Question

How to calculate coefficient of linear expansion

Answer 1

The coefficient of linear expansion, often denoted as \( \alpha \), is a material constant that quantifies how much a material expands per unit length for each degree of temperature increase. It is defined by the equation:

\[
\alpha = \frac{1}{L_0} \frac{\Delta L}{\Delta T}
\]

where:
- \( \alpha \) is the coefficient of linear expansion (in units of \( \text{°C}^{-1} \) or \( \text{K}^{-1} \)),
- \( L_0 \) is the original length of the material,
- \( \Delta L \) is the change in length,
- \( \Delta T \) is the change in temperature.

### Steps to Calculate the Coefficient of Linear Expansion:

1. **Measure the Original Length**:
- Record the initial length \( L_0 \) of the material at a known temperature.

2. **Change the Temperature**:
- Heat or cool the material to a new temperature, allowing it to reach thermal equilibrium.

3. **Measure the New Length**:
- Measure the new length \( L \) of the material after the temperature change.

4. **Calculate the Change in Length**:
- Compute the change in length \( \Delta L \):
\[
\Delta L = L - L_0
\]

5. **Determine the Change in Temperature**:
- Calculate the change in temperature \( \Delta T \):
\[
\Delta T = T_{\text{final}} - T_{\text{initial}}
\]

6. **Substitute Values into the Coefficient Formula**:
- Finally, substitute \( \Delta L \), \( \Delta T \), and \( L_0 \) into the formula for \( \alpha \):
\[
\alpha = \frac{\Delta L}{L_0 \cdot \Delta T}
\]

### Example Calculation:

- Suppose you have a metal rod that is originally 1.0 m long at 20 °C.
- Upon heating it to 100 °C, the new length is measured to be 1.002 m.

1. \( L_0 = 1.0 \, \text{m} \)
2. \( L = 1.002 \, \text{m} \)
3. \( \Delta L = 1.002 \, \text{m} - 1.0 \, \text{m} = 0.002 \, \text{m} \)
4. \( T_{\text{initial}} = 20 \, \text{°C} \)
5. \( T_{\text{final}} = 100 \, \text{°C} \)
6. \( \Delta T = 100 - 20 = 80 \, \text{°C} \)

Now, calculate \( \alpha \):

\[
\alpha = \frac{0.002 \, \text{m}}{1.0 \, \text{m} \cdot 80 \, \text{°C}} = \frac{0.002}{80} = 0.000025 \, \text{°C}^{-1} \text{ or } 2.5 \times 10^{-5} \, \text{°C}^{-1}
\]

This is the coefficient of linear expansion for the metal rod.