You invent a new musical instrument, the phonitar, which combines strings and pi

Question

You invent a new musical instrument, 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 placed under equal tension of 600 N.

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.

a. What fundamental notes do the two strings on your instrument produce on this cold day?

b. What fundamental note does the closed pipe play?

c. What chord does your instrument now play with the fundamental notes?

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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