What will be the speed of sound in a perfectly rigid rod?

Options

A. 332 m/s
B. les than 332 m/s
C. Infinite
D. Zero

Answer

To find the speed of sound in a perfectly rigid rod, we can follow these steps:

### Step 1: Understand the formula for the speed of sound
The speed of sound in a medium can be expressed using the formula:
\[ V_w = \sqrt{\frac{Y}{\rho}} \]
where:
- \( V_w \) is the speed of sound,
- \( Y \) is Young's modulus of the material,
- \( \rho \) is the density of the material.

### Step 2: Define Young's modulus
Young's modulus \( Y \) is defined as the ratio of stress to strain. Mathematically, it is given by:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
where:
- Stress = Force/Area,
- Strain = Change in length/Original length (or change in volume/original volume).

### Step 3: Analyze the case of a perfectly rigid rod
In a perfectly rigid rod, there is no deformation when a force is applied. This means that:
- The change in length (or change in volume) is zero.

### Step 4: Calculate Young's modulus for a perfectly rigid rod
Since strain is defined as the change in length/original length, and the change in length is zero:
\[ \text{Strain} = \frac{0}{\text{Original length}} = 0 \]
Thus, Young's modulus \( Y \) becomes:
\[ Y = \frac{\text{Stress}}{0} \]
This implies that \( Y \) approaches infinity.

### Step 5: Substitute Young's modulus into the speed of sound formula
Now, substituting \( Y \) into the speed of sound formula:
\[ V_w = \sqrt{\frac{Y}{\rho}} = \sqrt{\frac{\infty}{\rho}} = \infty \]

### Conclusion
Therefore, the speed of sound in a perfectly rigid rod is:
\[ \text{Speed of sound} = \infty \]

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