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Elasticity revision notes

Every idea in the Elasticity chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 25 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Stress, strain and Hooke's law

Stress
6× asked, last 2025
stress=FA(N m−2=Pa)\text{stress}=\frac{F}{A}\qquad(\text{N m}^{-2}=\text{Pa})
Restoring force per unit area. Normal stress: FF perpendicular to AA (tensile or compressive). Shear stress: FF along AA.
  • Hollow column with radii r1<r2r_1<r_2: the load-bearing area is π(r22−r12)\pi(r_2^2-r_1^2). A load shared equally by nn identical columns puts F/nF/n on each.
  • Not πd2\pi d^2. Area is πr2=πd2/4\pi r^2=\pi d^2/4. Halve a diameter, but use a given radius as it is. Convert mm or cm to m before squaring.
Strain
2× asked, last 2024
ΔLL (longitudinal),ΔVV (volumetric),θ≈Δxh (shear)\frac{\Delta L}{L}\ (\text{longitudinal}),\qquad \frac{\Delta V}{V}\ (\text{volumetric}),\qquad \theta\approx\frac{\Delta x}{h}\ (\text{shear})
Change divided by original size, so no unit. Shear strain is the tilt angle in radians.
FFLLΔL\Delta L
Longitudinal ΔL/L\Delta L/L
FFθ\theta
Shear: tilt θ≈Δx/h\theta\approx\Delta x/h
Volumetric ΔV/V\Delta V/V
Hooke's law
4× asked, last 2024
stress=E×strain\text{stress}=E\times\text{strain}
Valid only up to the proportional limit. EE = modulus of elasticity, unit N m⁻² like stress.
  • Elasticity comes from interatomic forces that pull displaced atoms back. So a modulus depends only on the material and temperature: changing length or area changes the extension, not YY. YY falls as temperature rises.
Stress–strain curve
1× asked, last 2023
Slope of the straight part = YY. Within the elastic limit, area under the curve = energy stored per unit volume.
strainstress12345brittle: breaksslope =Y=Y
1 proportional limit, 2 elastic limit, 3 yield point, 4 ultimate stress, 5 fracture.
Ductile (blue): long plastic stretch; brittle (red) breaks just past the elastic limit
  • In order: proportional limit (Hooke's law ends), elastic limit (full recovery below it), yield point (plastic flow starts), ultimate stress (maximum), fracture point.
  • Ductile metals (copper, mild steel) have a long plastic region; brittle ones (glass, cast iron) break just past the elastic limit. Elastomers (rubber) stretch hugely and recover, but do not obey Hooke's law over most of it.
  • Not 'rubber is more elastic than steel'. More elastic means a larger modulus (more stress for the same strain). Steel's high YY and high elastic limit suit buildings and bridges.
  • Not stored energy beyond the elastic limit. There the area under the curve is the work done, and part of it is lost as heat.
Also remember
  • Interatomic potential U(r)U(r): equilibrium separation is where F=−dU/dr=0F=-dU/dr=0, the minimum of UU. The depth of that minimum is the bond (dissociation) energy.

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Practise Elasticity

21 more concepts in 6 topics

Young's modulus and stretched wires, Bulk modulus and compressibility, Shear modulus and Poisson's ratio, Elastic potential energy and Thermal stress, plus ideas from waves. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Young's modulus and stretched wires

  • Extension of a wire16×
  • Comparing two wires10×
  • Natural length from two loads7×
  • Find the tension first11×
  • Wires in series2×
  • Rod hanging under its own weight2×
  • Rod rotating about one end
  • Breaking stress and longest hanging wire4×
  • Searle's experiment3×
  • Sag of a beam loaded at its centre

Bulk modulus and compressibility

  • Bulk modulus12×
  • Compression at a depth5×
  • Density and side length under pressure3×
  • Bulk modulus of a gas1×

Shear modulus and Poisson's ratio

  • Modulus of rigidity3×
  • Poisson's ratio1×
  • Volume change of a stretched wire
  • Relations between the moduli2×

Elastic potential energy

  • Energy stored in a stretched wire2×
  • Energy density5×

Thermal stress

  • Rod held between rigid supports4×
  • Also remember1

Also asked here: waves

  • Key points1

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