Two identical strings are stretched at tensions `T_(A)` and `T_(B)`. A truning fork is used to set them in vibration. A vibrates in its fundamental mode and `B` in its second harmonic mode.

Options

A. `T_(A)=2T_(B)`
B. `T_(A)=4T_(B)`
C. `2T_(A)=T_(B)`
D. `4T_(A)=T_(B)`

Answer

To solve the problem, we need to establish the relationship between the tensions \( T_A \) and \( T_B \) in the two strings vibrating at different modes.

### Step-by-Step Solution:

1. **Understanding the Modes of Vibration**:
- String A is vibrating in its **fundamental mode**.
- String B is vibrating in its **second harmonic mode**.

2. **Frequency of the Tuning Fork**:
- Let \( N \) be the frequency of the tuning fork used to set both strings in vibration. Since both strings are vibrating due to the same tuning fork, their frequencies will be equal.

3. **Frequency Formula for String A**:
- For string A vibrating in its fundamental mode, the frequency is given by:
\[ N = \frac{1}{2L} \sqrt{\frac{T_A}{M}} \]
where \( L \) is the length of the string and \( M \) is the mass per unit length.

4. **Frequency Formula for String B**:
- For string B vibrating in its second harmonic mode, the frequency is given by:
\[ N = 2 \times \frac{1}{2L} \sqrt{\frac{T_B}{M}} = \frac{1}{L} \sqrt{\frac{T_B}{M}} \]

5. **Setting the Frequencies Equal**:
- Since both frequencies are equal, we can set the two equations equal to each other:
\[ \frac{1}{2L} \sqrt{\frac{T_A}{M}} = \frac{1}{L} \sqrt{\frac{T_B}{M}} \]

6. **Simplifying the Equation**:
- Multiply both sides by \( 2L \):
\[ \sqrt{\frac{T_A}{M}} = 2 \sqrt{\frac{T_B}{M}} \]
- Squaring both sides gives:
\[ \frac{T_A}{M} = 4 \frac{T_B}{M} \]

7. **Cancelling \( M \)**:
- Since \( M \) is the same for both strings, we can cancel it out:
\[ T_A = 4 T_B \]

8. **Final Relationship**:
- Therefore, the relationship between the tensions in the two strings is:
\[ \frac{T_A}{T_B} = 4 \]

### Conclusion:
Thus, the tension in string A is four times the tension in string B, or \( T_A = 4 T_B \).