Question

A 1.5-m string is held fixed at both ends. When driven by a 180-Hz source, the string vibrates in 4 distinct segments. What is the natural fundamental frequency of the string?

Answer 1

If the string vibrates in 4 distinct segments, then it has 5 nodes (4 segments + 2 fixed ends) where the amplitude is zero. The wavelength of the fundamental mode is twice the length of the string, so:

wavelength = 2 x length = 2 x 1.5 m = 3.0 m

The speed of the wave on the string is given by the formula:

speed = frequency x wavelength

Solving for the frequency, we get:

frequency = speed / wavelength

The speed of the wave depends on the tension and mass per unit length of the string, which are not given in the problem. However, we can assume that the tension and mass per unit length are constant, so the speed of the wave is also constant.

Therefore, if the string vibrates at 180 Hz in the 4th harmonic, we can find the frequency of the fundamental mode by dividing 180 Hz by 4:

fundamental frequency = 180 Hz / 4 = 45 Hz

So the natural fundamental frequency of the string is 45 Hz.