Question

A violin emits sound of frequency \[510\;{\rm{Hz}}\]. How far will the sound waves reach when the string completes 250 vibrations? The velocity of sound is \[340{\rm{ }}{{\rm{m}} {\left/
 {\vphantom {{\rm{m}} {\rm{s}}}} \right. } {\rm{s}}}\].

Answer 1

Hint: We know that the time period of vibration of the violin string is inverse of its frequency. Using this concept, we will find the time period of vibrations per revolution. We will also use the distance travelled by the sound, given by the product of velocity and time taken by it.

Complete step by step answer:
Given:
Violin emits the sound of a frequency of \[f = 510\;{\rm{Hz}}\].
The number of vibrations to be taken into consideration is \[n = 250\].
The velocity of sound is \[v = 340{\rm{ }}{{\rm{m}} {\left/
 {\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}\].

As know that the time period of vibration is inverse of the frequency of sound emitted by violin so we can write:
\[T = \dfrac{1}{f}\]
Here T is the time period of vibration.

We also know that the time period's value is given for one revolution of the violin string. For n number of revolutions, we can write:
\[T' = nT\]
Here T' is the time taken by the string to complete $n$ number of revolutions.

We will substitute \[\dfrac{1}{f}\] for T in the above expression to get the final equation of the total time period taken by the violin's string to complete n revolution.
\[\begin{array}{c}
T' = n\left( {\dfrac{1}{f}} \right)\\
 = \dfrac{n}{f}
\end{array}\]

We will substitute 250 for n and \[510\;{\rm{Hz}}\] for f in the above expression.
\[\begin{array}{c}
T' = \dfrac{{250}}{{510\;{\rm{Hz}}}}\\
 = 0.49{\rm{ s}}
\end{array}\]

Let us write the expression for distance travelled by the sound waves to complete 250 revolutions.
\[d = v \times T'\]
Here d is the distance travelled.

We will substitute \[340{\rm{ }}{{\rm{m}} {\left/
{\vphantom {{\rm{m}} {\rm{s}}}} \right. } {\rm{s}}}\] for v and \[0.49{\rm{ s}}\] for T'.
\[\begin{array}{c}
d = \left( {340{\rm{ }}{{\rm{m}} {\left/
{\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}} \right)\left( {0.49{\rm{ s}}} \right)\\
 = 166.6{\rm{ m}}
\end{array}\]

Therefore, the distance travelled by the sound wave to complete 250 revolutions is equal to \[166.6{\rm{ m}}\].

Note:
We know that the S.I. The unit of frequency is Hertz, which is the inverse of the second. We can say that frequency is measured per unit second; therefore, we substituted second for Hertz while calculating the vibrations' time period.