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

Options

A. `11xx10` cm longer
B. `11xx10` cm shorter
C. `11xx10` cm shorter
D. `11xx10` cm longer

Answer

To find the length of the iron bar at 19°C, we can use the formula for linear thermal expansion:

\[ L = L_0 (1 + \alpha \Delta T) \]

Where:
- \( L \) is the final length of the bar.
- \( L_0 \) is the initial length of the bar.
- \( \alpha \) is the coefficient of linear expansion of the material.
- \( \Delta T \) is the change in temperature.

### Step-by-Step Solution:

1. **Identify the given values:**
- Initial length \( L_0 = 10 \, \text{cm} \)
- Initial temperature \( T_0 = 20^\circ C \)
- Final temperature \( T_f = 19^\circ C \)
- Coefficient of linear expansion for iron \( \alpha = 11 \times 10^{-6} \, \text{°C}^{-1} \)

2. **Calculate the change in temperature (\( \Delta T \)):**
\[ \Delta T = T_f - T_0 = 19^\circ C - 20^\circ C = -1^\circ C \]

3. **Substitute the values into the thermal expansion formula:**
\[ L = L_0 (1 + \alpha \Delta T) \]
\[ L = 10 \, \text{cm} \left( 1 + (11 \times 10^{-6}) \times (-1) \right) \]

4. **Calculate the expression inside the parentheses:**
\[ L = 10 \, \text{cm} \left( 1 - 11 \times 10^{-6} \right) \]
\[ L = 10 \, \text{cm} \left( 1 - 0.000011 \right) \]
\[ L = 10 \, \text{cm} \times 0.999989 \]

5. **Calculate the final length:**
\[ L \approx 10 \, \text{cm} - 0.00011 \, \text{cm} = 9.99989 \, \text{cm} \]

6. **Determine the change in length:**
\[ \Delta L = L_0 - L = 10 \, \text{cm} - 9.99989 \, \text{cm} \approx 0.00011 \, \text{cm} \]

### Final Answer:
The length of the iron bar at 19°C is approximately **9.99989 cm**, which is about **0.00011 cm shorter** than at 20°C.

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