Question

A tuning fork vibrating with a sonometer having $20 \,cm$ wire produces $5$ beats/s. The beat frequency does not change when the length of the wire is changed to $21 \,cm$. The frequency of tuning fork is :

• A. 200
• B. 210
• C. 205
• D. 215

Answer 1

Answer: (C)

The frequency of sonometer wire is given by
$n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$
where $l$ is length,
$T$ the tension
and $m$ the mass per unit length.
When the length of wire is $0.20 \,m $.
$n_{1}=\frac{1}{2 \times 0.2} \sqrt{\frac{T}{m}}$
When length is $0.21\, m$,
then $n_{2}=\frac{1}{0.21 \times 2} \sqrt{\frac{T}{m}}$
Let frequency of tuning fork be $N$,
$ \frac{n_{1}}{n_{2}}=\frac{N+5}{N-5}$
$ \Rightarrow \frac{0.21}{0.20}=\frac{N+5}{N-5} $
$\Rightarrow $ $0.21 \,N-1.05=0.2\, N+1$
$ \Rightarrow 0.01\, N=2.05 $
$\Rightarrow $ $N=\frac{2.05}{0.01}=205 \,Hz$