A sonometer wire oflength \(1.5\ m\) is made of steel. The tension in it produces an elastic strain of \(1 \%\). What is the fundamental frequency of steel if density and elasticity of steel are \(7.7 \times 10^3 \) \(kg/m^3\) and \(2.2 \times 10^{11}\) \(N/m^2\) respectively?

\(f=\frac{v}{2 \ell}=\frac{1}{2 \ell} \sqrt{\frac{T}{\mu}}=\frac{1}{2 \ell} \sqrt{\frac{T}{A \rho}}\left[\because v=\sqrt{\frac{T}{\mu}} \text { and } \mu=\frac{m}{\ell}\right]\)

Also, \(Y=\frac{T \ell}{A \Delta \ell} \Rightarrow \frac{T}{A}=\frac{Y \Delta \ell}{\ell} \Rightarrow f=\frac{1}{2 \ell} \sqrt{\frac{\gamma \Delta \ell}{\ell \rho}}\)         \(...(i)\)

Putting the value of \(\ell, \frac{\Delta \ell}{\ell}, \rho\) and \(\gamma\) in eq \(^{n} .\) \((i)\) we get,

\(f=\sqrt{\frac{2}{7}} \times \frac{10^{3}}{3}\) or, \(f \approx 178.2 \mathrm{Hz}\)