When the wire of a sonometer is 73 cm long, it is in resonance with a tuning fork. On shortening the wire by 0.5 cm, it makes 3 beats per second with the same fork. Calculate the frequency of the tuning fork?

Answer

To solve the problem step by step, we will use the relationship between the frequency of a vibrating wire and its length. The frequency of a wire is inversely proportional to its length.

### Step-by-Step Solution:

1. **Identify the initial conditions**:
- The original length of the wire, \( L = 73 \, \text{cm} \).
- The new length of the wire after shortening, \( L' = 73 \, \text{cm} - 0.5 \, \text{cm} = 72.5 \, \text{cm} \).

2. **Define the frequencies**:
- Let \( f \) be the frequency of the tuning fork (which is also the frequency of the wire when it is 73 cm long).
- Let \( f' \) be the frequency of the wire when it is shortened to 72.5 cm.
- According to the problem, \( f' = f + 3 \, \text{Hz} \) (since it makes 3 beats per second with the tuning fork).

3. **Use the relationship between frequency and length**:
- The frequency of the wire is inversely proportional to its length:
\[ f \propto \frac{1}{L} \]
- Therefore, we can write:
\[ \frac{f}{f'} = \frac{L'}{L} \]
- Substituting the values of \( L \) and \( L' \):
\[ \frac{f}{f + 3} = \frac{72.5}{73} \]

4. **Cross-multiply to solve for \( f \)**:
- Cross-multiplying gives:
\[ 73f = (f + 3) \times 72.5 \]
- Expanding the right side:
\[ 73f = 72.5f + 217.5 \]

5. **Rearranging the equation**:
- Bringing all terms involving \( f \) to one side:
\[ 73f - 72.5f = 217.5 \]
- Simplifying:
\[ 0.5f = 217.5 \]

6. **Solving for \( f \)**:
- Dividing both sides by 0.5:
\[ f = \frac{217.5}{0.5} = 435 \, \text{Hz} \]

7. **Conclusion**:
- The frequency of the tuning fork is \( 435 \, \text{Hz} \).