A steel rod of length `25cm` has a cross-sectional area of `0.8cm^(2)` . The force required to stretch this rod by the same amount as the expansion produced by heating it through `10^(@)C` is `(alpha_(steel)=10^(-5)//^(@)C` and `Y_(steel)=2xx10^(10)N//m^(2))`

Explanation

To solve the problem step by step, we will follow the principles of Young's modulus and linear expansion.

### Step 1: Understanding the Problem
We need to find the force required to stretch a steel rod by the same amount as the expansion produced by heating it through \(10^\circ C\). We are given:
- Length of the rod, \(L = 25 \, \text{cm} = 0.25 \, \text{m}\)
- Cross-sectional area, \(A = 0.8 \, \text{cm}^2 = 0.8 \times 10^{-4} \, \text{m}^2\)
- Coefficient of linear expansion of steel, \(\alpha = 10^{-5} \, \text{°C}^{-1}\)
- Young's modulus of steel, \(Y = 2 \times 10^{10} \, \text{N/m}^2\)

### Step 2: Calculate the Change in Length due to Thermal Expansion
The change in length (\(\Delta L\)) due to thermal expansion can be calculated using the formula:
\[ \Delta L = \alpha \cdot L \cdot \Delta T \]
where \(\Delta T = 10^\circ C\).

Substituting the values:
\[ \Delta L = (10^{-5} \, \text{°C}^{-1}) \cdot (0.25 \, \text{m}) \cdot (10 \, \text{°C}) = 2.5 \times 10^{-6} \, \text{m} \]

### Step 3: Calculate the Strain
Strain (\( \epsilon \)) is defined as the change in length divided by the original length:
\[ \epsilon = \frac{\Delta L}{L} = \frac{2.5 \times 10^{-6} \, \text{m}}{0.25 \, \text{m}} = 1 \times 10^{-5} \]

### Step 4: Calculate the Stress
Using Young's modulus, we know that:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \implies \text{Stress} = Y \cdot \epsilon \]
Substituting the values:
\[ \text{Stress} = (2 \times 10^{10} \, \text{N/m}^2) \cdot (1 \times 10^{-5}) = 2 \times 10^{5} \, \text{N/m}^2 \]

### Step 5: Calculate the Force
Force (\(F\)) can be calculated using the formula:
\[ F = \text{Stress} \cdot A \]
Substituting the values:
\[ F = (2 \times 10^{5} \, \text{N/m}^2) \cdot (0.8 \times 10^{-4} \, \text{m}^2) = 16 \, \text{N} \]

### Final Result
The force required to stretch the rod by the same amount as the expansion produced by heating it through \(10^\circ C\) is:
\[ F = 16 \, \text{N} \]