Question

A tuning fork produces $5\frac{beats}{sec}$. with a sonometer wire of lengths $40cm$ and $44cm$, other factors remaining unchanged. The frequency of the tuning fork is

• A. $80Hz$
• B. $88Hz$
• C. $160Hz$
• D. $105Hz$

Answer 1

The correct option is D

$105Hz$

Step 1. Given Data:

Tuning Fork produces beats $b=5\frac{beats}{sec}$

Length of the first sonometer wire $l_1=40cm$

Length of the second sonometer wire $l_2=44cm$

Step 2. Find the frequency of the tuning fork:

If $f$is the frequency $l$ is the length of the string then,

We know, $f\propto \frac{1}{l}$

Since the Tuning fork produces $5\frac{beats}{sec}$,then the possible frequencies will be $f\pm 5Hz$

From the above relation, it is evident that the frequency and length are inversely related,

This means that shorter lengths will have a higher frequency.

Therefore, for the first sonometer $f_1=f+5$

and for the second one $f_2=f-5$

Now, for the beats to form $f_1{l}_1=f_2{l}_2$

$\Rightarrow$ $\left(f+5\right)40=\left(f-5\right)44$

$\Rightarrow$ $40f+200=44f-220$

$\Rightarrow$ $4f=420$

$\Rightarrow$ $f=105Hz$

Hence option (D) is correct.