If a metal wire of length L, having area of cross-section A and young' s modulus Y . Behave as a spring of spring constant K. The value of K is

Options

A. `(YA)/(L)`
B. `(YA)/(2L)`
C. `(2YA)/(L)`
D. `(YL)/(A)`

Explanation

To find the spring constant \( K \) of a metal wire of length \( L \), cross-sectional area \( A \), and Young's modulus \( Y \), we can use the relationship between stress, strain, and Young's modulus.

### Step-by-Step Solution:

1. **Understanding Young's Modulus**:
Young's modulus \( Y \) is defined as the ratio of stress to strain. Mathematically, it can be expressed as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{\frac{F}{A}}{\frac{x}{L}} \]
where \( F \) is the force applied, \( A \) is the cross-sectional area, \( x \) is the extension, and \( L \) is the original length of the wire.

2. **Rearranging the Young's Modulus Formula**:
From the definition, we can rearrange the formula to express \( F \):
\[ Y = \frac{F \cdot L}{A \cdot x} \]
This can be rearranged to find \( F \):
\[ F = \frac{Y \cdot A \cdot x}{L} \]

3. **Identifying the Spring Constant**:
The spring constant \( K \) is defined in Hooke's law as:
\[ F = K \cdot x \]
By comparing this with the expression we derived for \( F \), we can equate the two:
\[ K \cdot x = \frac{Y \cdot A \cdot x}{L} \]

4. **Solving for the Spring Constant \( K \)**:
We can simplify this equation by dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ K = \frac{Y \cdot A}{L} \]

### Final Result:
Thus, the spring constant \( K \) of the metal wire is given by:
\[ K = \frac{Y \cdot A}{L} \]