AIIMS 2004 Physics Sound Waves MCQ Question

To solve the problem, we need to understand the behavior of sound waves in an organ pipe that is closed at one end. The key concepts involved here are the fundamental frequency of the pipe and the nature of overtones.

Explanation of the Correct Answer

1. Fundamental Frequency and Overtones: An organ pipe closed at one end produces a fundamental frequency, which is the lowest frequency at which it can resonate. The fundamental frequency for this pipe is given as $f_1 = 1500 \, \text{Hz}$.

   For a pipe closed at one end, the harmonics (or overtones) can only be odd multiples of the fundamental frequency. The frequencies of the harmonics can be expressed as follows:

   • Fundamental frequency (1st harmonic): $f_1 = 1500 \, \text{Hz}$
   • 3rd harmonic (1st overtone): $f_3 = 3f_1 = 3 \times 1500 = 4500 \, \text{Hz}$
   • 5th harmonic (2nd overtone): $f_5 = 5f_1 = 5 \times 1500 = 7500 \, \text{Hz}$
   • 7th harmonic (3rd overtone): $f_7 = 7f_1 = 7 \times 1500 = 10500 \, \text{Hz}$
   • 9th harmonic (4th overtone): $f_9 = 9f_1 = 9 \times 1500 = 13500 \, \text{Hz}$
   • 11th harmonic (5th overtone): $f_{11} = 11f_1 = 11 \times 1500 = 16500 \, \text{Hz}$
   • 13th harmonic (6th overtone): $f_{13} = 13f_1 = 13 \times 1500 = 19500 \, \text{Hz}$

   Here, we can see that the first six harmonics (including the fundamental) are:

   • $f_1 = 1500 \, \text{Hz}$ (fundamental)
   • $f_3 = 4500 \, \text{Hz}$ (1st overtone)
   • $f_5 = 7500 \, \text{Hz}$ (2nd overtone)
   • $f_7 = 10500 \, \text{Hz}$ (3rd overtone)
   • $f_9 = 13500 \, \text{Hz}$ (4th overtone)
   • $f_{11} = 16500 \, \text{Hz}$ (5th overtone)
   • $f_{13} = 19500 \, \text{Hz}$ (6th overtone)

2. Hearing Range: A normal person can typically hear sounds in the frequency range of about 20 Hz to 20,000 Hz. From our calculations:

   • The 1st harmonic (1500 Hz) is audible.
   • The 1st overtone (4500 Hz) is audible.
   • The 2nd overtone (7500 Hz) is audible.
   • The 3rd overtone (10500 Hz) is audible.
   • The 4th overtone (13500 Hz) is audible.
   • The 5th overtone (16500 Hz) is audible.
   • The 6th overtone (19500 Hz) is audible, but it is approaching the upper limit of human hearing.

   Thus, the maximum number of overtones that can be distinctly heard, when counting the fundamental frequency, is 6 total (1 fundamental + 5 overtones). Therefore, the maximum number of overtones that a normal person can hear is 6.

Clarification of Other Options

• Option A (14): This is incorrect because it exceeds the hearing range of a normal person. Even if we consider higher harmonics, they will eventually fall outside the audible range.
• Option B (13): Similarly, this option is also incorrect for the same reason. It implies that there would be more than 6 overtones at frequencies within the audible range.
• Option D (9): This is incorrect as well. This suggests counting more harmonics than what can be audibly perceived by an average human.

Conclusion

The correct answer is C) 6 because a closed organ pipe can only produce odd harmonics, and the maximum number of distinct audible frequencies