A metal rod has a length of 100cm at 200°C. At what temperature will its length be 99.4cm. If the linear…

Analysis

To solve this problem, we need to use the formula for linear expansion, which describes how the length of a solid changes when its temperature changes.

Concept

The formula for linear expansion is:

Where:

• $L_1$ = Initial length (100 cm)
• $L_2$ = Final length (99.4 cm)
• $\alpha$ = Linear expansivity ($2 \times 10^{-5} \, ^\circ\text{C}^{-1}$)
• $\Delta\theta$ = Change in temperature ($\theta_2 - \theta_1$)
• $\theta_1$ = Initial temperature ($200^\circ\text{C}$)
• $\theta_2$ = Final temperature (what we need to find)

Step-by-Step Solution

1. Identify the change in length ($\Delta L$):

2. Use the linear expansivity formula:

3. Solve for $\Delta\theta$:

4. Find the final temperature ($\theta_2$): Since $\Delta\theta = \theta_2 - \theta_1$:

Conclusion

The temperature at which the rod will have a length of 99.4 cm is $-100^\circ\text{C}$.

Exam Tip

Always pay attention to the sign of your change in length ($\Delta L$). If the length decreases, $\Delta L$ must be negative, which will lead to a negative temperature change. Don't let the negative sign confuse you—it simply means the temperature dropped!