Class 11 · Chapter 9

Mechanical Properties of Solids

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Mechanical Properties of Solids Formula Sheet

9 formulas across 3 topics in Mechanical Properties of Solids.

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Updated 2026-07-03 · v1.0.0

Stress & Strain

Stress is the internal restoring force per unit area; strain is the fractional deformation. Their ratio — within the elastic limit — defines the material.

Stress = F / A

Stress

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
F	internal restoring force (equals applied force in equilibrium)	N	[MLT⁻²]
A	cross-sectional area	m²	[L²]

Valid when

Unit N/m² (pascal) — same unit and dimensions as pressure, but stress is not a scalar like pressure

Common mistakes

Longitudinal (tensile/compressive), shear (tangential) and volumetric stress are three different animals — identify which before picking a modulus

Read the concept →

Strain = ΔL / L (longitudinal); ΔV/V (volumetric); tanφ ≈ φ (shear)

Strain (three types)

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
ΔL	change in length	m	[L]
L	original length	m	[L]
φ	shear angle	rad	[M⁰L⁰T⁰]

Valid when

Dimensionless — it is a ratio

Read the concept →

Worth remembering

Hooke's law (stress ∝ strain) holds only up to the proportional limit — the stress-strain curve then bends through elastic limit, yield point, and fracture
Elastomers (rubber) stretch enormously WITHOUT obeying Hooke's law and without permanent set

Elastic Moduli

Three moduli for three deformation modes: Young's for stretching, bulk for squeezing, shear for twisting. All share the unit of stress.

Y = (F/A) / (ΔL/L) = FL / (A ΔL)

Young's modulus

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
Y	Young's modulus of the materialsteel ≈ 2 × 10¹¹ N/m² — property of MATERIAL, not of the wire	N/m²	[ML⁻¹T⁻²]

Valid when

Within the proportional limit
Solids only

Common mistakes

A thicker or longer wire has the SAME Y — geometry changes ΔL, never Y
Y(steel) > Y(copper) > Y(rubber): larger Y means HARDER to stretch, so steel is more elastic than rubber in physics language

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ΔL = FL / (AY) = MgL / (πr²Y)

Elongation of a loaded wire

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Rearranging the Young's modulus definition

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
ΔL	extension	m	[L]
r	wire radius	m	[L]

Valid when

Wire stretching under its OWN weight: ΔL = ρgL²/2Y (half the naive value, because tension varies along the wire)

Common mistakes

Same load, wire of double radius → ΔL becomes ¼ (A ∝ r²) — ratio questions live here

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B = −ΔP / (ΔV/V); compressibility k = 1/B

Bulk modulus & compressibility

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
B	bulk modulus	N/m²	[ML⁻¹T⁻²]
ΔP	change in pressure	N/m²	[ML⁻¹T⁻²]
ΔV/V	fractional volume change	dimensionless	[M⁰L⁰T⁰]

Valid when

Minus sign keeps B positive (pressure up, volume down)
Defined for solids, liquids AND gases

Common mistakes

For gases B depends on the process: isothermal B = P, adiabatic B = γP

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η = (F/A) / φ

Shear modulus (modulus of rigidity)

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
η	shear modulus	N/m²	[ML⁻¹T⁻²]
φ	shear angle	rad	[M⁰L⁰T⁰]

Valid when

Solids only — fluids cannot sustain shear stress at rest (that is what makes them fluids)

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σ = − (lateral strain) / (longitudinal strain)

Poisson's ratio

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
σ	Poisson's ratiotheoretical range −1 to 0.5; most metals ≈ 0.3	dimensionless	[M⁰L⁰T⁰]

Valid when

σ = 0.5 means volume stays constant on stretching (ideal rubber)

Read the concept →

Worth remembering

Y, B and η exist only within elastic behaviour; beyond the yield point the material flows plastically and no modulus applies

Elastic Potential Energy

Stretching a wire stores energy exactly like compressing a spring — the wire IS a very stiff spring with k = YA/L.

u = ½ × stress × strain = ½ Y (strain)²

Elastic energy density

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Work done per unit volume = area under the stress-strain curve

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
u	energy stored per unit volume	J/m³	[ML⁻¹T⁻²]

Common mistakes

The factor ½ exists because force GROWS from zero — using full F × ΔL doubles the answer

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k(wire) = YA / L

Effective spring constant of a wire

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Comparing F = (YA/L)ΔL with F = kx

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
k(wire)	equivalent spring constant	N/m	[MT⁻²]

Valid when

Wires in series/parallel combine exactly like springs

Read the concept →

Worth remembering

Breaking force depends on AREA only, not length — a longer rope is not weaker; breaking STRESS is the material property

NEET keeps to direct modulus and energy-density substitutions; JEE builds composite rods and wire-block oscillation systems from k = YA/L.

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