An iron bar 10 cm in length is kept at `20^@C`. If the coefficient of linear expansion of iron is `alpha = 11 xx 10^(-6).^(@)C^(-1)`, then at `19^(@)C` it will be

Options

A. `11 xx 10^(-6)cm` longer
B. `11 xx 10^(-6)cm` shorter
C. `11 xx 10^(-5)cm` shorter
D. `11 xx 10^(-5)cm` longer

Answer

To solve the problem of how the length of an iron bar changes with temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Original length of the iron bar, \( L_0 = 10 \, \text{cm} \)
- Initial temperature, \( T_1 = 20^\circ C \)
- Final temperature, \( T_2 = 19^\circ C \)
- Coefficient of linear expansion of iron, \( \alpha = 11 \times 10^{-6} \, ^\circ C^{-1} \)

2. **Calculate the Change in Temperature**:
\[ \Delta T = T_2 - T_1 = 19^\circ C - 20^\circ C = -1^\circ C \]
(The negative sign indicates a decrease in temperature.)

3. **Use the Formula for Linear Expansion**:
The change in length (\( \Delta L \)) due to temperature change can be calculated using the formula:
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta T \]

4. **Substitute the Values into the Formula**:
\[ \Delta L = 10 \, \text{cm} \cdot (11 \times 10^{-6} \, ^\circ C^{-1}) \cdot (-1^\circ C) \]

5. **Calculate the Change in Length**:
\[ \Delta L = 10 \cdot 11 \times 10^{-6} \cdot (-1) = -11 \times 10^{-5} \, \text{cm} \]

6. **Determine the New Length**:
Since the length decreases, the new length \( L \) can be calculated as:
\[ L = L_0 + \Delta L = 10 \, \text{cm} - 11 \times 10^{-5} \, \text{cm} \]

7. **Final Calculation**:
\[ L = 10 \, \text{cm} - 0.00011 \, \text{cm} = 9.99989 \, \text{cm} \]
The decrease in length is \( 11 \times 10^{-5} \, \text{cm} \).

### Conclusion:
The length of the iron bar at \( 19^\circ C \) will be \( 9.99989 \, \text{cm} \), and it decreases by \( 11 \times 10^{-5} \, \text{cm} \).