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Find the frequencies of the first three harmonics of a piano string of length 1.5m,...

Physics
JAMB 2001

Find the frequencies of the first three harmonics of a piano string of length 1.5m, if the velocity of the wave on the string is 120ms-1

  • A. 40Hz, 80Hz, 120Hz
  • B. 80Hz, 160Hz, 240Hz
  • C. 180Hz, 360Hz, 540Hz
  • D. 369Hz, 180Hz, 90Hz
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Correct Answer: Option A
Explanation

in vibrations in string, the fundamental frequency happens to be the first harmonic, and is given as

f\(_0\) = \(\frac{V}{2l}\)

Where V = Velocity of the sound wave, I = Length of the string.

The second harmonic is the first overtone given by

f1 =  \(\frac{V}{l}\) = 2f0 ;

and the third harmonic happens to be the second overtone, given by

f2 =    \(\frac{3V}{2l}\) = 2f0.

Thus if V = 120m/s and l = 1.5m
Thus f1 = 2f0 = 2 x 40 = 80Hz
Thus f2 = 3f0 = 3 x 40 = 120Hz
Thus we have in order, 40Hz, 80Hz, 120Hz

There is an explanation video available below.


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Explanation Video

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