Question

$25$ tuning forks are arranged in decreasing order of frequency. Any two successive forks produce $3$ beats/sec. If the frequency of the first tuning fork is the octave of last, then the frequency of ${21}^{st}$ fork is

• A. $72\text{Hz}$
• B. $288\text{Hz}$
• C. $84\text{Hz}$
• D. $87\text{Hz}$

Answer 1

The correct option is C $84\text{Hz}$

Given that,
Total number of tuning fork, $N=25$
Let the frequency of ${25}^{th}$ tuning fork is $n_{25}=n$
Frequency of the $1^{st}$ tuning fork is $n_1=2n$
Number of beats between any two successive fork ${\Delta }_n=3\text{beats/sec}$
Frequency of the $2^{nd}$ tuning fork, $n_2=2n-(2-1)\times {\Delta }_n=2n-3$
Frequency of the $3^{rd}$ tuning fork, $n_3=2n-(3-1)\times {\Delta }_n=2n-6$
Similarly, frequency of ${25}^{th}$ tuning fork, $n_{25}=2n-(25-1)\times {\Delta }_n=2n-72$
According to question,
$n=2n-72$
$\Rightarrow n=72\text{Hz}$

Frequency of the ${21}^{st}$ tuning fork, $n_{21}=2n-(21-1)\times {\Delta }_n=2n-60$
$\Rightarrow n_{21}=2\times 72-60$
$\Rightarrow n_{21}=84\text{Hz}$