Question

A tuning fork of frequency $246\text{Hz}$ produces 6 beats per second with a wire of length $30\text{cm}$ vibrating in its fundamental mode. The beat frequency decreases when the length is slightly shortened. What could be the minimum length by which the wire be shortened so that it produces no beats with the tuning fork?

• A. $1.4\text{cm}$
• B. $0.7\text{cm}$
• C. $2\text{cm}$
• D. $2.1\text{cm}$

Answer 1

The correct option is B $0.7\text{cm}$

Frequency of wire can be either $252\text{Hz or}240\text{Hz}$ according to the given data.

As $f\propto \frac{1}{l}$, by decreasing the length of wire, frequency of wire will increase.

But, the beat frequency is decreasing.

Hence, original frequency was, $(246-6)\text{Hz}$ or $240\text{Hz}$

Now, for no beats we have to make it $246\text{Hz}$.

So,

$\frac{f_1}{f_2}=\frac{l_2}{l_1}$

$\Rightarrow l_2=\frac{f_1}{f_2}\times l_1=\frac{240}{246}\times 30=29.27\approx 29.3\text{cm}$

So,

$\Delta l=l_1-l_2=30-29.3=0.7\text{cm}$

Hence, option $\left(B\right)$ is the correct answer.