Question

You have a string with a mass of 0.0121 kg. You stretch the string with a force of 9.51 N, giving it a length of 1.89 m. Then you vibrate the spring transversely at precisely the frequency that corresponds to its fourth normal mode, that is at its fourth harmonic. What is the wavelength of the standing wave you create in the string? What is the frequency?

Answer 1

The wavelength of the standing wave in the string is approximately 0.945 m, and the frequency is approximately 5.68 Hz. These values are calculated using the formulas for wavelength and frequency of standing waves.

Step-by-step explanation:

To find the wavelength of the standing wave created in the string, we can use the formula:

wavelength = 2L/n

where L is the length of the string and n is the mode number. In this case, since the string is vibrating in its fourth harmonic, n = 4. Substituting the given values, we get:

wavelength = 2 * 1.89 m / 4 = 0.945 m

To find the frequency of the standing wave, we can use the formula:

frequency = (v/2L) * n

where v is the speed of the wave. In this case, we'll need to first find the speed of the wave using the formula:

v = sqrt(Ft/μ)

where Ft is the tension force and μ is the linear mass density of the string. Substituting the given values, we get:

v = sqrt(9.51 N / 0.0121 kg) ≈ 26.96 m/s

Now we can use the speed and the formula for frequency:

frequency = (26.96 m/s / (2 * 1.89 m)) * 4 = 5.68 Hz