A sonometer wire of length $$1.5\,m$$  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 ?

Solution :
Fundamental frequency,
$$\eqalign{ & f = \frac{v}{{2\ell }} = \frac{1}{{2\ell }}\sqrt {\frac{T}{\mu }} = \frac{1}{{2\ell }}\sqrt {\frac{T}{{A\rho }}} \cr & \left[ {\because v = \sqrt {\frac{T}{\mu }} \,\,{\text{and}}\,\,\mu = \frac{m}{\ell }} \right] \cr & {\text{Also,}}\,\,Y = \frac{{T\ell }}{{A\Delta \ell }} \Rightarrow \frac{T}{A} = \frac{{Y\Delta \ell }}{\ell } \cr & \Rightarrow f = \frac{1}{{2\ell }}\sqrt {\frac{{\gamma \Delta \ell }}{{\ell \rho }}} \,......\left( {\text{i}} \right) \cr & {\text{Putting}}\,{\text{the}}\,{\text{value of }}\,\,\ell ,\frac{{\Delta \ell }}{\ell },\rho \,\,{\text{and}}\,\gamma \,\,{\text{in}}\,{\text{e}}{{\text{q}}^{\text{n}}}{\text{.}}\left( {\text{i}} \right)\,{\text{we}}\,{\text{get}} \cr & f = \sqrt {\frac{2}{7}} \times \frac{{{{10}^3}}}{3} \cr & {\text{or,}}\,\,f \approx 178.2\,Hz \cr} $$