A chord attached to a viberating tunning fork divides it into 6loops, when its tension is 36N. The tensin at which it will viberate in 4loops is

Options

A. 24N
B. 36N
C. 64N
D. 81N

Answer

To solve the problem step by step, we will use the relationship between the wavelength, tension, and the number of loops formed by a vibrating string.

### Step-by-Step Solution:

1. **Understand the relationship between loops and wavelength**:
- When a string vibrates, the number of loops (or segments) formed is related to the wavelength (λ) of the wave. For n loops, the length of the string (L) can be expressed as:
\[ L = n \cdot \frac{\lambda}{2} \]
- This means that for 6 loops, the relationship is:
\[ L = 6 \cdot \frac{\lambda_1}{2} \implies L = 3\lambda_1 \]

2. **Determine the wavelength for 4 loops**:
- For 4 loops, the relationship is:
\[ L = 4 \cdot \frac{\lambda_2}{2} \implies L = 2\lambda_2 \]
- From this, we can express the wavelength for 4 loops as:
\[ \lambda_2 = \frac{L}{2} \]

3. **Relate the wavelengths**:
- Now we have:
\[ \lambda_1 = \frac{L}{3} \quad \text{and} \quad \lambda_2 = \frac{L}{2} \]
- To find the ratio of the wavelengths:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{L}{3}}{\frac{L}{2}} = \frac{2}{3} \]

4. **Use the relationship between wavelength and tension**:
- The wavelength is related to the tension (T) in the string by the formula:
\[ \lambda \propto \sqrt{T} \]
- Therefore, we can write:
\[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{T_1}{T_2}} \]
- Substituting the ratio of wavelengths:
\[ \frac{2}{3} = \sqrt{\frac{T_1}{T_2}} \]

5. **Square both sides**:
- Squaring both sides gives:
\[ \left(\frac{2}{3}\right)^2 = \frac{T_1}{T_2} \implies \frac{4}{9} = \frac{T_1}{T_2} \]

6. **Substitute the known tension**:
- Given that \( T_1 = 36 \, \text{N} \):
\[ \frac{4}{9} = \frac{36}{T_2} \]

7. **Solve for \( T_2 \)**:
- Cross-multiplying gives:
\[ 4T_2 = 36 \cdot 9 \]
\[ 4T_2 = 324 \]
\[ T_2 = \frac{324}{4} = 81 \, \text{N} \]

### Final Answer:
The tension at which the string will vibrate in 4 loops is **81 N**.