Resonance Data That Do Not Fit the Predicted String
A student investigates standing waves on a 1.50 m string fixed at both ends. The string has a linear mass density of 2.0 g/m and is under 45 N of tension. The student predicts a wave speed of 150 m/s and a fundamental frequency of 50 Hz. Using a mechanical oscillator and a frequency generator, the student finds large-amplitude standing waves at 75 Hz and 125 Hz. No other resonances are found between 20 Hz and 200 Hz, although the student does not check carefully near 50 Hz and 100 Hz. The student concludes that the string's fundamental frequency is 25 Hz because the difference between the two observed frequencies is 50 Hz. A classmate says the data are consistent with the prediction. The teacher asks the group to test both interpretations against the harmonic model and to identify what should be measured next.
1. Test whether 75 Hz and 125 Hz can be consecutive harmonics of one string. What fundamental frequency would they imply, and what wave speed would that require?
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Consecutive harmonics differ by f1. If 75 Hz and 125 Hz were consecutive, f1 would be 50 Hz, but 75/50 = 1.5, not an integer, so they cannot be consecutive harmonics of a 50 Hz fundamental. If they belong to one harmonic series, their frequencies must be integer multiples of a common f1. The greatest common divisor is 25 Hz, so they could be the 3rd and 5th harmonics with f1 = 25 Hz. That would imply v = 2Lf1 = 2(1.50 m)(25 Hz) = 75 m/s, not the predicted 150 m/s. The data therefore do not fit the 50 Hz prediction unless the string tension or linear density is very different from the stated values.
2. The classmate claims the data are consistent with the 50 Hz prediction. Evaluate this claim.
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The classmate's claim is not supported. A 50 Hz fundamental would produce resonances at 50, 100, 150 Hz, and so on, not at 75 and 125 Hz. The observed pair is consistent with a 25 Hz fundamental (3rd and 5th harmonics), but that conflicts with the predicted 150 m/s speed and 50 Hz fundamental. Possible causes include a tension lower than 45 N-11.25 N would give 75 m/s with the same linear density-a linear density about four times larger than 2.0 g/m, a different effective length, or the oscillator or support resonating rather than the string. The group should not average the two interpretations; it should check the assumptions.
3. Describe a measurement plan to resolve the discrepancy, including at least two specific checks.
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Measure the tension directly with a spring scale or a calibrated hanging mass over a pulley, and measure linear density by weighing a known length of string. Scan frequency slowly in small steps around 50 and 100 Hz to see whether weak resonances exist. Use a driver with adjustable coupling and a vibration sensor at the string center. Check the support distance and whether the string slips at the ends. If the tension is actually near 11 N, the 25 Hz interpretation becomes plausible; if tension is near 45 N, the observed 75 Hz and 125 Hz peaks are probably not string harmonics.