A violin string vibrates at 440 Hz when unfingered. At what frequency will it vibrate if it is fingered one fourth of the way down from the end?

the string will be 3/4 as long, so it will vibrate with 4/3 the open frequency.

440 * 4/3 = 587 Hz

To find the frequency at which the violin string will vibrate when it is fingered one fourth of the way down, we need to apply the rule of vibrating strings.

The frequency of a vibrating string is inversely proportional to its length. Mathematically, the relationship can be expressed as:

f ∝ 1/L

Where:
f = frequency
L = length of the vibrating string

In this case, when the string is unfingered, we can assume that the full length of the string (L) is responsible for the frequency of 440 Hz.

Now, if the string is fingered one fourth of the way down, we can assume that the length of the vibrating portion will be three-fourths of the original length (L).

Therefore, we can set up a proportion to find the new frequency (f'):

f/440 = 1/(3/4)L

Simplifying the equation:

f/440 = 4/3L

Now, we can rearrange the equation to solve for f':

f = (4/3L) * 440

Since we have the original frequency (440 Hz) and know that the original length (L) corresponds to this frequency, we can substitute these values into the equation to find the new frequency:

f = (4/3 * L) * 440

Before continuing, we need to determine the length of the vibrating portion when the string is fingered one fourth of the way down. Let's assume the full length of the string is represented by L0, and the length of the vibrating portion (L') can be calculated as:

L' = L0 - (1/4)L0

L' = 3/4L0

Now, let's substitute this value into the equation to find the new frequency:

f = (4/3 * 3/4L0) * 440

f = (16/12L0) * 440

Simplifying further:

f = (4/3 * 440L0)

f = (1760/3)L0

Therefore, the frequency at which the violin string will vibrate when it is fingered one fourth of the way down from the end is (1760/3)L0.

To find the frequency at which the violin string will vibrate when fingered one fourth of the way down from the end, we need to first understand the relationship between the length of a string and its frequency of vibration.

The frequency of vibration of a string is inversely proportional to its length. This means that as the length of the string decreases, the frequency increases, and vice versa.

In this case, if the string is fingered one fourth of the way down from the end, it means that the length of the vibrating portion of the string is three-fourths (3/4) of the total length.

To calculate the frequency, we can use the formula for the frequency of vibration of a string:

frequency = (speed of wave propagation) / (2 * length of string)

Since the speed of wave propagation remains constant for a given string, the frequency is directly proportional to the reciprocal of the length of the string.

Let's denote the original frequency when unfingered as f1, and the frequency when fingered as f2. Similarly, let's denote the original length of the string as L1, and the length of the string when fingered as L2.

According to the formula, we have:

f1 / L1 = f2 / L2

Since the fingered length is three-fourths (3/4) of the original length, we can substitute the values into the equation:

f1 / L1 = f2 / (3/4 * L1)

f1 / L1 = (4/3) * f2 / L1

Simplifying the equation, we can cancel out L1:

f1 = (4/3) * f2

Now, we rearrange the equation to find f2:

f2 = (3/4) * f1

Given that the original frequency f1 is 440 Hz, we substitute the value into the equation:

f2 = (3/4) * 440 Hz

Calculating this expression, we get:

f2 = 330 Hz

Therefore, the string will vibrate at a frequency of 330 Hz when fingered one fourth of the way down from the end.