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The fundamental frequency of a vibrating string fixed at both ends is f. Will the 5th harmonic vibrate with the same wavelength as that of fundamental?
(A) Yes
(B) No
(C) Depends on the tension in the string.
(D) Depends on the linear density of the string.

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Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or tension of the string is correctly adjusted, the sound produced is a musical tone. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos.

Complete step by step answer:
1. The lowest resonant frequency of a vibrating object is called its fundamental frequency. Most vibrating objects have more than one resonant frequency and those used in musical instruments typically vibrate at harmonics of the fundamental. A harmonic is defined as an integer (whole number) multiple of the fundamental frequency. Vibrating strings, open cylindrical air columns, and conical air columns will vibrate at all harmonics of the fundamental. Cylinders with one end closed will vibrate with only odd harmonics of the fundamental. Vibrating membranes typically produce vibrations at harmonics, but also have some resonant frequencies which are not harmonics. It is for this class of vibrators that the term overtone becomes useful - they are said to have some non-harmonic overtones.
2, The nth harmonic = n $\times$ the fundamental frequency. 5th harmonic refers to 5f.
Here, n=5. So, the 5th harmonic will be 5f.
3. The frequency has increased to 5 times, while the velocity of the wave remains the same. Thus wavelength will decrease by 5 fold.

The correct option is (b)

Note The shorter the string, the higher the frequency of the fundamental. The higher the tension, the higher the frequency of the fundamental. The lighter the string, the higher the frequency of the fundamental.