A guitar string vibrates at
82.4 Hz
when its full length is allowed to vibrate. If it is now held so that four-fifths of it is allowed to vibrate in the same harmonic, what frequency will it produce?
To determine the frequency produced when four-fifths of a guitar string is allowed to vibrate, we need to consider the relationship between the length of the string and its frequency of vibration.
The frequency of a vibrating string is inversely proportional to its length. In other words, as the length of the string decreases, the frequency increases, and vice versa.
Given that the full length of the string vibrates at a frequency of 82.4 Hz, we can use this information to find the frequency when four-fifths of the string is allowed to vibrate.
Let's consider the proportion between the full length (L) and the length that vibrates (4/5L):
Full Length (L) : Vibrating Length (4/5L)
We can set up a proportion to find the relationship between the frequencies:
Frequency (f1) : Frequency (f2)
Since we know the frequency for the full length (82.4 Hz) and we want to find the frequency when four-fifths of the string vibrates, we can assign the values as follows:
82.4 Hz : Frequency (f2)
Now, let's set up the proportion:
L : 4/5L = 82.4 Hz : f2
To find f2, we cross-multiply and solve for f2:
L * f2 = 82.4 Hz * (4/5L)
Simplifying the equation:
f2 = (82.4 Hz * 4) / 5
f2 = 65.92 Hz
Therefore, when four-fifths of the guitar string is allowed to vibrate, it will produce a frequency of 65.92 Hz.