A wire of length 100.0m at 30o has linear expansive of 2x10-5K-1 Calculate the length of the wire at a temperature of - 10oC

Analysis

To solve this problem, we need to calculate the change in length of the wire due to a change in temperature. This is a classic application of the Linear Expansivity formula.

Concept

The formula for linear expansion is:

Where:

• $L_1$ = Initial length = $100.0\text{ m}$
• $L_2$ = Final length
• $\alpha$ = Linear expansivity = $2 \times 10^{-5}\text{ K}^{-1}$
• $\Delta\theta$ = Change in temperature = $(\theta_2 - \theta_1)$

Step-by-Step Solution

1. Identify the variables:

• $L_1 = 100.0\text{ m}$
• $\alpha = 2 \times 10^{-5}\text{ K}^{-1}$
• $\theta_1 = 30^\circ\text{C}$
• $\theta_2 = -10^\circ\text{C}$

2. Calculate the change in temperature ($\Delta\theta$):

(Note: A change of $1^\circ\text{C}$ is equivalent to a change of $1\text{ K}$, so we can use $-40\text{ K}$ directly.)

3. Apply the formula:

Conclusion

The calculation shows that the wire contracts as the temperature drops from $30^\circ\text{C}$ to $-10^\circ\text{C}$. The final length is $99.92\text{ m}$.

Exam Tip

Always pay attention to the sign of your temperature change ($\Delta\theta$). If the final temperature is lower than the initial temperature, the change is negative, which means the material will contract (the length will decrease). If the temperature increases, the change is positive, and the material will expand.