to this sliding. Liquids have G = 0 (they can’t resist sliding — that’s roughly what being liquid means).
Bulk Modulus: The Squeeze Rating
Its reciprocal is compressibility. Water’s high B (~2×10⁹ Pa) makes hydraulic brakes work — the fluid refuses to shrink. Solids’ B is larger still; gases are the soft ones.
Poisson’s Ratio: The Thinning Effect
Stretch a rubber band and it visibly thins: lengthwise stretch comes with crosswise contraction. Poisson’s ratio σ = −(lateral strain)/(longitudinal strain), typically ~0.3 for metals, 0.5 for rubber (volume-preserving). Nothing stretches for free.
Elastic Potential Energy
Work done stretching a Hookean object = area under the force-extension triangle:
Thermal Stress
Heat a rail that’s welded tight: it wants to expand (ΔL = αLΔT) but can’t — so strain becomes stress: σ_thermal = YαΔT. Engineless but real: railways have expansion gaps, bridges have roller joints for exactly this reason.
Solved Examples
U = ½kx² = ½ × 10⁵ × (0.002)² = 0.2 J.
✔
Answer: 0.2 J
u = ½ × stress × strain = ½ × 2×10⁸ × 10⁻³ = 10⁵ J/m³.
✔
Answer: 10⁵ J/m³
σ = YαΔT = 2×10¹¹ × 1.2×10⁻⁵ × 30 = 7.2×10⁷ Pa — over a third of steel’s elastic limit, from sunshine alone.
Now every expansion gap in a rail line makes sense. ✔
Answer: 7.2 × 10⁷ Pa
- Forgetting the minus sign in B. Pressure increase gives volume DECREASE: the sign keeps B positive.
- Using ½stress×strain for non-elastic stretches. The ½ only exists where Hooke’s straight line holds.
- Thermal stress without expansion blocked. A free rod just grows longer (zero stress); stress appears only when the growth is refused.
- Confusing G and B. G = slide resistance (shape change, volume same); B = squeeze resistance (volume change, shape same).
This Physics in Your Daily Life
- Railway expansion gaps and bridge roller joints — engineered refusals of YαΔT: leave room or the steel makes its own (buckled tracks on hot days).
- Hydraulic brakes and hydraulic presses — work because fluid compressibility is tiny: push here, move there, nothing squished in between.
- Archery bows and catapults — elastic energy storage weapons: ½(stress)(strain)V released in milliseconds.
- Memory-foam pillows — low shear modulus materials that flow around your head slowly: comfort as tuned G.
- Balloon rubber thinning as it inflates — Poisson’s ratio near 0.5 in action: volume nearly preserved as shape transforms.
Stretch a slingshot and the rubber visibly loads up — let go and it spends everything on the stone. Nothing moved while you held it, yet the energy sat there, in trillions of stretched atomic bonds, waiting. A deformed solid is a loaded spring that forgot to look like one.
0.2 J sounds small — but released in 1 ms it’s 200 W of power delivered to a pebble. A longbow at full draw (~100 J over 0.5 m) throws an arrow through armour: elastic storage converts slow muscle work into explosive delivery.
Graph force against extension: a straight rising line. The stored energy is the shaded triangle under it (½Fx). Release, and the area converts into kinetic energy: the graph’s geometry IS the weapon’s ballistic budget.
Practice set (answers hidden — try first)
(NEET-level) Shear modulus of liquids:
(JEE Main-level) u = ½×2×10⁷×4×10⁻⁴ =
(NEET-level) Compressibility is the reciprocal of:
(Concept) A freely expanding heated rod develops:
(JEE Main-level) Y = 10¹¹, α = 2×10⁻⁵, ΔT = 40: thermal stress =
- G resists shear (liquids: G = 0)
- B resists volume squeeze; 1/B = compressibility
- Poisson: stretch thins (σ ≈ 0.3 metals)
- U = ½FΔL = ½(stress)(strain)V
- thermal stress = YαΔT when expansion is blocked
- 🔁 shear vs bulk modulus
- 🔁 Poisson’s ratio meaning
- 🔁 elastic energy formulas
- 🧠 Chant: ‘slide G, squeeze B, store one-half’.
- 🧠 Thermal: ‘refused expansion becomes stress’.
- 🏠 Daily: rail gaps exist because of YαΔT.
- 🏠 Daily: bows and slingshots = ½stress×strain×V weapons.
Quick revision
- Shear modulus G = shear stress/strain — resistance to sliding layers
- Bulk modulus B = pressure/(ΔV/V) — resistance to volume squeeze; its reciprocal is compressibility
- Poisson’s ratio: stretch a wire, it thins — σ links the two
- Elastic PE stored: U = ½ × stress × strain × volume = ½FΔL (the spring’s ½kx² in disguise)
- Thermal stress: a clamped rod that can’t expand develops stress YαΔT
- Shear modulus: the slide rating
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