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  4. A 70 cm long sonometer wire is in unison with a tuning fork. If the length of the wire is decreased by 1.0 cm, it produces 4 beats per second with the same tuning fork. The frequency of the tuning fork is

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Q. A $70 \,cm$ long sonometer wire is in unison with a tuning fork. If the length of the wire is decreased by $1.0\, cm$, it produces $4$ beats per second with the same tuning fork. The frequency of the tuning fork is

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Solution:

Let the frequency of the fork be $n$.
In the first case, the length of the wire is $70\, cm$.
Therefore, $n=\frac{1}{2 \times 70} \sqrt{\frac{T}{m}} \ldots$ (i)
On decreasing the length of the wire, its frequency will increase.
Hence, in the second case when the length is $69\, cm$,
then $n+4=\frac{1}{2 \times 69} \sqrt{\frac{T}{m}} \ldots$ (ii)
Dividing Eqs. (i) by (ii), we get
$\frac{n}{n+4}=\frac{1 / 70}{1 / 69} $
$\Rightarrow \frac{n}{n+4}=\frac{69}{70} $
$70 n-69 n=69 \times 4 $
$n=276 \,Hz$