A tuning fork vibrating with a sonometer having a wire of length $20 \,cm$ produces $5$ beats per second. The beats frequency does not change if the length of the wire is changed to $21 \,cm$. The frequency of the tuning fork must be ............ $Hz$
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(c)
Let frequency of tuning fork be $f$.
From question we know the frequencies of sonometer wire $\left(\frac{v}{21}\right)$ are $\frac{v}{2 \times 0.2}$ and $\frac{v}{2 \times 0.21}$
$\operatorname{Now} \frac{v}{0.4}-f=8$
$f-\frac{v}{0.42}=5 \quad \dots(ii)$
Solving $(i)$ and $(ii)$, answer is $(c)$
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