A wire of length L and cross-sectional area A is made of a material of Young's modulus Y. IF the wire is stretched by an amount x, the workdone is

Options

A. `(Y Ax^(2))/(2L)`
B. `(Y A x)/(2L^(2))`
C. `(Y A x)/(2L)`
D. `(Y A x^(2))/(L )`

Answer

To find the work done in stretching a wire of length \( L \) and cross-sectional area \( A \) made of a material with Young's modulus \( Y \) by an amount \( x \), we can follow these steps:

### Step 1: Understand Young's Modulus
Young's modulus \( Y \) is defined as the ratio of stress to strain:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]
Where:
- \( F \) is the force applied,
- \( A \) is the cross-sectional area,
- \( \Delta L \) is the change in length (which is \( x \)),
- \( L \) is the original length of the wire.

### Step 2: Rearranging Young's Modulus
From the definition of Young's modulus, we can rearrange the equation to find the force \( F \):
\[ F = Y \cdot \frac{A \cdot x}{L} \]

### Step 3: Work Done in Stretching the Wire
The work done \( W \) in stretching the wire through a distance \( x \) can be expressed as:
\[ dW = F \cdot dx \]
Substituting the expression for \( F \):
\[ dW = \left( Y \cdot \frac{A \cdot x}{L} \right) dx \]

### Step 4: Integrating to Find Total Work Done
To find the total work done in stretching the wire from \( 0 \) to \( x \), we need to integrate \( dW \):
\[ W = \int_{0}^{x} Y \cdot \frac{A \cdot x}{L} \, dx \]
Since \( Y \), \( A \), and \( L \) are constants, we can take them out of the integral:
\[ W = Y \cdot \frac{A}{L} \int_{0}^{x} x \, dx \]

### Step 5: Evaluating the Integral
The integral of \( x \) is:
\[ \int x \, dx = \frac{x^2}{2} \]
Evaluating this from \( 0 \) to \( x \):
\[ W = Y \cdot \frac{A}{L} \cdot \left[ \frac{x^2}{2} - 0 \right] = Y \cdot \frac{A}{L} \cdot \frac{x^2}{2} \]

### Step 6: Final Expression for Work Done
Thus, the total work done in stretching the wire by an amount \( x \) is:
\[ W = \frac{Y A x^2}{2L} \]

### Summary
The work done in stretching the wire is given by:
\[ W = \frac{Y A x^2}{2L} \]