Question

Two tuning forks $P$ and $Q$ sounded together and $6$ beats per second are heard. $P$ is in unison with a $30cm$ air column open at both ends and $Q$ is in resonance when length of air column is increased by $2cm$. The frequencies of forks $P$ and $Q$ are

• A. $90Hz$ and $84Hz$
• B. $100Hz$ and $106Hz$
• C. $96Hz$ and $90Hz$
• D. $206Hz$ and $200Hz$

Answer 1

The correct option is C $96Hz$ and $90Hz$

We know that $f=\frac{v}{2I}$
(Fundamental frequency of an open organ pipe).
Number of beats heard $=6$
$\Rightarrow f_P-f_Q=6$
or $\frac{v}{2\left(0.3\right)}-\frac{v}{2\left(0.32\right)}=6$
or $v=\frac{2\times 0.3\times 0.32\times 6}{0.32-0.3}$
Now, $f_P=\frac{v}{2\left(0.3\right)}=\frac{0.32\times 6}{0.02}=96Hz$
and $f_Q=\frac{v}{2\left(0.32\right)}=\frac{0.3\times 6}{0.02}=90Hz$.