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Elasticity formulas

Master Elasticity through 22 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Elasticity, every formula

22 formulas, typeset and free. Print it, or keep it open beside your practice.

Stress
$stress = \frac{F}{A}$
Longitudinal strain
$strain = \frac{Δ L}{L}$
Hooke's law
$stress \propto strain$
within elastic limit
Q1MCQHooke's law
Within the elastic limit, Hooke's law states:
- A
  stress $\propto$ strain
- B
  stress $\propto$ strain$^{2}$
- C
  stress = constant
- D
  strain = 0
Young's modulus
$Y = \frac{stress}{longitudinal strain} = \frac{F L}{A Δ L}$
Q1MCQYoung's modulus
Young's modulus is:
- A
  $\dfrac{FL}{A\,\Delta L}$
- B
  $\dfrac{A\,\Delta L}{FL}$
- C
  $\dfrac{F}{A}$
- D
  $\dfrac{\Delta L}{L}$
Bulk modulus
$B = - \frac{Δ P}{Δ V / V}$
Q1MCQBulk modulus
The bulk modulus is:
- A
  $-\dfrac{\Delta P}{\Delta V/V}$
- B
  $\dfrac{\Delta V/V}{\Delta P}$
- C
  $\dfrac{F}{A}$
- D
  $\dfrac{\Delta L}{L}$
Compressibility
$K = \frac{1}{B}$
Q1MCQCompressibility
Compressibility equals:
- A
  $\dfrac{1}{B}$
- B
  $B$
- C
  $B^{2}$
- D
  $\dfrac{B}{2}$
Shear (rigidity) modulus
$η = \frac{shear stress}{shear strain} = \frac{F / A}{θ}$
Q1MCQShear modulus
The rigidity (shear) modulus is:
- A
  $\dfrac{F/A}{\theta}$
- B
  $\dfrac{\theta}{F/A}$
- C
  $\dfrac{FL}{A\,\Delta L}$
- D
  $\dfrac{\Delta P}{\Delta V/V}$
Poisson's ratio
$σ = - \frac{lateral strain}{longitudinal strain}$
Q1MCQPoisson's ratio
Poisson's ratio is defined as:
- A
  $-\dfrac{\text{lateral strain}}{\text{longitudinal strain}}$
- B
  $\dfrac{\text{stress}}{\text{strain}}$
- C
  $\dfrac{F}{A}$
- D
  $\dfrac{\Delta L}{L}$
Elastic potential energy
$U = \frac{1}{2} \times stress \times strain \times volume$
Q1MCQElastic PE
The elastic potential energy in a stretched wire is:
- A
  $\tfrac12\times\text{stress}\times\text{strain}\times V$
- B
  $\text{stress}\times\text{strain}$
- C
  $\tfrac12 F\Delta L^{2}$
- D
  $YV$
Energy density
$u = \frac{1}{2} \frac{stress ^{2}}{Y} = \frac{1}{2} Y (strain)^{2}$
Q1MCQEnergy density
The elastic energy density is:
- A
  $\tfrac12\dfrac{\text{stress}^{2}}{Y}$
- B
  $Y\,\text{strain}$
- C
  $\dfrac{\text{stress}}{Y}$
- D
  $\dfrac{F}{A}$
Force constant of a wire
$k = \frac{Y A}{L}$
Q1MCQForce constant
The force constant of a wire is:
- A
  $\dfrac{YA}{L}$
- B
  $\dfrac{YL}{A}$
- C
  $YAL$
- D
  $\dfrac{L}{YA}$
Extension under load
$Δ L = \frac{F L}{A Y}$
Q1MCQExtension
The extension of a wire under load $F$ is:
- A
  $\dfrac{FL}{AY}$
- B
  $\dfrac{AY}{FL}$
- C
  $\dfrac{F}{AY}$
- D
  $FLY$
Thermal stress
$stress = Y α Δ T$
Q1MCQThermal stress
The thermal stress in a clamped rod heated by $\Delta T$ is:
- A
  $Y\alpha\Delta T$
- B
  $\alpha\Delta T$
- C
  $\dfrac{Y}{\alpha\Delta T}$
- D
  $Y\Delta T$
Work done in stretching
$W = \frac{1}{2} F Δ L$
Q1MCQWork stretching
The work done in stretching a wire by $\Delta L$ (force $F$) is:
- A
  $\tfrac12 F\Delta L$
- B
  $F\Delta L$
- C
  $\tfrac12 F\Delta L^{2}$
- D
  $2F\Delta L$
Elongation due to own weight
$Δ L = \frac{ρ g L ^{2}}{2 Y}$
Q1MCQOwn-weight elongation
The elongation of a wire due to its own weight is:
- A
  $\dfrac{\rho g L^{2}}{2Y}$
- B
  $\dfrac{\rho g L}{Y}$
- C
  $\dfrac{\rho g L^{2}}{Y}$
- D
  $\dfrac{\rho g}{2Y}$
Poisson's ratio range
$- 1 \leq σ \leq 0.5$
Q1MCQPoisson range
The theoretical range of Poisson's ratio is:
- A
  $-1$ to $0.5$
- B
  $0$ to $1$
- C
  $-0.5$ to $1$
- D
  $0$ to $0.5$ only
Volume strain
$\frac{Δ V}{V} = (1 - 2 σ) \frac{Δ L}{L}$
Q1MCQVolume strain
The volume strain in terms of $\sigma$ and longitudinal strain is:
- A
  $(1-2\sigma)\dfrac{\Delta L}{L}$
- B
  $(1+2\sigma)\dfrac{\Delta L}{L}$
- C
  $\sigma\dfrac{\Delta L}{L}$
- D
  $\dfrac{\Delta L}{L}$
Breaking stress
$stress_{m a x} = \frac{F _{b r e ak}}{A}$
Q1MCQBreaking stress
The breaking stress of a wire depends on:
- A
  the material, not the length
- B
  the length only
- C
  the weight only
- D
  nothing
Interatomic force constant
$k = Y r_{0}$
r_0 = interatomic spacing
Q1MCQInteratomic constant
The interatomic force constant is:
- A
  $Y r_0$
- B
  $\dfrac{Y}{r_0}$
- C
  $Y r_0^{2}$
- D
  $\dfrac{r_0}{Y}$
Torsion of a wire
$τ = \frac{π η r ^{4}}{2 L} θ$
Q1MCQTorsion
The restoring torque per unit twist of a wire is:
- A
  $\dfrac{\pi\eta r^{4}}{2L}$
- B
  $\dfrac{\pi\eta r^{2}}{2L}$
- C
  $\dfrac{2L}{\pi\eta r^{4}}$
- D
  $\eta r$
Elastic after-effect
$delay in returning to original state$
Q1MCQElastic after-effect
The delay in a body returning to its original shape after removing stress is called:
- A
  elastic after-effect
- B
  elastic fatigue
- C
  plasticity
- D
  ductility
Stress–strain: elastic limit
$limit beyond which permanent deformation occurs$
Q1MCQElastic limit
The stress beyond which a body does not return to its original shape is the:
- A
  elastic limit
- B
  breaking stress
- C
  yield strain
- D
  bulk modulus

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Elasticity formulas

Elasticity, every formula

Stress

Longitudinal strain

Hooke's law

Young's modulus

Bulk modulus

Compressibility

Shear (rigidity) modulus

Poisson's ratio

Elastic potential energy

Energy density

Force constant of a wire

Extension under load

Thermal stress

Work done in stretching

Elongation due to own weight

Poisson's ratio range

Volume strain

Breaking stress

Interatomic force constant

Torsion of a wire

Elastic after-effect

Stress–strain: elastic limit

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