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:
- 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 \(f_0, 2f_0, 3f_0, \dots\).
- 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 \(f_0, 2f_0, 3f_0, \dots\).
- 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 \(f_0, 3f_0, 5f_0, \dots\).
- 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.
Important Questions from Oscillations & Waves
The frequency (f), wavelength (λ) and speed (v) of a sound wave are related as
The length of a simple pendulum is increased four times to its previous value while the mass is doubled. What is the ratio of the new and previous time period of the pendulum?
The frequency (f), wavelength (λ) and speed (v) of a sound wave are related as
The length of a simple pendulum is increased four times to its previous value while the mass is doubled. What is the ratio of the new and previous time period of the pendulum?
- Which of the following best describes the shape of a sinusoidal wave signal during one complete cycle?
