A metallic wire of length `l` is held between two rigid supports. If the wire is cooled through a temperature t. (Y= Young's modulus of elasticity of wire, `rho=` density, `alpha=` thermal coefficient of linear expansion). Then the frequency of oscillation is proportional to

Options

A. `1/l`
B. `sqrt(Y)`
C. `sqrt(Yalpha//rho)`
D. `sqrt(l)`

Answer

To determine how the frequency of oscillation of a metallic wire held between two rigid supports changes when the wire is cooled, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A metallic wire of length \( l \) is held between two rigid supports. When the wire is cooled, it contracts, and this contraction generates tension in the wire.
- We need to find how the frequency of oscillation relates to Young's modulus \( Y \), density \( \rho \), and the thermal coefficient of linear expansion \( \alpha \).

2. **Establishing the Relationship**:
- The tension \( F \) in the wire due to cooling can be expressed as:
\[ F = Y \cdot A \cdot \alpha \cdot \Delta T \]
where \( A \) is the cross-sectional area of the wire, and \( \Delta T \) is the change in temperature.

3. **Frequency of Oscillation**:
- The frequency \( f \) of a vibrating wire can be expressed using the formula:
\[ f \propto \frac{1}{2l} \sqrt{\frac{F}{\mu}} \]
where \( \mu \) is the mass per unit length of the wire, given by \( \mu = \rho \cdot A \).

4. **Substituting for Tension**:
- Substitute the expression for tension \( F \) into the frequency formula:
\[ f \propto \frac{1}{2l} \sqrt{\frac{Y \cdot A \cdot \alpha \cdot \Delta T}{\mu}} \]

5. **Replacing \( \mu \)**:
- Since \( \mu = \rho \cdot A \), we can rewrite the frequency as:
\[ f \propto \frac{1}{2l} \sqrt{\frac{Y \cdot A \cdot \alpha \cdot \Delta T}{\rho \cdot A}} \]
- The area \( A \) cancels out:
\[ f \propto \frac{1}{2l} \sqrt{\frac{Y \cdot \alpha \cdot \Delta T}{\rho}} \]

6. **Final Expression**:
- Thus, the frequency of oscillation is proportional to:
\[ f \propto \sqrt{\frac{Y \cdot \alpha}{\rho}} \]

### Conclusion:
The frequency of oscillation of the metallic wire, when cooled, is proportional to \( \sqrt{\frac{Y \cdot \alpha}{\rho}} \).