Resonant System Frequency Analysis

The problem states a resonant system has a fundamental frequency, denoted as \(f_0\), equal to 140 Hz. The next observed resonant frequencies are 280 Hz and 420 Hz.

Harmonic Series Identification

• Fundamental frequency: \(f_0 = 140\) Hz.
• First higher frequency: \(f_1 = 280\) Hz.
• Second higher frequency: \(f_2 = 420\) Hz.

We observe the relationship between these frequencies:

• \(f_1 = 280 \text{ Hz} = 2 \times 140 \text{ Hz} = 2 f_0\)
• \(f_2 = 420 \text{ Hz} = 3 \times 140 \text{ Hz} = 3 f_0\)

This indicates that the resonant system supports frequencies that are integer multiples of the fundamental frequency. The series is \(f_0, 2f_0, 3f_0, \dots\), which includes all harmonics.

Analyzing Resonant System Types

Let's examine the harmonic series produced by the given options:

1. Organ pipe closed at both ends: Theoretically produces frequencies \(f_n = n \frac{v}{2L}\) (where \(n=1, 2, 3, \dots\)). This results in the series \(f_0, 2f_0, 3f_0, \dots\).
2. Organ pipe open at both ends: Produces frequencies \(f_n = n \frac{v}{2L}\) (where \(n=1, 2, 3, \dots\)). This also results in the series \(f_0, 2f_0, 3f_0, \dots\).
3. Organ pipe closed at one end and open at the other: Produces only odd harmonics, \(f_n = (2n-1) \frac{v}{4L}\) (where \(n=1, 2, 3, \dots\)). This results in the series \(f_0, 3f_0, 5f_0, \dots\).
4. Taut string vibrating between two fixed points: Produces frequencies \(f_n = n \frac{v}{2L}\) (where \(n=1, 2, 3, \dots\)). This results in the series \(f_0, 2f_0, 3f_0, \dots\).

Conclusion

The observed frequency pattern (\(f_0, 2f_0, 3f_0, \dots\)) matches the harmonic series produced by an organ pipe open at both ends, a pipe closed at both ends, and a taut string.

Given the options and the typical representation in physics problems, an organ pipe open at both ends is a standard example for this type of harmonic series.