When a string fixed at its both ends vibrates in 1 loop, 2 loops, 3 loops and 4 loops, the frequencies are in the ratio

Options

A. 1 : 1 : 1 : 1
B. 1 : 2 : 3 : 4
C. 4 : 3 : 2 : 1
D. 1 : 4 : 9 : 16

Explanation

To solve the problem of finding the ratio of frequencies when a string fixed at both ends vibrates in different modes (1 loop, 2 loops, 3 loops, and 4 loops), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Vibrating String**:
- A string fixed at both ends can vibrate in different modes, which are characterized by the number of loops (or antinodes) formed. The first mode has 1 loop, the second has 2 loops, and so on.

2. **Formula for Frequency**:
- The frequency of vibration of a string fixed at both ends is given by the formula:
\[ f_n = \frac{n v}{2L} \]
where:
- \( f_n \) is the frequency of the nth harmonic,
- \( n \) is the number of loops (or harmonics),
- \( v \) is the speed of the wave on the string,
- \( L \) is the length of the string.

3. **Calculating Frequencies for Each Mode**:
- For **1 loop (1st harmonic)**:
\[ f_1 = \frac{1 \cdot v}{2L} = \frac{v}{2L} \]
- For **2 loops (2nd harmonic)**:
\[ f_2 = \frac{2 \cdot v}{2L} = \frac{2v}{2L} = \frac{v}{L} \]
- For **3 loops (3rd harmonic)**:
\[ f_3 = \frac{3 \cdot v}{2L} = \frac{3v}{2L} \]
- For **4 loops (4th harmonic)**:
\[ f_4 = \frac{4 \cdot v}{2L} = \frac{4v}{2L} = \frac{2v}{L} \]

4. **Finding the Ratio of Frequencies**:
- Now we can express the frequencies in terms of \( v \) and \( L \):
- \( f_1 = \frac{v}{2L} \)
- \( f_2 = \frac{v}{L} \)
- \( f_3 = \frac{3v}{2L} \)
- \( f_4 = \frac{2v}{L} \)

- The ratio of the frequencies \( f_1 : f_2 : f_3 : f_4 \) can be calculated as:
\[ f_1 : f_2 : f_3 : f_4 = \frac{v}{2L} : \frac{v}{L} : \frac{3v}{2L} : \frac{2v}{L} \]

- To simplify, we can multiply each term by \( 2L/v \) to eliminate \( v \) and \( L \):
\[ 1 : 2 : 3 : 4 \]

5. **Final Answer**:
- Therefore, the frequencies are in the ratio:
\[ 1 : 2 : 3 : 4 \]