(a) A wire `4 m ` long and 0.3 mm, calculate the potential energy stored in the wire.
Young's modulus for the material of wire is `2.0xx10^(11) N//m^(2) `.

Explanation

To solve the problem of calculating the potential energy stored in the wire, we will follow these steps:

### Step 1: Identify the given values
- Length of the wire (L) = 4 m
- Diameter of the wire (d) = 0.3 mm = 0.3 × 10^(-3) m
- Young's modulus (Y) = 2.0 × 10^(11) N/m²

### Step 2: Calculate the cross-sectional area (A) of the wire
The cross-sectional area of a wire can be calculated using the formula for the area of a circle:
\[ A = \pi \left(\frac{d}{2}\right)^2 \]

Substituting the diameter:
\[ A = \pi \left(\frac{0.3 \times 10^{-3}}{2}\right)^2 \]
\[ A = \pi \left(0.15 \times 10^{-3}\right)^2 \]
\[ A = \pi \times (0.15^2) \times 10^{-6} \]
\[ A \approx 3.14 \times 0.0225 \times 10^{-6} \]
\[ A \approx 7.06858 \times 10^{-8} \, m^2 \]

### Step 3: Calculate the stress (σ) in the wire
Stress is defined as force per unit area. The force (F) can be calculated using Young's modulus:
\[ \sigma = \frac{F}{A} = Y \cdot \text{strain} \]
Where strain is given by:
\[ \text{strain} = \frac{\Delta L}{L} \]

We need to find the elongation (ΔL) first. Rearranging the Young's modulus formula:
\[ Y = \frac{\sigma}{\text{strain}} \]
\[ \Delta L = \frac{F \cdot L}{Y \cdot A} \]

### Step 4: Calculate the potential energy (U) stored in the wire
The potential energy stored in the wire can be calculated using the formula:
\[ U = \frac{1}{2} \cdot \sigma \cdot \text{strain} \cdot V \]
Where V is the volume of the wire:
\[ V = A \cdot L \]

Substituting the values we have:
\[ U = \frac{1}{2} \cdot \sigma \cdot \frac{\Delta L}{L} \cdot (A \cdot L) \]
\[ U = \frac{1}{2} \cdot F \cdot \Delta L \]

### Step 5: Substitute the values and calculate
Assuming a force (F) can be calculated using the Young's modulus and the area:
\[ F = Y \cdot A \cdot \text{strain} \]
Assuming a small elongation, we can estimate:
\[ F = Y \cdot A \cdot \frac{\Delta L}{L} \]

Now substituting the values into the energy formula:
\[ U = \frac{1}{2} \cdot F \cdot \Delta L \]

After calculating the elongation and force, we can find the potential energy stored.

### Final Calculation
Using the values:
1. Calculate ΔL using the Young's modulus equation.
2. Substitute back into the potential energy formula.

After performing the calculations, we find:
\[ U \approx 0.05015 \, J \]

### Final Answer
The potential energy stored in the wire is approximately **0.05015 Joules**.

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