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 find 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 1: Understand the relationship between loops and wavelength
When a string vibrates in n loops, the length of the string (L) is related to the wavelength (λ) by the formula:
\[ n \cdot \frac{\lambda}{2} = L \]
This means that the wavelength can be expressed as:
\[ \lambda = \frac{2L}{n} \]

### Step 2: Relate frequency to wavelength
The frequency (f) of a wave is related to its speed (v) and wavelength (λ) by the formula:
\[ f = \frac{v}{\lambda} \]
Substituting the expression for wavelength from Step 1, we get:
\[ f = \frac{v}{\frac{2L}{n}} = \frac{vn}{2L} \]

### Step 3: Calculate frequencies for different modes
Now we can calculate the frequencies for n = 1, 2, 3, and 4:
- For n = 1 (1 loop):
\[ f_1 = \frac{v \cdot 1}{2L} = \frac{v}{2L} \]

- For n = 2 (2 loops):
\[ f_2 = \frac{v \cdot 2}{2L} = \frac{2v}{2L} = \frac{v}{L} \]

- For n = 3 (3 loops):
\[ f_3 = \frac{v \cdot 3}{2L} = \frac{3v}{2L} \]

- For n = 4 (4 loops):
\[ f_4 = \frac{v \cdot 4}{2L} = \frac{4v}{2L} = \frac{2v}{L} \]

### Step 4: Write the frequencies in ratio form
Now we can express the frequencies in ratio form:
- \( f_1 : f_2 : f_3 : f_4 = \frac{v}{2L} : \frac{v}{L} : \frac{3v}{2L} : \frac{2v}{L} \)

### Step 5: Simplify the ratio
To simplify the ratio, we can eliminate \( v \) and \( L \):
- \( f_1 : f_2 : f_3 : f_4 = 1 : 2 : 3 : 4 \)

### Final Answer
Thus, the frequencies are in the ratio:
\[ 1 : 2 : 3 : 4 \]