Question

A steel wire of length 1 m, mass 0.1 kg and uniform cross-sectional area ${10}^{-7}{m}^2$ is rigidly fixed at both ends. The temperature of the wire is decreased by $80/3^{\circ }C$. If transverse waves are set up in the wire, the frequency (in Hz) is ___ .

The Young’s modulus of steel = $2\times {10}^{11}N{m}^{-2}$ and coefficient of linear expansion of steel = $1.2\times {10}^{-5}pe{r}^{\circ }C$.

Answer 1

Contraction $\Delta L=\alpha L\Delta \theta$

Young's modulus $Y=\frac{TL}{A\Delta L};T$ = tension

$\therefore T=\frac{YA\Delta L}{L}=YA\alpha \Delta \theta$

$=2\times {10}^{11}\times {10}^{-7}\times 1.2\times {10}^{-5}\times \frac{80}{3}$

$=6.4N$

Frequency of fundamental mode is

$V=\frac{2}{2L}\sqrt{\frac{T}{\mu }};\mu$ = mass per unit length of wire = $\frac{1}{2\times 1}\sqrt{\frac{6.4}{0.1}}=4Hz$