Question

Assume that the displacement (s) of air is proportional to the pressure difference ($\Delta p$) created by a sound wave. Displacement (s) further depends on the speed of sound (v), density of the air (ρ) and the frequency (f). If $\vartriangle p \sim 10pa$ , $v \sim 300m{s^{ - 1}}$ , $\rho \sim 1kg{m^{ - 3}}$ and $f \sim 1000Hz$, then s will be of the order of (take the multiplicative constant to be $1)$
(A) $1mm$
(B) $10mm$
(C) $\dfrac{1}{{10}}mm$
(D) $\dfrac{3}{{100}}mm$

Answer 1

Hint: In order to solve this question, we will first form the exact formula for the displacement using dimensional analysis and then we will solve for displacement to find the order using the value of the given parameter.

Complete answer:
We have given that Displacement (s) is proportional to Pressure difference ($\vartriangle p$) , speed of sound (v), density of air (ρ) and frequency (f), So Let s can be written mathematically as $S \propto {(\vartriangle p)^a}{(v)^b}{(\rho )^c}{(f)^d}$ It’s given that proportionality constant one so
$S = {(\vartriangle p)^a}{(v)^b}{(\rho )^c}{(f)^d}$

Using Dimensions of each quantity as
$
  \left [S\right ] = \left [L\right ] \\
  \left [\vartriangle p\right ] = \left [M{L^{ - 1}}{T^{ - 2}}\right ] \\
  \left [v\right ] = \left [L{T^{ - 1}}\right ] \\
  \left [\rho \right ] = \left [M{L^{ - 3}}\right ] \\
  \left [f\right ] = \left [{T^{ - 1}}\right ] \\
 $
so, $\left [L\right ] = \left [{M^{a + c}}{L^{ - a + b - 3c}}{T^{ - 2a - b - d}}\right ]$

On comparing the powers we get,
$
  a = 1 \\
  b = c = d = - 1 \\
 $
so, we get the exact formula of displacement of air S in terms of velocity of sound, density of air, frequency and pressure difference as
$ \Rightarrow S = \dfrac{{\vartriangle p}}{{\rho vf}}$

Now, on putting the values as $\vartriangle p \sim 10pa$ , $v \sim 300m{s^{ - 1}}$ , $\rho \sim 1kg{m^{ - 3}}$ and $f \sim 1000Hz$ we get,
$
  S = \dfrac{{10}}{{1 \times 300 \times 1000}}m \\
  S = \dfrac{{10}}{{300}}mm \\
  S \sim \dfrac{3}{{100}}mm \\
 $
So, Displacement of air is of the order of $\dfrac{3}{{100}}mm$

Hence, the correct answer is option (D) $\dfrac{3}{{100}}mm$.

Note:It should be remembered that the multiplicative constant is the constant term introduced when the proportional sign is replaced and here the basic conversion units are used as ${10^{ - 3}}m = 1mm$ always, be careful while changing units as even one decimal error in units can lead to an incorrect answer.