Two open organ pipes of lengths $50 \mathrm{~cm}$ and $51 \mathrm{~cm}$ are totally immersed in a medium.…

Two open organ pipes of lengths $50 \mathrm{~cm}$ and $51 \mathrm{~cm}$ are totally immersed in a medium. They are found to give 40 beats in $10 \mathrm{~s}$ when each is sounding at its fundamental note. The speed of sound in this medium is
  1. $275 \mathrm{~ms}^{-1}$
  2. $310 \mathrm{~ms}^{-1}$
  3. $258 \mathrm{~ms}^{-1}$
  4. $204 \mathrm{~ms}^{-1}$

Solution

We know that, fundamental frequency $ \begin{aligned} f_0 & =\frac{v}{2 l} \\ \text { Beat } & =f_1-f_2 \\ \Rightarrow \frac{40}{10} & =\frac{v}{2 l_1}-\frac{v}{2 l_2} \Rightarrow 4=\frac{v}{2}\left(\frac{1}{l_1}-\frac{1}{l_2}\right) \\ \Rightarrow \quad 4 & =\frac{v}{2}\left(\frac{1}{50}-\frac{1}{51}\right) \times \frac{1}{10^{-2}} \\ \Rightarrow 8 \times 10^{-2} & =v\left(\frac{51-50}{50 \times 51}\right) \\ \Rightarrow 8 \times 10^{-2} & =\frac{v}{2550} \\ \Rightarrow \quad v & =2550 \times 8 \times 10^{-2} \\ & =204 \mathrm{~ms}^{-1} \end{aligned} $

Asked in: AP EAMCET 2022 (06 Jul Shift 2)

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