1.A heavy uniform rod of mass $m$, length $L$, and cross-sectional area $A$ hangs vertically from a ceiling. Find the total elongation $\Delta L$ due to its own weight.
Sol: Tension at distance $y$ from bottom: $T(y) = \frac{m g}{L} y$. $\Delta L = \int_0^L \frac{T(y)}{A Y} dy = \frac{m g}{A Y L} \int_0^L y dy = \frac{m g L}{2 A Y}$.
2.A steel wire of radius $r$ and length $L$ is fixed at one end and rotated in a horizontal circle with angular velocity $\omega$ carrying a mass $M$ at its free end. Find total elongation of the wire.
Sol: Centrifugal tension $T = M \omega^2 L$. Elongation $\Delta L = \frac{F L}{A Y} = \frac{(M \omega^2 L) L}{\pi r^2 Y} = \frac{M \omega^2 L^2}{\pi r^2 Y}$.
