Question

Speed of transverse wave on a straight wire (mass $6.0g$, length $60cm$ and area of cross-section $1.0m{m}^2$) is $90m/s$. If the Young’s modulus of wire is $16\times {10}^{11}N{m}^{-2}$, the extension of wire over its natural length is

• A. $0.03mm$
• B. $0.02mm$
• C. $0.04mm$
• D. $0.01mm$

Answer 1

The correct option is A $0.03mm$

Given, $M=6grams=6\times {10}^{-3}kg$
$$\begin{array}{cc}& L=60cm=0.6m\\ & A=1m{m}^2=1\times {10}^{-6}{m}^2\\ & \text{Using the relation,}{v}^2=\frac{T}{\mu }\\ & \Rightarrow T=\mu v^2=v^2\times \frac{M}{L}\\ \text{As Young’s modulus,}& Y=\frac{stress}{strain}\\ & strain=\frac{stress}{Y}=\frac{T}{AY}\\ & strain=\frac{\Delta L}{L}=\frac{v^2\frac{M}{L}}{AY}=v^2\frac{M}{AYL}\\ & \Rightarrow \Delta L=\frac{v^{2}M}{AY}\\ & \Delta L=\frac{8100\times 6\times {10}^{-3}}{1\times {10}^{-6}\times 16\times {10}^{11}}\\ & \Delta L=0.03mm\end{array}$$