Question

Two tuning forks produce $4$ beats per second when they are sounded together. If both the forks are moving towards the observer in same direction with same speed $u$, the beat frequency heard becomes $5Hz$.
Consider the speed of sound to be $v$ $m{s}^{-1}$and choose the correct statements.

• A. The value of $u$ is $\frac{v}{4}$
• B. The value of $u$ is $\frac{v}{5}$
• C. In the above situation, in addition to forks motion, if observer also begins to run with speed $u$ towards both the forks, the beat frequency he hears now is $5Hz$.
• D. In the above situation, in addition to forks motion, if observer also begins to run with speed $u$ towards both the forks, the beat frequency he hears now is $6Hz$.

Answer 1

Answer: (B), (D)

The correct options are
B The value of $u$ is $\frac{v}{5}$
D In the above situation, in addition to forks motion, if observer also begins to run with speed $u$ towards both the forks, the beat frequency he hears now is $6Hz$.

The frequencies of two tuning forks can be considerd as $f$ and $f+4$.
When forks are moving towards the observer, apparent frequencies measured are,
$f_1=f\left[\frac{v}{v-u}\right],f_2=(f+4)\left[\frac{v}{v-u}\right]$
$\therefore \Delta f=f_2-f_1=4\left[\frac{v}{v-u}\right]$
$\therefore 5=4\left[\frac{v}{v-u}\right]\Rightarrow u=\frac{v}{5}$

When the observer also moves towards tuning forks, the frequencies received by him will be,
$f_1^{\prime }=f\left[\frac{v+u}{v-u}\right],f_2^{\prime }=(f+4)\left[\frac{v+u}{v-u}\right]$
$\therefore \Delta f^{\prime }=f_2^{\prime }-f_1^{\prime }=4\left[\frac{v+u}{v-u}\right]$
$\Delta f^{\prime }=4\left[\frac{v+\frac{v}{5}}{v-\frac{v}{5}}\right]=4\left(\frac{6}{4}\right)=6Hz$