What force should be applied to the ends of steel rod of cross sectional area `10 cm^(2)` to prevent it form elongation when heated form 273 K to 303 k ? `(alpha " of steel" 10^(-5).^(@)C ^(-1),Y=2xx 10^(11) NH^(-2))`

Options

A. `2 xx 10^(4)N`
B. `3 xx 10^(4)N`
C. `6 xx 10^(4)N`
D. `12 xx 10^(4)N`

Explanation

To solve the problem of determining the force that should be applied to the ends of a steel rod to prevent elongation when heated from 273 K to 303 K, we can follow these steps:

### Step 1: Identify the given values
- Cross-sectional area \( A = 10 \, \text{cm}^2 = 10 \times 10^{-4} \, \text{m}^2 = 10^{-3} \, \text{m}^2 \)
- Coefficient of linear expansion \( \alpha = 10^{-5} \, \text{°C}^{-1} \)
- Young's modulus \( Y = 2 \times 10^{11} \, \text{N/m}^2 \)
- Initial temperature \( T_1 = 273 \, \text{K} \)
- Final temperature \( T_2 = 303 \, \text{K} \)

### Step 2: Calculate the change in temperature
\[ \Delta T = T_2 - T_1 = 303 \, \text{K} - 273 \, \text{K} = 30 \, \text{K} \]

### Step 3: Relate stress, strain, and force
The stress \( \sigma \) in the rod can be expressed as:
\[ \sigma = Y \times \text{strain} \]
where strain can be defined in terms of thermal expansion:
\[ \text{strain} = \alpha \times \Delta T \]

### Step 4: Substitute the expression for strain into the stress formula
Thus, we can write:
\[ \sigma = Y \times (\alpha \times \Delta T) \]

### Step 5: Relate stress to force
Stress is also defined as force per unit area:
\[ \sigma = \frac{F}{A} \]
where \( F \) is the force and \( A \) is the cross-sectional area.

### Step 6: Combine the equations to find force
Equating the two expressions for stress:
\[ \frac{F}{A} = Y \times (\alpha \times \Delta T) \]
From this, we can solve for \( F \):
\[ F = A \times Y \times (\alpha \times \Delta T) \]

### Step 7: Substitute the known values into the equation
Substituting the values we have:
\[ F = (10 \times 10^{-4} \, \text{m}^2) \times (2 \times 10^{11} \, \text{N/m}^2) \times (10^{-5} \, \text{°C}^{-1}) \times (30 \, \text{K}) \]

### Step 8: Calculate the force
Calculating this step by step:
1. Calculate \( \alpha \times \Delta T \):
\[ 10^{-5} \times 30 = 3 \times 10^{-4} \]
2. Now substitute into the force equation:
\[ F = (10 \times 10^{-4}) \times (2 \times 10^{11}) \times (3 \times 10^{-4}) \]
\[ F = 10 \times 2 \times 3 \times 10^{-4} \times 10^{11} \times 10^{-4} \]
\[ F = 60 \times 10^{3} \, \text{N} = 6 \times 10^{4} \, \text{N} \]

### Final Answer
The force that should be applied to the ends of the steel rod to prevent elongation when heated is:
\[ F = 6 \times 10^{4} \, \text{N} \]