A guitar string is .75m and has a mass of .005kg. A standing wave is produced when the string is plucked.
a) what is the fifth harmonic if the tension in the string is 90N?
v=sqrt(F_t/mass density)= 116m/s
f1=v/2L = 116m/s/(2*.75m) =77.3Hz
fn=n*f1
f5=5*77.3Hz =386.5Hz
and to find the wavelength
wavelength_5 = v/f_5
=116m/s/386.5Hz
=.3m
b) What is the frequency and wavelength of the sound produced by the subsequent vibrations of the sound box of this guitar. Assume room temperature (v=340m/s)
Wouldn't frequency be the same because of resonance. But what about the wavelenght is that also the same?
To find the frequency of the sound produced by the subsequent vibrations of the sound box of the guitar, we can assume that the guitar string transmits the vibrations to the sound box efficiently and that the sound box resonates at the same frequency as the string. Therefore, the frequency would indeed be the same as the fifth harmonic of the guitar string, which is 386.5 Hz.
However, it's important to note that the speed of sound in air is different from the speed of vibrations in the guitar string. Given that the speed of sound in air is 340 m/s at room temperature, we can calculate the wavelength of the sound produced using the formula:
wavelength = speed of sound / frequency
wavelength = 340 m/s / 386.5 Hz
wavelength ≈ 0.88 m
So, the wavelength of the sound produced by the subsequent vibrations of the sound box of the guitar would be approximately 0.88 meters.