One string of a certain musical instrument is 50 cm long and has a mass of 11.75 g. It is being played in a room where the speed of sound is 343 m/s. The speed of a wave travelling on a stretched string of mass per unit length μ and under tension F is v = √F/μ
a) To what tension must you adjust the string so that, when vibrating in its fourth harmonic, it produces sound of wavelength 1.2 m?
b) What frequency sound does this string produce in its fundamental mode of vibration?
To answer these questions, we need to use the wave equation for a vibrating string and some basic physics principles. Let's break down the steps for each question.
a) To determine the tension required for the string to produce sound of a certain wavelength in its fourth harmonic, we need to use the wave equation:
v = sqrt(F/μ)
where:
v is the wave speed,
F is the tension in the string, and
μ is the mass per unit length of the string.
First, we need to calculate the wave speed using the given speed of sound in the room:
v = 343 m/s
Next, we calculate the mass per unit length of the string using the given length and mass:
μ = mass/length
μ = 11.75 g / (50 cm) = (11.75 g) / (0.5 m) = 23.5 g/m
Now, we can substitute the values of v and μ into the wave equation:
343 m/s = sqrt(F / (23.5 g/m)
To solve for F, we square both sides of the equation:
(343 m/s)^2 = F / (23.5 g/m)
F = (343 m/s)^2 * (23.5 g/m) = 2580.635 N
Therefore, the tension in the string must be adjusted to approximately 2580.635 N in order to produce sound of wavelength 1.2 m in its fourth harmonic.
b) To determine the frequency of the sound produced by the string in its fundamental mode of vibration, we can use the formula:
v = λf
where:
v is the wave speed (which we know as 343 m/s),
λ is the wavelength of the sound, and
f is the frequency of the sound.
We can rearrange the formula to solve for f:
f = v / λ
Substituting the given values:
f = 343 m/s / 1.2 m = 285.833 Hz
Therefore, the string will produce a sound with a frequency of approximately 285.833 Hz in its fundamental mode of vibration.