To calculate the average coefficient of linear expansion ($\alpha$) of the rod's material, we use the formula:

$\alpha = \frac{\Delta L}{L_0 \Delta T}$

Where:

• $\Delta L$ is the change in length of the rod,
• $L_0$ is the initial length of the rod,
• $\Delta T$ is the change in temperature.

Given:

• Initial length ($L_0$) = $40.125 \, \text{cm}$,
• Final length ($L$) = $40.423 \, \text{cm}$,
• Initial temperature ($T_0$) = $20.0^\circ \text{C}$,
• Final temperature ($T$) = $45.0^\circ \text{C}$.

Step 1: Calculate $\Delta L$ and $\Delta T$

$\Delta L = L - L_0 = 40.423 \, \text{cm} - 40.125 \, \text{cm} = 0.298 \, \text{cm}$

$\Delta T = T - T_0 = 45.0^\circ \text{C} - 20.0^\circ \text{C} = 25.0^\circ \text{C}$

Step 2: Substitute into the formula

$\alpha = \frac{\Delta L}{L_0 \Delta T} = \frac{0.298 \, \text{cm}}{40.125 \, \text{cm} \times 25.0^\circ \text{C}}$

$\alpha = \frac{0.298}{1003.125} \, \text{°C}^{-1}$

$\alpha \approx 2.97 \times 10^{-4} \, \text{°C}^{-1}$

Final Answer:

The average coefficient of linear expansion of the rod's material is approximately:

$\boxed{\alpha \approx 2.97 \times 10^{-4} \, \text{°C}^{-1}}$

---

Would you like further explanation or have additional questions?

5 Related Questions:

1. How would the coefficient change if the temperature range increased?
2. What are some common materials with similar coefficients of linear expansion?
3. How does this concept apply to bimetallic strips in thermostats?
4. What is the formula for volumetric expansion, and how does it relate to linear expansion?
5. How does temperature affect the elasticity of materials along with expansion?

Tip:

Always ensure units are consistent (e.g., centimeters vs meters) to avoid calculation errors.