The string of a violin has a fundamental frequency of 440 Hz. If the violin string is shortend by one fifth, itd fundamental frequency will be changed to

Options

A. 440 Hz
B. 880 Hz
C. 550 Hz
D. 2200 Hz

Answer

To solve the problem of how the fundamental frequency of a violin string changes when the string is shortened by one-fifth, we can follow these steps:

### Step 1: Understand the relationship between frequency and length
The fundamental frequency (f) of a vibrating string is inversely proportional to its length (L). This relationship can be expressed by the formula:
\[ f \propto \frac{1}{L} \]

### Step 2: Define the original length and frequency
Let the original length of the string be \( L_1 \) and the original frequency be \( f_1 = 440 \, \text{Hz} \).

### Step 3: Calculate the new length
If the string is shortened by one-fifth, the new length \( L_2 \) can be calculated as:
\[ L_2 = L_1 - \frac{1}{5}L_1 = \frac{4}{5}L_1 \]

### Step 4: Relate the new frequency to the new length
Using the inverse relationship between frequency and length, we can express the new frequency \( f_2 \) as:
\[ f_2 \propto \frac{1}{L_2} \]

### Step 5: Substitute the new length into the frequency equation
Substituting \( L_2 \) into the frequency equation gives:
\[ f_2 = k \cdot \frac{1}{L_2} = k \cdot \frac{1}{\frac{4}{5}L_1} = k \cdot \frac{5}{4L_1} \]
Where \( k \) is a constant that relates frequency and length.

### Step 6: Relate the new frequency to the original frequency
Since \( f_1 = k \cdot \frac{1}{L_1} \), we can express \( k \) in terms of \( f_1 \):
\[ k = f_1 \cdot L_1 \]
Thus, we can express \( f_2 \) in terms of \( f_1 \):
\[ f_2 = f_1 \cdot \frac{5}{4} \]

### Step 7: Substitute the original frequency
Now, substituting \( f_1 = 440 \, \text{Hz} \):
\[ f_2 = 440 \cdot \frac{5}{4} = 440 \cdot 1.25 = 550 \, \text{Hz} \]

### Final Answer
The new fundamental frequency of the violin string after it is shortened by one-fifth is **550 Hz**.

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