A bar of iron is 10 cm at `20^(@)C`. At `19^(@)C` it will be `(alpha_(Fe)=11xx10^(-6)//.^(@)C)`

Options

A. `11xx10^(-6)`cm longer
B. `11xx10^(-6)cm` cm shorter
C. `11xx10^(-5)cm` shorter
D. `11xx10^(-5)cm` longer

Answer

To solve the problem, we need to calculate the change in length of the iron bar when the temperature decreases from 20°C to 19°C. We will use the formula for linear thermal expansion:

\[ \Delta L = L \cdot \alpha \cdot \Delta T \]

Where:
- \(\Delta L\) = change in length
- \(L\) = original length of the bar
- \(\alpha\) = coefficient of linear expansion
- \(\Delta T\) = change in temperature

### Step-by-Step Solution:

1. **Identify the given values:**
- Original length \(L = 10 \, \text{cm}\)
- Coefficient of linear expansion \(\alpha = 11 \times 10^{-6} \, \text{°C}^{-1}\)
- Initial temperature \(T_1 = 20 \, \text{°C}\)
- Final temperature \(T_2 = 19 \, \text{°C}\)

2. **Calculate the change in temperature (\(\Delta T\)):**
\[ \Delta T = T_2 - T_1 = 19 \, \text{°C} - 20 \, \text{°C} = -1 \, \text{°C} \]

3. **Substitute the values into the formula:**
\[ \Delta L = L \cdot \alpha \cdot \Delta T \]
\[ \Delta L = 10 \, \text{cm} \cdot (11 \times 10^{-6} \, \text{°C}^{-1}) \cdot (-1 \, \text{°C}) \]

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

5. **Interpret the result:**
The negative sign indicates that the length of the iron bar decreases. Therefore, the change in length is:
\[ \Delta L = -11 \times 10^{-5} \, \text{cm} \]

### Final Answer:
The bar of iron will be shorter by \(11 \times 10^{-5} \, \text{cm}\) at \(19 \, \text{°C}\).

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