Vardaan Institute - Detailed Solutions: Master Sheet (Mechanical Properties of Solids)

Section A: Stress-Strain Solutions

Stress-Strain graph features.

Sol: Proportional limit satisfies Hooke's Law ($\sigma \propto \epsilon$). Yield point marks permanent set. Ultimate strength is max stress; fracture point is where material breaks.

A steel wire of $L=2\text{ m}, A=2\text{ mm}^2$ under $F=100\text{ N}$. Find stress, strain, elongation.

Sol: Stress $= \frac{F}{A} = \frac{100}{2 \times 10^{-6}} = 5 \times 10^7\text{ Pa}$. Strain $= \frac{\text{Stress}}{Y} = \frac{5 \times 10^7}{2 \times 10^{11}} = 2.5 \times 10^{-4}$. $\Delta L = \text{Strain} \times L = 2.5 \times 10^{-4} \times 2 = 0.5\text{ mm}$.

Section B: Elastic Moduli & Energy Solutions

Derive elastic potential energy density stored in a stretched wire.

Sol: Work $W = \frac{1}{2} F \Delta L$. Volume $V = A L$. Energy density $u = \frac{W}{V} = \frac{1}{2} \left(\frac{F}{A}\right) \left(\frac{\Delta L}{L}\right) = \frac{1}{2} \times \text{stress} \times \text{strain}$.

Calculate fractional change in volume of a sphere under $P = 10^7\text{ Pa}$ ($B = 1.4 \times 10^{11}\text{ Pa}$).

Sol: $B = \frac{P}{\Delta V/V} \implies \frac{\Delta V}{V} = \frac{P}{B} = \frac{10^7}{1.4 \times 10^{11}} = 7.14 \times 10^{-5}$. Percentage volume change $= 0.00714\%$.

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