NEET Physics — Chapter 9

Mechanical Properties of Solids

Stress, strain, Hooke's law, Young's modulus, bulk modulus, shear modulus, stress-strain curve, Poisson's ratio — complete NEET notes with diagrams and exam traps.

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1. Elasticity — Stress, Strain, and Hooke's Law

Elasticity is the property of a body to regain its original shape and size after removal of deforming forces. A body is elastic if it recovers completely; it is plastic if it does not.

Stress is the restoring force per unit area developed inside a body:

ext{Stress} = rac{F}{A}

SI unit: N/m² = Pascal (Pa). Dimensional formula: $[ML^{-1}T^{-2}]$ .

Types: Tensile/Compressive stress (normal force), Shear stress (tangential force).

Strain is the fractional change in dimension (dimensionless):

Longitudinal strain = $Delta L / L$ (change in length / original length)

Volumetric strain = $Delta V / V$ (change in volume / original volume)

Shear strain = $anphi approx phi$ (angular deformation for small $phi$ )

Hooke's Law: Within the elastic limit, stress is directly proportional to strain:

ext{Stress} propto ext{Strain} implies ext{Stress} = E imes ext{Strain}

where $E$ is the modulus of elasticity. Hooke's Law fails beyond the elastic limit.

NEET tip: Stress has the same units as pressure (Pa = N/m²). Strain is dimensionless. Modulus of elasticity (E, G, K) has the same dimensions as stress:

[ML^{-1}T^{-2}]

2. Elastic Moduli — Young's, Bulk, and Shear

Young's Modulus (Y): Relates longitudinal stress to longitudinal strain (for wires/rods):

Y = rac{ ext{Longitudinal Stress}}{ ext{Longitudinal Strain}} = rac{F/A}{Delta L/L} = rac{FL}{ADelta L}

For a wire of length $L$ , cross-sectional area $A$ , extension $Delta L$ under force $F$ .

Bulk Modulus (B or K): Relates volume stress (pressure) to volumetric strain:

B = - rac{Delta P}{Delta V/V} = -V rac{dP}{dV}

Negative sign: increase in pressure → decrease in volume. Liquids and gases have B but no Y or G.

Compressibility: $K = 1/B$

Shear Modulus / Modulus of Rigidity (G or η): Relates shear stress to shear strain:

G = rac{ ext{Shear Stress}}{ ext{Shear Strain}} = rac{F/A}{ anphi} approx rac{F}{Aphi}

Relative magnitudes: For most solids, $Y > B > G$ (approximately). Gases have $B$ only; liquids have $B$ and $G approx 0$ (very small).

Modulus	Stress type	Applies to
Young's (Y)	Longitudinal	Solids only
Bulk (B)	Volumetric (pressure)	Solids, liquids, gases
Shear (G)	Tangential	Solids only

3. Stress–Strain Curve for a Ductile Material

The stress–strain graph reveals the elastic and plastic behaviour of a material:

Key points on the curve:

O→A (Proportional limit): Hooke's law holds — linear, perfectly elastic
A→B (Elastic limit): Not perfectly linear but still elastic (returns to original shape)
B (Yield point): Permanent deformation begins — upper yield point
B→C (Plastic region): Strain increases with little stress — material "flows"
C→D (Strain hardening): Material strengthens — stress must increase again
D (Ultimate stress/Tensile strength): Maximum stress the material can withstand
D→E (Necking and fracture): Material narrows (necking) and breaks at E

NEET tip: Slope of the linear portion O→A = Young's modulus. Area under the stress-strain curve = elastic potential energy per unit volume stored in the material.

4. Elastic PE, Poisson's Ratio, and Thermal Stress

Elastic Potential Energy stored in a stretched wire:

U = rac{1}{2} imes ext{Stress} imes ext{Strain} imes ext{Volume} = rac{1}{2} rac{F cdot Delta L}{1} = rac{FDelta L}{2}

Energy per unit volume = $rac{1}{2} imes ext{stress} imes ext{strain} = rac{ ext{stress}^2}{2Y} = rac{Y imes ext{strain}^2}{2}$

Poisson's Ratio ( $sigma$ or $ u$): When a wire is stretched longitudinally, it contracts laterally:

sigma = - rac{ ext{lateral strain}}{ ext{longitudinal strain}} = - rac{Delta D/D}{Delta L/L}

Theoretical range: $-1 leq sigma leq 0.5$ . For most materials: $0 < sigma < 0.5$ . For rubber: $sigma approx 0.5$ (incompressible). Cork: $sigma approx 0$ (no lateral change — used in wine bottles).

Thermal Stress: If a rod of length $L$ is clamped at both ends and temperature changes by $Delta T$ :

ext{Thermal strain} = alphaDelta T quadimpliesquad ext{Thermal stress} = YalphaDelta T

$alpha$ = coefficient of linear thermal expansion. The clamped rod exerts compressive force if heated, tensile force if cooled.

Pro tip: For a wire: spring constant

k = YA/L

. So a thicker wire (larger A) is stiffer, and a longer wire (larger L) is more compliant. When two wires of the same material are joined in series, their effective k:

1/k = 1/k_1 + 1/k_2

5. NEET Traps & Formula Summary

Trap 1 — Steel is more elastic than rubber: Elasticity is the ability to recover — not the ability to stretch. Steel returns to original shape under large stress; rubber can be stretched easily but still returns. Steel has higher Young's modulus = less strain for same stress = more "elastic" in physics.

Trap 2 — Bulk modulus of gas depends on process: For isothermal process:

B = P

(pressure). For adiabatic process:

B = gamma P

. This is why sound travels faster in adiabatic conditions.

Trap 3 — Elastic PE is not stored in plastic deformation: Energy in plastic deformation is converted to heat/deformation — not recoverable. Only energy in the elastic region is recoverable.

Trap 4 — Poisson's ratio range:

sigma

is always between −1 and 0.5 theoretically, and between 0 and 0.5 for real materials. Values above 0.5 are physically impossible (violates energy conservation).

Formula Sheet:

Stress	$sigma = F/A$
Young's modulus	$Y = FL/(ADelta L)$
Bulk modulus	$B = -Delta P/({Delta V/V})$
Shear modulus	$G = F/(Aphi)$
Elastic PE/vol	$u = rac{1}{2}sigma arepsilon = rac{sigma^2}{2Y}$
Poisson's ratio	$ u = -(Delta D/D)/(Delta L/L)$
Wire spring const.	$k = YA/L$
Thermal stress	$= YalphaDelta T$

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