A string, 6.0 meters in length, is fixed at both ends and tightened until the wave speed is 100 m/s. What is the frequency of the standing wave shown in the figure?

Question

BlackTom AI · Accepted Answer

Start by identifying the general relationship for standing waves on a string fixed at both ends. The allowed wavelengths are λ_n = 2L/n, where L is the string length and n is the harmonic number (n = 1, 2, 3, ...). The corresponding frequency is f_n = v/λ_n = n v /(2L).

Given: L = 6.0 m and v = 100 m/s, so f_n = n × 100 / (2 × 6) = n × 100 / 12 ≈ 8.33 n Hz.

Now evaluate each option by matching it to a permissible harmonic value:
- 200 Hz: This would require n ≈ 200 / 8.33 ≈ 24, which is far beyond the typical visible standing wave in the figure and does not correspond to a simple small-number harmonic shown. Unlikely for a 6 m string pictured in this context.
- 50 Hz: This would require n ≈ 50 / 8.33 ≈ 6, which is a possible harmonic in general, but the drawing does not show six antinodes or the corresponding wavelength pattern consistent with n = 6 for a 6 m string.
- 100 Hz: This would require n ≈ 100 / 8.33 ≈ 12, which would imply many more nodes/antinodes than is depicted and is inconsistent with the shown pattern.
- 25 Hz: This would require n ≈ 25 / 8.33 ≈ 3, which corresponds to the third harmonic. The standing wave pattern in the figure appears to have three clear antinodes (three lobes) along the length, matching n = 3.

From the direct formula f = n v /(2L) and the visual count of antinodes in the figure, only the third harmonic yields a frequency of 25 Hz. The other options require harmonics that do not align with the observed mode or would imply more/less nodes than shown.

Therefore, the frequency consistent with the given length, wave speed, and the pictured standing wave is the value corresponding to n = 3, which gives f = 25 Hz. The incorrect choices either correspond to harmonics not depicted or do not match the geometry of the shown mode.

SPHY001 Practice Quiz - Unit 17 | LA057

A string, 6.0 meters in length, is fixed at both ends and tightened until the wave speed is 100 m/s. What is the frequency of the standing wave shown in the figure?

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