Question

A piano tuner has a tuning fork that sounds with a frequency of $250$Hz. The tuner strikes the fork and plays a key that sounds with a frequency of $200$Hz. What is the frequency of the beats that the piano tuner hears?
(a) 0 Hz
(b) 0.8 Hz
(c) 1.25 Hz
(d) 50 Hz
(e) 450 Hz

Answer 1

Hint:In this solution, we are going to use the difference in two frequencies to find the frequency of beats. Both the frequencies are given in question, we just need to find their difference.

Complete Step by Step Answer:Given:
Frequency of the tuning fork ${f_1} = 250$Hz
Frequency of the piano key played ${f_2} = 200$Hz
The frequency of beats$f$that the piano tuner hears is nothing but the difference of the frequencies of the two waves that are interfering to produce the Beats. In this question, the two waves are of tuning fork and piano key.
The formula to find the frequency of beats$f$is given as:
The frequency of beats$f = \left| {{f_1} - {f_2}} \right|\;\;\;...........(1)$
Substituting the value of${f_1} = 250$Hz and ${f_2} = 200$Hz in equation (1), we get,
$\begin{array}{c}
f = \left| {250\;{\rm{Hz}} - 200\;{\rm{Hz}}} \right|\\
 = 50\;{\rm{Hz}}
\end{array}$
Therefore, the frequency of beats is $50$Hz. So option (d) is our correct option.
Additional Information: A piano tuner uses the interesting phenomenon of Beats to tune his piano using a tuning fork. The tuner produces the sound of tuning fork and piano at the same time, and the two sounds are heard distinctively, we call it beats. The tuner then changes the tension of piano strings such that the beats frequency becomes zero, and one common sound must be heard when the tuning fork and the piano are played together.

Note: The modulus (two vertical bars) in the formula of frequency of the beats represent that the difference of the two frequencies of the interfering notes will always be positive (as frequency can never be negative). So, be aware that you can calculate the difference of frequencies as $\left( {{f_1} - {f_2}} \right)$ or$\left( {{f_2} - {f_1}} \right)$ but the resulting frequency of the beats must always be given with a positive sign.