A closed organ pipe $P_{1}$ vibrating in its first overtone and another open organ pipe $P_{2}$ vibrating in its third overtone, both pipes are in resonance with the tuning fork of frequency 12Hz. In column-II, the beat frequency in Hz is given when produced in conditions provided in column-I. Match the column-I and II

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Solution

Let $f_{1}$ and $f_{2}$ be the fundamental frequencies of $P_{1}$ and $P_{2}$ .
$3 f_{1} = 4 f_{2} = 12 H z$
$\Rightarrow$ $= f_{1} = 4 H z$ & $f_{2} = 3 H z .$
(A) $| 5 f_{1} - 4 f_{2} | = | 20 - 12 | = 8$ (r)
(B) $| 3 f_{1} - 3 f_{2} | = | 12 - 9 | = 3$ (q)
(C) $| 9 f_{1} - 8 f_{2} | = | 36 - 24 | = 12$ (s)
(D) $| 5 f_{1} - 5 f_{2} | = | 20 - 15 | = 5$ (t)

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Let f1 and f2 be the fundamental frequencies of P1 and P2.3f1=4f2=12Hz⇒ =f1=4Hz & f2=3Hz.(A) |5f1−4f2|=|20−12|=8 (r)(B) |3f1−3f2|=|12−9|=3 (q)(C) |9f1−8f2|=|36−24|=12 (s)(D) |5f1−5f2|=|20−15|=5 (t)