MHTCET 26 Apr 2025 Shift 1 Paper

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Question : 43 of 150

Marks: +1, -0

An air column is of length 17 cm . The ratio of frequencies of

5^{\text {th }}

overtone if the air column is closed at one end to that open at both ends is (velocity of sound in air

=340 \mathrm{~ms}^{-1}

$\frac{9}{11}$
$\frac{5}{7}$
$\frac{11}{12}$
$\frac{13}{9}$
Closed Pipe (One End Closed):
When an air column is closed at one end, it produces only odd harmonics. This means only certain frequencies are possible.
The formula for the $n$ th overtone in a closed pipe is that it is actually the $(2 n+1)$ th harmonic.
The $5^{\text {th }}$ overtone in this case is the $11^{\text {th }}$ harmonic. (Because $2 \times 5+1=11$ )
The frequency formula for a closed pipe is: $f_{\text {closed }}=\frac{11 V}{4 L}$
Open Pipe (Both Ends Open):
When the pipe is open at both ends, it can produce all harmonics (both odd and even).
The formula for the $n$ th overtone in an open pipe is that it is the $(n+1)^{\text {th }}$ harmonic.
The $5^{\text {th }}$ overtone here is the $6^{\text {th }}$ harmonic. (Because $5+1=6$ )
The frequency formula for an open pipe is: $f_{\text {open }}=\frac{6 V}{2 L}$
Finding the Ratio:
To get the answer, divide the formula for closed pipe frequency by the formula for open pipe frequency:
$\frac{f_{\text {closed }}}{f_{\text {open }}}=\frac{11 V}{4 L} \times \frac{2 L}{6 V}=\frac{11 \times 2}{4 \times 6}=\frac{22}{24}=\frac{11}{12}$
So, the ratio of frequencies is $\frac{11}{12}$ .

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