A steel rod 100 cm long is clamped at its midpoint. The fundamental frequency of longitudinal vibrtions of the rod is 3 kHz. What is the speed of the sound in the rod?

Options

A. `3xx10^(3)ms^(-1)`
B. `4xx10^(3)ms^(-1)`
C. `5xx10^(3)ms^(-1)`
D. `6xx10^(3)ms^(-1)`

Explanation

To find the speed of sound in the steel rod, we can follow these steps:

### Step 1: Understand the setup
The steel rod is 100 cm long and is clamped at its midpoint. This means that the rod is effectively divided into two equal halves, each 50 cm long. When clamped at the midpoint, the fundamental mode of vibration will have nodes at the clamped ends and antinodes at the open ends.

### Step 2: Determine the effective length for fundamental frequency
Since the rod is clamped at the midpoint, the effective length (L) for the fundamental frequency of longitudinal vibrations is half the length of the rod:
\[ L = \frac{100 \text{ cm}}{2} = 50 \text{ cm} = 0.5 \text{ m} \]

### Step 3: Use the relationship for fundamental frequency
The fundamental frequency (f) of longitudinal vibrations in a rod is given by the formula:
\[ f = \frac{v}{\lambda} \]
where \(v\) is the speed of sound in the rod and \(\lambda\) is the wavelength. For the fundamental frequency in a rod clamped at both ends, the wavelength is given by:
\[ \lambda = 2L \]
Thus, substituting for \(\lambda\):
\[ f = \frac{v}{2L} \]

### Step 4: Rearranging the formula to find speed
We can rearrange the formula to find the speed of sound in the rod:
\[ v = 2Lf \]

### Step 5: Substitute the known values
We know:
- \(L = 0.5 \text{ m}\)
- \(f = 3 \text{ kHz} = 3000 \text{ Hz}\)

Now substituting these values into the equation:
\[ v = 2 \times 0.5 \text{ m} \times 3000 \text{ Hz} \]
\[ v = 1 \text{ m} \times 3000 \text{ Hz} \]
\[ v = 3000 \text{ m/s} \]

### Final Answer
The speed of sound in the steel rod is:
\[ v = 3000 \text{ m/s} \]

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