You have a string with a mass of 0.0133 kg. You stretch the string with a force of 8.89 N, giving it a length of 1.97 m. Then, you vibrate the string transversely at precisely the frequency that corresponds to its fourth normal mode; that is, at its fourth harmonic. What is the wavelength 4 of the standing wave you create in the string? What is the frequency f4?

(i) The wavelength is 0.985 m

(ii) The frequency of the wave is 36.84 Hz

Explanation:

Given;

mass of the string, m = 0.0133 kg

tensional force on the string, T = 8.89 N

length of the string, L = 1.97 m

Velocity of the wave is:

(i) The wavelength:

Fourth harmonic of a string with two nodes, the wavelength is given as,

L = 2λ

λ = L/2

λ = 1.97 / 2

λ = 0.985 m

(ii) Frequency of the wave is:

v = fλ

f = v / λ

f = 36.29 / 0.985

f = 36.84 Hz

A) The wavelength of the standing wave at fourth harmonic is; λ = 0.985 m

B) The frequency of the wave at the calculated wavelength is; f = 36.84 Hz

We are given;

mass of string; m = 0.0133 kg

Force on the string; F = 8.89 N

Length of string; L = 1.97 m

A) We want to find the wavelength at the fourth normal node.

At the fourth harmonic, there will be 2 nodes.

Thus, the wavelength will be;

λ = L/2

λ = 1.97/2

λ = 0.985 m

B) We need to first find the velocity of the wave from the formula;

v = √(F/(m/L)

Plugging in the relevant values gives;

v = √(8.89/(0.0133/1.97)

v = 36.2876 m/s

Now, formula for frequency here is;

f = v/λ

f = 36.2876/0.985

f = 36.84 Hz

Read more about harmonics of standing waves at; link