A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod is given to be 2.53k Hz. What is the speed of sound in steel?

Options

A. `6.2 km//s`
B. `5.06 km//s`
C. `7.32 km//s`
D. `7.45 km//s`

Answer

To find the speed of sound in steel based on the given information, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a steel rod that is 100 cm long, which is clamped at its middle. This means that the rod can vibrate in a fundamental mode of vibration, where the length of the rod is effectively divided into two segments.

2. **Convert Length to Meters**: The length of the rod is given as 100 cm. We convert this to meters:
\[ L = 100 \, \text{cm} = 1 \, \text{m} \]

3. **Identify the Vibration Mode**: Since the rod is clamped at its middle, it will vibrate in a fundamental mode where the length of the rod corresponds to half of the wavelength (λ) of the sound wave. Therefore, we can express this relationship as:
\[ L = \frac{\lambda}{2} \]

4. **Calculate the Wavelength (λ)**: Rearranging the equation gives us:
\[ \lambda = 2L = 2 \times 1 \, \text{m} = 2 \, \text{m} \]

5. **Use the Frequency to Find Speed**: The fundamental frequency (f) of the rod is given as 2.53 kHz. We convert this to Hz:
\[ f = 2.53 \, \text{kHz} = 2.53 \times 10^3 \, \text{Hz} \]

6. **Calculate the Speed of Sound (v)**: The speed of sound can be calculated using the formula:
\[ v = f \times \lambda \]
Substituting the values we have:
\[ v = (2.53 \times 10^3 \, \text{Hz}) \times (2 \, \text{m}) = 5.06 \times 10^3 \, \text{m/s} \]

7. **Convert to km/s**: To express the speed in kilometers per second:
\[ v = 5.06 \, \text{km/s} \]

### Final Answer:
The speed of sound in steel is approximately **5.06 km/s**.