The top string of a guitar has a fundamental frequency of 300 Hz when it is allowed to vibrate as a whole, along all its 60.0 cm length from the neck to the bridge. A fret is provided for limiting vibration to just the lower two-thirds of the string. The guitarist can play a "natural harmonic" by gently touching the string at this fret and plucking the string at about one sixth of its length from the bridge. What frequency will be heard then?

300x3 = 900 Hz

To determine the frequency that will be heard when playing a natural harmonic on the guitar string, we need to understand the concept of harmonics and how they relate to the length of the vibrating string.

In the case of a vibrating string, the fundamental frequency (the lowest frequency produced) is determined by the length of the string. Each harmonic above the fundamental frequency is an integer multiple of the fundamental frequency. Harmonics are produced by dividing the vibrating string into equal parts and allowing these divisions to vibrate.

In this specific scenario, the fundamental frequency of the guitar string is given as 300 Hz. This frequency corresponds to the entire 60.0 cm length of the string (from the neck to the bridge).

When a guitarist gently touches the string at the specified fret, they limit the vibration to just the lower two-thirds of the string. This means that the vibrating length of the string is now two-thirds of its original length.

To find the frequency of the harmonic heard when plucking the string at one-sixth of its length from the bridge, we need to calculate the vibrating length of the string at this point.

Given:
- Original string length (L): 60.0 cm
- Vibrating length when the fret is applied (L'): two-thirds of L
- Plucking position (P): one-sixth of L

Let's calculate the new vibrating length (L'):

Vibrating length (L') = (2/3) * L
= (2/3) * 60.0 cm
= 40.0 cm

Now, we can calculate the frequency of the natural harmonic using the concept of harmonics:

Frequency of harmonic = (Fundamental frequency) * (Harmonic number)

In this case, the harmonic number corresponds to the ratio of the plucking position to the vibrating length. So, the harmonic number (N) can be calculated as:

N = P / L'
= (1/6) * L' / L'
= 1/6

Finally, let's substitute the values into the formula to find the frequency:

Frequency of harmonic = (Fundamental frequency) * (Harmonic number)
= 300 Hz * (1/6)
= 50 Hz

Therefore, the frequency that will be heard when playing this specific natural harmonic on the guitar string will be 50 Hz.