One end of a long metallic wire of length (L) is tied to the ceiling. The other end is tied to a massless spring of spring constant . (K.A) mass (m) hangs freely from the free end of the spring. The area of cross- section and the Young's modulus of the wire are (A) and (Y) respectively. If the mass is slightly pulled down and released, it will oscillate with a time period (T) equal to :

Options

A. `2pisqrt((m)/(k))`
B. `2pisqrt((m(YA+kL))/(YAK))`
C. `2pisqrt((m(YA+kL))/(AK))`
D. `2pisqrt((m(Y+kL))/(YAK))`

Explanation

To solve the problem, we need to find the time period \( T \) of the oscillation of a mass \( m \) attached to a spring and a wire. We will consider the spring constant of the wire and the spring in series to find the equivalent spring constant.

### Step-by-step Solution:

1. **Identify the Components**:
- We have a mass \( m \) hanging from a spring with spring constant \( K \).
- The mass is also attached to a wire of length \( L \), cross-sectional area \( A \), and Young's modulus \( Y \).

2. **Calculate the Spring Constant of the Wire**:
- The spring constant \( K_1 \) of the wire can be derived from the formula for Young's modulus:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]
- Rearranging gives:
\[ F = Y \cdot A \cdot \frac{\Delta L}{L} \]
- The spring constant \( K_1 \) is defined as the force per unit extension:
\[ K_1 = \frac{F}{\Delta L} = \frac{Y \cdot A}{L} \]

3. **Combine the Spring Constants**:
- The effective spring constant \( K_{eq} \) for two springs in series (the spring and the wire) is given by:
\[ \frac{1}{K_{eq}} = \frac{1}{K} + \frac{1}{K_1} \]
- Substituting \( K_1 \):
\[ \frac{1}{K_{eq}} = \frac{1}{K} + \frac{L}{Y \cdot A} \]
- Taking the reciprocal gives:
\[ K_{eq} = \frac{K \cdot (Y \cdot A)}{Y \cdot A + K \cdot L} \]

4. **Calculate the Time Period**:
- The time period \( T \) of the oscillation is given by:
\[ T = 2\pi \sqrt{\frac{m}{K_{eq}}} \]
- Substituting \( K_{eq} \):
\[ T = 2\pi \sqrt{\frac{m \cdot (Y \cdot A + K \cdot L)}{K \cdot (Y \cdot A)}} \]

### Final Expression for Time Period:
The time period \( T \) can be expressed as:
\[ T = 2\pi \sqrt{\frac{m \cdot (Y \cdot A + K \cdot L)}{K \cdot (Y \cdot A)}} \]