Question

When a sound wave of frequency $300Hz$ passes through a medium the maximum displacement of the medium is$0.1cm$. The maximum velocity of the particle is equal to
(A) $60m/s$
(B) $30m/s$
(C) $30\pi cm/s$
(D) $60\pi cm/s$

Answer 1

Hint
The particle in the medium is in simple harmonic motion. This question is solved by using the formula for maximum velocity in a simple harmonic motion. The maximum displacement is the maximum amplitude, and we are given the frequency of the sound wave. By converting the frequency of sound into angular frequency. We can substitute this data in the formula and find the maximum velocity.
Maximum velocity of particle, ${v_{max}} = A\omega$
Here, Maximum amplitude is represented by $A$, Angular frequency is represented by $\omega$, Maximum velocity is represented by ${v_{max}}$, and Frequency is represented by $f$

Complete step by step answer
The sound wave traveling in the medium causes the particles in the medium to vibrate. These vibrations are in the form of a simple harmonic motion. Thus, the particle in the medium is in a simple harmonic motion.
We know that the formula for maximum velocity is ${v_{max}} = A\omega$
Frequency is given as $300Hz$
Converting frequency into the angular frequency $(\omega = 2\pi f)$ we get
$\Rightarrow \omega = 300 \times 2 \times \pi$
$\Rightarrow \omega = 600\pi$
Substituting the values of angular frequency and maximum amplitude we get,
$\Rightarrow {v_{max}} = A\omega$
$\Rightarrow {v_{max}} = 0.1cm \times 600\pi$
$\Rightarrow {v_{max}} = 60\pi cm/s$
The maximum velocity of the particle is $60\pi cm/s$.
Thus, option (D) $60\pi cm/s$ is the correct answer.

Note
Often students forget to convert frequency into angular frequency. Always remember to convert the value of frequency into the angular frequency and then substitute the values in the formula.
Simple harmonic motion is a special type of periodic motion where the restoring force on the moving object is directly proportional to the object's displacement magnitude and acts towards the object's equilibrium position. It results in an oscillation which, if uninhibited by friction or any other dissipation of energy, continues indefinitely.