1.Derive expression for elastic potential energy density $u = \frac{1}{2} \times \text{stress} \times \text{strain}$ stored in a stretched wire.
Sol: Work done $W = \int_0^x F dx = \int_0^x \left(\frac{YA}{L} x\right) dx = \frac{1}{2} \frac{YA}{L} x^2$. Energy density $u = \frac{W}{A L} = \frac{1}{2} \left(\frac{F}{A}\right) \left(\frac{x}{L}\right) = \frac{1}{2} \times \text{stress} \times \text{strain}$.
2.A spherical ball of volume $V=0.5\text{ m}^3$ is subjected to a uniform pressure $P=10^7\text{ Pa}$. If Bulk modulus $B = 1.6 \times 10^{11}\text{ Pa}$, find volume change $\Delta V$.
Sol: $B = \frac{P}{\Delta V / V} \implies \Delta V = \frac{P V}{B} = \frac{10^7 \times 0.5}{1.6 \times 10^{11}} = 3.125 \times 10^{-5}\text{ m}^3 = 31.25\text{ cm}^3$.
