World-renowned bluegrass musician, Frog Newton, has invited you to join his band

Question

World-renowned bluegrass musician, Frog Newton, has invited you to join his band. It is apparent that he is only looking for eye-candy on the stage, because you are only asked to play one note, the D# at a frequency of 622 Hz. You make your own instrument using a string that is 0.4 meters long and has a mass of 1 grams. Now you need to calculate the necessary tension on the string, And you will be ready to jam!

What is the wavelength of the first harmonic (also called the fundamental frequency) of standing wave created by the vibrating string
m

At what speed would a wave propagate on this string?
m/s

What is the linear density of the string?
kg/m

What is the tension needed to produce the D# note?
N

Explanation / Answer

fundamental frequency = sqrt(tension / (mass / length)) / 2L

622 = sqrt(tension / (mass / length)) / 2L

622 = sqrt(tension / (1 * 10^-3 / 0.4)) / 2 * 0.4

(622 * 2 * 0.4)^2 = tension / (1 * 10^-3 / 0.4)

(622 * 2 * 0.4)^2 * (1 * 10^-3 / 0.4) = tension

tension = 619.0144 N

wave velocity = sqrt(619.0144 / (1 * 10^-3 / 0.4))

wave velocity = 497.6 m/s

wave velocity = frequency * wavelength

497.6 = 622 * wavelength

wavelength = 0.8 m

linear density = 1 * 10^-3 / 0.4

linear density = 0.0025 kg/m

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