Question

Two vibrating tuning forks produce progressive waves given by ${y_1} = 4\sin 500\pi t$and ${y_2} = 2\sin 506\pi t$. Number of beats produced per minute is
${\text{A}}{\text{. 360}} \\\ {\text{B}}{\text{. 180}} \\\ {\text{C}}{\text{. 3}} \\\ {\text{D}}{\text{. 60}} \\\$

Answer 1

Hint: In this question first we have to find the angular frequency and then frequency of each given wave. And then use the formula of beat frequency to get the number of beats produced per minute.

Complete step-by-step answer:

Formula used: Beat Frequency = $|{f_2}{\text{ - }}{f_1}{\text{ }}|$

We have, ${y_1} = 4\sin 500\pi t$
And ${y_2} = 2\sin 506\pi t$

We know standard form of wave is $y = {\text{A}}\sin wt$

A= Amplitude
On comparing above equation with eq.1 and eq.2, we get

$\Rightarrow {w_1} = 500\pi$
$\Rightarrow {w_2} = 506\pi$

We know, Beats are the periodic and repeating fluctuations heard in the intensity of a sound when two sound waves of very similar frequencies interface with one another. The beat frequency refers to the rate at which the volume is heard to be oscillating from high to low volume.

Beat Frequency = $|{f_2}{\text{ - }}{f_1}{\text{ }}|$
We know, $w = 2\pi f$

Then,
$\Rightarrow 2\pi {f_1} = 500\pi \\\ \Rightarrow {f_1} = 250{\text{ eq}}{\text{.1}} \\\$

And

$\Rightarrow 2\pi {f_2} = 506\pi \\\ \Rightarrow {f_2} = 253{\text{ eq}}{\text{.2}} \\\$

Then beat frequency per second = $|{f_2}{\text{ - }}{f_1}{\text{ }}|$
= ${\text{| 253 - 250 }}|$
= 3 beats/sec

Now, the beat per minute = $3 \times 60$ beats/min
= 180 beats/min
Hence, option B. is correct.

Note: Whenever you get this type of question the key concept to solve is to learn the concept of beat frequency and formula of it. And one more thing to be remembered is that frequency is the number of waves that pass a fixed point in unit time or the number of cycles or vibrations undergone during one unit of time by a body in a periodic motion.