A machinist wishes to insert a copper rod with a diameter of 9 mm into a hole with a diameter of 8.998 mm. By how much would the machinist have to lower the temperature (in °C) of the rod to make it fit the hole?

To solve this problem, we need to calculate how much the temperature of the copper rod should be lowered so that its diameter contracts enough to fit into the hole.

### Formula for Linear Expansion:

The linear expansion of a material can be calculated using the formula:
\[
\Delta L = \alpha L_0 \Delta T
\]
Where:
- \(\Delta L\) is the change in length (or in this case, the change in diameter),
- \(\alpha\) is the coefficient of linear expansion of the material (for copper, \(\alpha = 16.5 \times 10^{-6} \, \text{°C}^{-1}\)),
- \(L_0\) is the initial length (or in this case, the initial diameter of the rod),
- \(\Delta T\) is the change in temperature.

### Step-by-Step Calculation:

1. **Initial Information:**
   - Diameter of the rod, \(L_0 = 9 \, \text{mm}\),
   - Diameter of the hole, \(L_{\text{hole}} = 8.998 \, \text{mm}\),
   - Change in diameter needed, \(\Delta L = L_0 - L_{\text{hole}} = 9 - 8.998 = 0.002 \, \text{mm}\).

2. **Rearrange the Formula:**
   We want to find \(\Delta T\), the change in temperature:
   \[
   \Delta T = \frac{\Delta L}{\alpha L_0}
   \]

3. **Substitute the Values:**
   \[
   \Delta T = \frac{0.002 \, \text{mm}}{(16.5 \times 10^{-6} \, \text{°C}^{-1}) \times 9 \, \text{mm}}
   \]

Let’s calculate this.The machinist would need to lower the temperature of the copper rod by approximately **13.47°C** to make it fit into the hole.

Would you like more details on the process or have any questions? Here are five related questions to consider:

1. How does the coefficient of linear expansion vary across different materials?
2. What happens to the properties of copper as temperature decreases?
3. How does volumetric expansion differ from linear expansion in three dimensions?
4. What is the significance of thermal contraction in manufacturing processes?
5. How could one account for the effects of temperature fluctuations in mechanical design?

**Tip:** When working with thermal expansion in metals, always check the coefficient of expansion, as it varies significantly between materials.

Learn how to calculate the temperature reduction required to fit a copper rod with a diameter of 9 mm into a hole of 8.998 mm. This problem uses the linear thermal expansion formula and involves calculating the change in temperature using copper's coefficient of linear expansion. Suitable for Grades 11-12.

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