Question

A wire of density $9\times {10}^{3}Kg/m^3$ is stretched between two clamps 1 m apart and is stretched to an extension of $4.9\times {10}^{-4}$ meter, Young's modulus of material is $9\times {10}^{10}N/m^2$. Then

• A. The lowest frequency of standing waveis 35 Hz
• B. The frequency of 1 st overtone is 70 Hz
• C. The frequency of 1 st overtone iis 105 Hz
• D. A and B both

Answer 1

The correct option is A The lowest frequency of standing waveis 35 Hz

$\begin{array}{c}\text{This is a fundamental standing wave}\\ \begin{array}{cc}\Rightarrow L& =\frac{\lambda }{2}\\ & \lambda =2L\\ \text{we know that,}& V=n\lambda \\ \text{Also}& Y=\sqrt{\frac{T}{\mu }}\end{array}\end{array}$

$\begin{array}{cc}{n}_0& =\frac{1}{2L}\sqrt{\frac{T}{\mu }}\\ \text{Given,}& \rho =a\times {10}^{3}kg/m^3\\ & l=1m\\ & \Delta l=4.9\times {10}^{-4}m\\ & Y=9\times {10}^{10}N/m^2\end{array}$

$\begin{array}{cc}{n}_0& =\frac{1}{2\times 1}\sqrt{\frac{VA\Delta l}{lA\rho }}\\ & =\frac{1}{2}\sqrt{\frac{9\times {10}^{10}\times 4.9\times {10}^{-4}}{1\times 9\times {10}^3}}10\\ n_0& =35Hz\end{array}$