Question

When a guitar string is sounded with a $440Hz$ tuning fork, a beat frequency of $5Hz$ is heard. If the experiment is repeated with a tuning fork of $437Hz$, the beat frequency is $8Hz$. The string frequency$\left( {Hz} \right)$ is
A. $445$
B. $435$
C. $429$
D. $448$

Answer 1

Hint:When two waves of nearly same frequencies interfere with each other and produce maximum or minimum intensity is called as beats. Consider the beats produced by the combinations of the guitar with the two tuning forks, apply the definition of beats to find the frequency of the guitar string.

Complete step by step answer:
One beat is one cycle of maximum and minimum intensities, meaning that the frequency of beats is given by the difference in the individual frequencies of the two waves. We have the frequency of one tuning fork as $440Hz$ and frequency of the other tuning fork as $437Hz$. Given that the beats heard when the first tuning fork is sounded with the guitar string is $5Hz$, mathematically we have,
$
\left| {{f_s} - 440} \right| = 5 \\
\Rightarrow{f_s} - 440 = \pm 5 \\
$
We can say that the frequency of the guitar string will be either ${f_s} = 445$ or ${f_s} = 435$.

Now, if we sound the guitar string with the tuning fork of frequency $437Hz$then the beats that are supposed to be heard have to be equal to $8$beats. If the frequency of the guitar string was $445Hz$, the beats produced will be equal to $445 - 437 = 8Hz$ which satisfies our condition.

Let us check for the other case. If the frequency of the guitar string was $435Hz$, the beats produced will be $437 - 435 = 2Hz$. As you can see that this does not satisfy our condition and hence, frequency of $435Hz$ is rejected. Therefore, the string frequency$\left( {Hz} \right)$ is $445$.

Hence,option A is correct.

Note:The beat frequency is given by mode of the difference between the individual frequencies. Mathematically, beats heard per second are given as $\left| {{f_1} - {f_2}} \right|$. Since we knew the frequencies of the tuning fork, we did not consider the mode sign. This question could have been asked in a different way. Such as, if the tuning fork was loaded with wax and with some conditions given, find out the frequency. In this case, remember that whenever a tuning fork is loaded with wax, its frequency will decrease. By applying the same analysis, you could have got the same correct answer.