A piano string $1.5\,m$ long is made of steel of density $7.7 \times 10^3 \,kg/m^3$ and Young’s modulus $2 \times 10^{11} \,N/m^2$. It is maintained at a tension which produces an elastic strain of $1\%$ in the string. The fundamental frequency of transverse vibrations of string is ......... $Hz$
Diffcult
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$\mathrm{Y}=\frac{\mathrm{F} \cdot \ell}{\mathrm{A} \cdot \Delta \ell} \Rightarrow \quad \mathrm{F}=\mathrm{YA} \cdot \frac{\Delta \ell}{\ell}$
$=2 \times 10^{11} \times \frac{1}{100} \times \mathrm{A}$
$\mathrm{n}=\frac{1}{2 \times 1.5} \sqrt{\frac{\mathrm{F}}{\mathrm{Ad}}}=\frac{1}{3} \sqrt{\frac{2 \times 10^{9} \mathrm{A}}{\mathrm{A} \times 7.7 \times 10^{8}}}$
$\mathrm{n}=170 \mathrm{Hz}$
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