Question

A piano string 1.5m long is made of steel of density 7.7x ${10}^3$ kg/ $m^3$ and Yong's modulus 2x${10}^{11}$ N/m .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

• A. 85Hz
• B. 170 Hz
• C. 340 Hz
• D. 310 Hz

Answer 1

The correct option is B 170 Hz

$\begin{array}{c}\frac{\lambda }{2}=L\Rightarrow \lambda =2L\\ V=n\lambda \\ \text{given strain}=1\%\text{,}Y=2\times {10}^{11}Pa\end{array}$

$\begin{array}{c}\begin{array}{cc}& \text{stress=}P=2\times {10}^{11}\times {10}^{-2}\\ & =2\times {10}^9\end{array}\\ \begin{array}{c}V=\sqrt{\frac{P}{\rho }}=\sqrt{\frac{2\times {10}^9}{7.7\times {10}^3}}=0.509\times {10}^3\\ n=\frac{509}{2\times 1.5}\approx 170Hz\end{array}\end{array}$