MechanicsHigh Yield (1-2 Qs in JEE)

Properties of Matter & Elasticity Formulas

Hooke's law, Young's modulus, bulk modulus, shear modulus, elastic potential energy, and thermal stress.

8 Formulas•413+ Questions in Bank

Core Formulas & Mathematical Identities

Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F / A}{\Delta L / L}, \quad B = -\frac{\Delta P}{\Delta V / V}, \quad \eta = \frac{F / A}{\theta}

Young's, Bulk, and Shear moduli.

Examiner Tip: Compressibility K = 1 / B.

U = \frac{1}{2} \times \text{Force} \times \Delta L, \quad u = \frac{U}{\text{Volume}} = \frac{1}{2} \times \text{Stress} \times \text{Strain} = \frac{1}{2} Y (\text{Strain})^2

Work done in stretching a wire stored as elastic energy.

Examiner Tip: Area under Stress-Strain curve up to elastic limit gives energy density.

\text{Stress} = Y \alpha \Delta T, \quad F = Y A \alpha \Delta T

Force developed when temperature of clamped wire changes by ΔT.

Examiner Tip: α is the linear coefficient of thermal expansion.

\sigma = -\frac{\Delta r / r}{\Delta L / L}, \quad Y = 3B(1 - 2\sigma) = 2\eta(1 + \sigma), \quad \frac{9}{Y} = \frac{1}{B} + \frac{3}{\eta}

Lateral contraction strain to longitudinal extension strain ratio, and master algebraic interrelations among Y, B, η, and σ.

Examiner Tip: Theoretical limits of σ are -1 to 0.5; practical engineering materials lie between 0 and 0.5.

\Delta L_{\text{self}} = \frac{M g L}{2AY} = \frac{\rho g L^2}{2Y}

Elongation produced purely by the wire's own gravitational weight (mass M, density ρ, length L).

Examiner Tip: Elongation under self-weight is exactly HALF the elongation produced by hanging a mass M at the bottom end.

\delta = \frac{W L^3}{48 Y I_g} = \frac{W L^3}{4 b d^3 Y} \quad (\text{Rectangular bar with breadth } b, \text{ depth } d)

Sagging depression δ at the center of a beam of length L supported at both ends loaded with weight W.

Examiner Tip: To minimize sagging, depth d is made large compared to breadth b (I-shaped girders in construction).

B = -V\frac{dP}{dV}, \quad K = \frac{1}{B}, \quad \rho(h) = \rho_0\left(1 + \frac{\rho_0 g h}{B}\right)

Volumetric elasticity and fractional increase in liquid density at depth h under hydrostatic pressure.

Examiner Tip: Minus sign indicates volume decreases as pressure increases.

C = \frac{\pi \eta r^4}{2L}, \quad W = \frac{1}{2}C\theta^2 = \frac{\pi \eta r^4 \theta^2}{4L}

Restoring couple C per unit angle of twist in a circular shaft of shear modulus η, radius r, and length L.

Examiner Tip: A hollow cylinder is much more resistant to twisting than a solid cylinder of identical mass and length.

Frequently tested misconceptions in Properties of Matter & Elasticity

•Breaking stress depends only on the material, INDEPENDENT of wire length or cross-sectional area.
•Energy density in stretched wire is (1/2) × Stress × Strain.