You invent a new musical instrsument, the phonitar, which combines strings and p

Question

You invent a new musical instrsument, the phonitar, which combines strings and pipes. You design it such that the fundamental frequencies of the two strings and one closed pipe form a G major chord (G-B-D) on a normal day (20 degrees C). The two strings both have a mass/length density of 0.0082 kg/m and are palced under equal tension of 600 N. a. How long must the bass string be in order to produce the note that is one obctave lower than the G above middle C as its fundamental frequency? b. How long must the other string be to produce the D above midddle C note as its fundamental frequency? c. How long must the closed pipe be to produce a B note as its fundamental frequency (B below middle C)? You are invited to perform a phonitar concerto at the Siberian Symphony located in Yakutsk. The temperature for today's outdoor performance is a balmy -12.7 degrees Celsius. d. What fundamental notes do the two strings on your instrument produce on this cold day? e. What fundamental note does the closed pipe play? And here is the table with the frequencies: Note Hz A# or Bb 233 B 247 Middle C 262 C# or Db 277 D 294 D# or Eb 311 E 330 F 349 F# or Gb 370 G 392 G# or Ab 415 A 440 A# or Bb 466 B 494 C%u2019 524

Explanation / Answer

given,

mass per unit length k = 0.0082 kg/m

tension T = 600 N

one octave lower than G is F

f = sqrt(T / k) / L

349 = sqrt(600 / 0.0082) / L

a) Length = 0.775 m

frequency of D = 294 Hz

294 = sqrt(600 / 0.0082) / L

b) Length = 0.92 m

let the speed of sound be v

so,

f = v / 4L

for B frequency = 247 Hz

247 = v / 4L

Length = v / 4 * 247

c) Length = v / 988

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