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

(C) $\frac{3}{100} \mathrm{~mm}$

(A) $\frac{1}{10} \mathrm{~mm}$

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Units & Measurement

Assume that the displacement (s) of air is proportional to the pressure difference (

Λ p

) created by a sound wave. Displacement (s) further depends on the speed of sound (v), density of air (

ρ

) and the frequency (f). If

Λ p \sim 10 Pa, v \sim 300 m / s, ρ \sim π 1 lg / m^{3}

and

f \sim 1000 Hz

, then s will be of the order of (take the multiplicative constant to be 1 )

Answer: (C)

Taking the displacement to depend on

Δ p

v

ρ

and

f

, dimensional balance singles out the combination

s \sim \frac{Δ p}{ρ v f}

, since pressure divided by density, speed and frequency carries the dimension of length. Numerically

s \sim \frac{10}{1 \times 300 \times 1000} = \frac{1}{3 \times 1 0 ^{4}}

\approx 0.033

mm, that is of order

\frac{3}{100}

mm.

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Single correct option