At 20°C, the gap between the tip of the hanging bar and the ground is 3.5mm. At what temperature that the bar touches the ground if the bar has a length of 10m, mass density of 7850kg/m³, elasticity of 200GPa, and a=11.7µm/(m·°C)?

To determine the temperature at which the bar will touch the ground, we need to calculate the change in temperature required to make up the 3.5 mm gap due to thermal expansion.

Given dataTo determine the temperature at which the bar will touch the ground, we need to calculate the change in temperature required to make up the 3.5 mm gap due to thermal expansion.

Given 
- Initial gap, \( \Delta L = 3.5 \, \text{mm} = 0.0035 \, \text{m} \)
- Initial length of the bar, \( L = 10 \, \text{m} \)
- Coefficient of linear expansion, \( \alpha = 11.7 \times 10^{-6} \, \text{m/(m} \cdot \degree \text{C)} \)
- Initial temperature, \( T_{\text{initial}} = 20 \degree \text{C} \)

The formula for thermal expansion is:
\[
\Delta L = \alpha L \Delta T
\]
where:
- \( \Delta L \) is the change in length,
- \( \alpha \) is the coefficient of linear expansion,
- \( L \) is the original length,
- \( \Delta T \) is the change in temperature.

We need to solve for \( \Delta T \), which represents the required change in temperature for the bar to expand by 3.5 mm.

1. **Rearrange the formula to solve for \( \Delta T \):**
   \[
   \Delta T = \frac{\Delta L}{\alpha L}
   \]

2. **Substitute the values:**
   \[
   \Delta T = \frac{0.0035 \, \text{m}}{(11.7 \times 10^{-6} \, \text{m/(m} \cdot \degree \text{C)}) \times 10 \, \text{m}}
   \]

3. **Calculate \( \Delta T \):**
   \[
   \Delta T = \frac{0.0035}{11.7 \times 10^{-5}}
   \]
   \[
   \Delta T \approx 29.91 \degree \text{C}
   \]

4. **Determine the final temperature:**
   \[
   T_{\text{final}} = T_{\text{initial}} + \Delta T
   \]
   \[
   T_{\text{final}} = 20 \degree \text{C} + 29.91 \degree \text{C}
   \]
   \[
   T_{\text{final}} \approx 49.91 \degree \text{C}
   \]

### Answer:
The bar will touch the ground at approximately \( 49.91 \degree \text{C} \).

Would you like further clarification or additional details on this calculation?

Here are five related questions you might consider:
1. How would the required temperature change if the bar were twice as long?
2. How does the mass density of the material influence thermal expansion?
3. What if the coefficient of expansion were different? How would that affect the result?
4. How does elasticity affect the behavior of materials under thermal stress?
5. What would happen if the initial gap were different?

**Tip:** When calculating thermal expansion, ensure units are consistent to avoid errors in temperature changes.

Learn how to calculate the temperature at which a hanging bar with an initial 3.5mm gap will touch the ground. Using the bar's length of 10 meters, coefficient of linear expansion, and initial temperature of 20°C, the bar reaches the ground at approximately 49.91°C. This problem covers key thermal expansion concepts and calculations. Suitable for students in Grades 11-12.

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