An organ pipe open from both ends produces 5 beats per second when vibrated with a source of frequency $200…

An organ pipe open from both ends produces 5 beats per second when vibrated with a source of frequency $200 \text{ Hz}$ in its fundamental mode. The second harmonic of the same pipe produces 10 beats per second with a source of frequency $420 \text{ Hz}$. The fundamental frequency of pipe is
  1. 195 Hz
  2. 205 Hz
  3. 190 Hz
  4. 210 Hz

Solution

Initially, number of beats per second $= 5$ $\therefore \text{Frequency of pipe} = 200 \pm 5 = 195\text{ Hz or } 205\text{ Hz}$ $\text{Frequency of second harmonic of the pipe} = 2n \text{ and number of beats in this case} = 10$ $\therefore f_1 = 2 \times 195 = 390\text{ Hz and } f_2 = 2 \times 205 = 410\text{ Hz}$ $\because \text{For } 10 \text{ beats with } 420\text{ Hz, it must by } f_2 = 410$ $\therefore f = 205\text{ Hz}$

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