Two sitar strings A and B playing the note 'G | vedclass

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Question

(A) Initial frequency of string $A$,$f_A = 324 \; Hz$.Let the frequency of string $B$ be $f_B$.The beat frequency is given by $n = |f_A - f_B| = 6 \;

Vedclass · Accepted Answer

$318$

Vedclass · Answer

(A) Initial frequency of string $A$,$f_A = 324 \; Hz$.Let the frequency of string $B$ be $f_B$.The beat frequency is given by $n = |f_A - f_B| = 6 \; Hz$.This implies $f_B = 324 \pm 6$,so $f_B = 330 \; Hz$ or $318 \; Hz$.When the tension in string $A$ is reduced,its frequency $f_A$ decreases (since $f \propto \sqrt{T}$).If $f_A$ was $324 \; Hz$ and it decreases,the new frequency $f_A'$ becomes less than $324 \; Hz$.Case $1$: If $f_B = 330 \; Hz$,the beat frequency $n' = |f_B - f_A'| = |330 - f_A'|$. As $f_A'$ decreases from $324$,the difference $|330 - f_A'|$ increases (e.g.,if $f_A' = 321$,$n' = 9 \; Hz$). This contradicts the given information that the beat frequency reduces to $3 \; Hz$.Case $2$: If $f_B = 318 \; Hz$,the beat frequency $n' = |f_A' - f_B| = |f_A' - 318|$. As $f_A'$ decreases from $324$ towards $318$,the difference decreases. For the beat frequency to become $3 \; Hz$,$f_A'$ must be $321 \; Hz$ $(321 - 318 = 3)$. This matches the condition.Therefore,the frequency of $B$ is $318 \; Hz$.

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