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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 the density and elasticity of steel are $7.7 \times 10^{3} kg / m^{3}$ and $2.2 \times 10^{11} N / m^{2}$ respectively?

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$770 Hz$

$251.5 Hz$

$178.2 Hz$

$200.5 Hz$

answer is B.

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Detailed Solution

$f = \frac{1}{2 l} \sqrt{\frac{T}{μ}} = \frac{1}{2 l} \sqrt{\frac{Stress}{Density}} = \frac{1}{2 I} \sqrt{\frac{γ \times strain}{Density}}$

$\Rightarrow f = \frac{1}{2 \times 1.5} \times \sqrt{\frac{2.2 \times 10^{11} \times \frac{1}{100}}{7.7 \times 10^{3}}}$

$= \frac{1}{3} \sqrt{\frac{2}{7} \times 10^{6}}$

$= \frac{1000}{3} \times \frac{\sqrt{2}}{\sqrt{7}} \approx 178.2 Hz$

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