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  4. A piano string 1.5m long is made of steel of density 7.7× 103kgm- 3 and Young's modulus 2× 1011Nm- 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

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Q. A piano string $1.5m$ long is made of steel of density $7.7\times 10^{3}kgm^{- 3}$ and Young's modulus $2\times 10^{11}Nm^{- 2}.$ 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

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Solution:

$Y=\frac{F . l}{A \cdot \Delta l}\Rightarrow F=YA.\frac{\Delta l}{l}$
$=2\times 10^{11}\times \frac{1}{100}\times A$
$n=\frac{1}{2 \times 1 . 5}\sqrt{\frac{F}{A d}}=\frac{1}{3}\sqrt{\frac{2 \times 10^{9} A}{ A \times 7 . 7 \times 10^{3}}}$
$n=170Hz$