A stretched string fixed at each end has a mass of 39.3 g and a length of 9.3 m. The tension in the string is 47.7 N. Given wavelength as 5.4 m, what is the vibration frequency for this harmonic?
To find the vibration frequency for this harmonic, we can use the wave equation:
v = λf
where:
v is the velocity of the wave,
λ (lambda) is the wavelength of the wave,
and f is the frequency of the wave.
In this case, we are given the wavelength (λ) as 5.4 m. To find the velocity of the wave (v), we need to use the definition of wave velocity:
v = √(T/μ)
where:
T is the tension in the string,
and μ (mu) is the linear mass density of the string.
The linear mass density (μ) can be calculated by dividing the mass (m) of the string by its length (L):
μ = m/L
Substituting the given values, we have:
μ = 39.3 g / 9.3 m = 4.22 g/m = 0.00422 kg/m
Now that we have the tension (T) which is given as 47.7 N, and the linear mass density (μ), we can calculate the velocity (v):
v = √(47.7 N / 0.00422 kg/m)
Calculating this gives us:
v ≈ 100.64 m/s
Finally, to find the frequency (f), we rearrange the wave equation and solve for f:
f = v/λ
Substituting the known values:
f = 100.64 m/s / 5.4 m
Calculating this gives us:
f ≈ 18.67 Hz
Therefore, the vibration frequency for this harmonic is approximately 18.67 Hz.