Question

Two tuning forks have frequencies $380\,{\text{Hz}}$ and $384\,{\text{Hz}}$ respectively. When they are sounded together, they produce 4 beats. After hearing the maximum sound, how long will it take to hear the minimum sound?
A. ${\text{1/4}}\,{\text{s}}$
B. ${\text{1/8}}\,{\text{s}}$
C. $1\,{\text{s}}$
D. $2\,{\text{s}}$

Answer 1

Use the formula for the bet period of the two sounds. This formula gives the relation between the beat period of the two sounds, frequency of the first sound and frequency of the second sound. Calculate the beat period of the sound when the two tuning forks are sounded together. Then use the formula for the time interval between two consecutive maximum and minimum sound to determine the required time.

Formula used:
The beat period $T$ of two sounds is given by
$T = \dfrac{1}{{{n_2} - {n_1}}}$ …… (1)
Here, ${n_1}$ is the frequency of the first sound and ${n_2}$ is the frequency of the second sound.

Complete step by step answer:
We have given that the frequency of the first tuning fork is $380\,{\text{Hz}}$ and the frequency of the second tuning fork is $384\,{\text{Hz}}$.
${n_1} = 380\,{\text{Hz}}$
$\Rightarrow{n_2} = 384\,{\text{Hz}}$
It is given that when these two tuning forks are sounded together the beat frequency of the sound produced is 4.
${n_2} - {n_1} = 4{{\text{s}}^{{\text{ - 1}}}}$

We have asked to calculate the time interval between the maximum sound and minimum sound produced by the sounding of two tuning forks together.Let us first calculate the beat period of the sound produced by the two tuning forks.Substitute $4{{\text{s}}^{{\text{ - 1}}}}$ for ${n_2} - {n_1}$ in equation (1).
$T = \dfrac{1}{{4{{\text{s}}^{{\text{ - 1}}}}}}$
$\Rightarrow T = \dfrac{1}{4}\,{\text{s}}$
Hence, the beat period of the sound produced by the two tuning forks is $\dfrac{1}{4}\,{\text{s}}$.

Let us now determine the time interval after which the minimum sound will produced after the maximum sound.The time interval $t$ between two consecutive maximum and minimum sounds is given by
$t = \dfrac{T}{2}$
Substitute $\dfrac{1}{4}\,{\text{s}}$ for $T$ in the above equation.
$t = \dfrac{{\dfrac{1}{4}\,{\text{s}}}}{2}$
$\therefore t = \dfrac{1}{8}\,{\text{s}}$
Therefore, the time interval after which one can hear the minimum sound is $\dfrac{1}{8}\,{\text{s}}$.

Hence, the correct option is B.

Note: The students should keep in mind that the beat that we have calculated is not the same as that of the general time period that we calculate for a sound wave. So, the students should not try to use the formula for the time period of the normal sound wave instead of that used in equation (1). That’s why the students should be careful while using the formula in case of beats produced in the sound.