Assume that the displacement (s) of air is proportional to the pressure difference $(ΔP)$ created by a sound wave. Displacement (S) further depends on the speed of sound $(v),$ density of air $(ρ)$ and the frequency $(f) .If ΔP ~ 10Pa, v ~ 300m/s,ρ ~ {1kg/m}^{3}$ and $f ~ 1000Hz$ then s will be of the order of ( take multiplicative constant to be 1)

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1mm

10mm

$\frac{3}{100} mm$

$\frac{1}{30} mm$

answer is B.

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Detailed Solution

$\begin{array}{l} Δ {P = BK S}_{0}, \\ B = {Bulk modulus = ρV}^{2} \\ \begin{array}{l} \Rightarrow Δ P = \frac{{ρV}^{2} 2 π {f S}_{0}}{V} (∵ K = \frac{ω}{V} = \frac{2 π f}{V}]) \\ \Rightarrow S_{0} = \frac{Δ P}{ρV 2 π f} \\ T a k e m u l t i p l i c a t i v e c o n s \tan t " 2 π " a s 1 \\ \Rightarrow S_{0} = \frac{10}{1 \times 300 \times 1000} = \frac{1}{30} mm \end{array} \end{array}$