A stretched string fixed at each end has a mass of 21 g and a length of 5.7 m. The tension in the string is 41.1 N. What is the vibration frequency for the third harmonic? Answer in Hz.
To find the vibration frequency for the third harmonic, we can use the formula:
f = (nv) / (2L)
where:
- f is the vibration frequency,
- n is the harmonic number,
- v is the wave speed in the string, and
- L is the length of the string.
First, let's find the wave speed in the string. The wave speed can be determined using the formula:
v = √(T / μ)
where:
- T is the tension in the string,
- μ is the linear mass density of the string.
To find the linear mass density (μ), we divide the mass of the string by its length:
μ = m / L
Now, we can substitute the given values into the formula for wave speed and find the linear mass density:
μ = 21 g / 5.7 m
= 0.037 g/m (since 1 g = 0.001 kg)
Next, substitute the tension (T) and linear mass density (μ) into the formula for wave speed:
v = √(41.1 N / 0.037 kg/m)
≈ 22.63 m/s
Finally, substitute the harmonic number (n), wave speed (v), and length (L) into the formula for frequency to find the vibration frequency for the third harmonic (n=3):
f = (3 * 22.63 m/s) / (2 * 5.7 m)
≈ 5.99 Hz
Therefore, the vibration frequency for the third harmonic in this stretched string is approximately 5.99 Hz.