# 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.