Sure, let's solve this step by step. We need to determine the temperature at which a steel wire that is initially 2.0 meters long at a temperature of [tex]\(50^\circ \text{C}\)[/tex] will have a length of 2.0024 meters. The coefficient of linear expansion for steel, [tex]\(\alpha\)[/tex], is [tex]\(1.2 \times 10^{-5} \, \text{(°C)}^{-1}\)[/tex].
1. Determine the change in length ([tex]\(\Delta L\)[/tex]):
[tex]\[
\Delta L = \text{Final length} - \text{Initial length}
\][/tex]
Using the given values:
[tex]\[
\Delta L = 2.0024 \, \text{m} - 2.0 \, \text{m} = 0.0024 \, \text{m}
\][/tex]
2. Use the linear expansion formula:
The linear expansion formula is given by:
[tex]\[
\Delta L = L_0 \times \alpha \times \Delta T
\][/tex]
Where:
- [tex]\(\Delta L\)[/tex] is the change in length
- [tex]\(L_0\)[/tex] is the initial length
- [tex]\(\alpha\)[/tex] is the coefficient of linear expansion
- [tex]\(\Delta T\)[/tex] is the change in temperature
3. Rearrange the formula to solve for [tex]\(\Delta T\)[/tex]:
[tex]\[
\Delta T = \frac{\Delta L}{L_0 \times \alpha}
\][/tex]
4. Substitute the known values into the formula:
[tex]\[
\Delta T = \frac{0.0024 \, \text{m}}{2.0 \, \text{m} \times 1.2 \times 10^{-5} \, \text{°C}^{-1}}
\][/tex]
5. Calculate [tex]\(\Delta T\)[/tex]:
[tex]\[
\Delta T = 100 \, \text{°C}
\][/tex]
6. Determine the final temperature:
The final temperature ([tex]\(T_f\)[/tex]) is the initial temperature plus the change in temperature:
[tex]\[
T_f = T_{\text{initial}} + \Delta T
\][/tex]
Where:
- [tex]\(T_{\text{initial}}\)[/tex] is the initial temperature
7. Substitute the values:
[tex]\[
T_f = 50 \, \text{°C} + 100 \, \text{°C} = 150 \, \text{°C}
\][/tex]
Therefore, the temperature at which the length of the steel wire will be 2.0024 meters is [tex]\(150\, \text{°C}\)[/tex].
