Question

The speed of a transverse wave, travelling on a wire having a length of $100\text{cm}$ and mass $10\text{g}$ is $160\text{m/s}$. The area of cross-section of the wire is $2{\text{mm}}^2$ and its Young's modulus is $32\times {10}^{11}{\text{N/m}}^2$. Find the extension of the wire from its natural length.

• A. $0.08\text{mm}$
• B. $0.04\text{mm}$
• C. $0.02\text{mm}$
• D. $1\text{mm}$

Answer 1

The correct option is B $0.04\text{mm}$

Linear mass density $\left(\mu \right)=\frac{10\times {10}^{-3}}{100\times {10}^{-2}}={10}^{-2}\text{kg/m}$
Wave speed $\left(V\right)=\sqrt{\frac{T}{\mu }}$
$\Rightarrow T=\mu V^2$
$={10}^{-2}\times 160\times 160=256\text{N}$
As we know, Young's modulus is given by
$Y=\frac{T/A}{\Delta L/L}$
So, $\Delta L=\frac{TL}{AY}$
$=\frac{256\times 1}{2\times {10}^{-6}\times 32\times {10}^{11}}=4\times {10}^{-5}$
$=0.04\text{mm}$