Question

An air column is of length 17 cm. Find the ratio of the frequency of the 5th overtone when the column is closed at one end to that when it is open at both ends. (Speed of sound in air = 340 m/s.)

Hint:

In MHT-CET, if a question asks for a ratio of frequencies of overtones for pipes of the same length, \(v\) and \(L\) always cancel. Don't waste time calculating the numerical values of the frequencies!

Answer 1

Step 1: Understanding the Question:
Air columns resonance frequencies depend on boundary conditions.
Step 2: Key Formula or Approach:
1. Closed pipe: \( f_n = (2n+1) \frac{v}{4L} \) (where \( n \) is overtone number).
2. Open pipe: \( f_n = (n+1) \frac{v}{2L} \) (where \( n \) is overtone number).
Step 3: Detailed Explanation:
For 5th overtone (\( n=5 \)):
Closed: \( f_C = (2 \cdot 5 + 1) \frac{v}{4L} = \frac{11v}{4L} \).
Open: \( f_O = (5 + 1) \frac{v}{2L} = \frac{6v}{2L} = \frac{12v}{4L} \).
Ratio:
\[ \frac{f_C}{f_O} = \frac{11v/4L}{12v/4L} = \frac{11}{12} \]
Step 4: Final Answer:
The ratio of frequencies is \( 11:12 \).