Question 5 of 23intermediate🔧 ApplyNumerical3 marks

A pipe, 30.0 cm long, is open at both ends. Which harmonic mode of the pipe resonates a 1.1 kHz source? Will resonance with the same source be observed if one end of the pipe is closed? Take the speed of sound in air as 330 m s−1^{-1}.

Correct Answer

(i) Second harmonic

(ii) No resonance will be observed

Exercise: EXAMPLES | Q: Example 14.5 | (Page 15)
For More Understanding

Explanation

This is Example 14.5 from NCERT Physics Class 11, Chapter 14 (Waves). The question tests understanding of resonance in open and closed pipes. For an open pipe, all harmonics are present, while for a closed pipe, only odd harmonics exist. The textbook solution demonstrates that the source frequency matches the second harmonic of the open pipe but corresponds to the fourth harmonic of the closed pipe, which is not a permissible mode.

Solution Steps

  1. Step 1: For open pipe, fundamental frequency v₁ = v/2L = 330/(2×0.32 \times 0.3) = 550 Hz

  2. Step 2: nth harmonic frequency vₙ = nv/2L = 550n Hz

  3. Step 3: Setting vₙ = 1100 Hz, we get n = 2. Hence, second harmonic resonates.

  4. Step 4: For closed pipe, fundamental frequency v₁ = v/4L = 330/(4×0.34 \times 0.3) = 275 Hz

  5. Step 5: Source frequency 1100 Hz corresponds to 1100/275 = 4, i.e., fourth harmonic

  6. Step 6: Since closed pipe has only odd harmonics (1st, 3rd, 5th...), fourth harmonic is not possible. Hence, no resonance will be observed.

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