A wire of length 10m is cooled from 50oC to 10oC. Calculate the decrease in the length of the wire. [c = 3 x 10-5K-1]

Analysis

To solve this problem, we need to calculate the change in length ($\Delta L$) of a material due to a change in temperature. This is a classic application of Linear Expansivity.

Concept

The formula for linear expansion is:

Where:

• $\Delta L$ = Change in length (what we are looking for)
• $L_0$ = Original length of the wire ($10\text{ m}$)
• $\alpha$ = Coefficient of linear expansivity ($3 \times 10^{-5}\text{ K}^{-1}$)
• $\Delta \theta$ = Change in temperature ($\theta_2 - \theta_1$)

Solution

Step 1: Identify the given values.

• $L_0 = 10\text{ m}$
• $\alpha = 3 \times 10^{-5}\text{ K}^{-1}$
• $\theta_1 = 50^\circ\text{C}$
• $\theta_2 = 10^\circ\text{C}$

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

(Note: A change in Celsius is equivalent to a change in Kelvin, so we can use $40\text{ K}$).

Step 3: Substitute the values into the formula.

Step 4: Perform the multiplication.

Step 5: Convert to standard form.

Conclusion

Comparing our result with the given options, we find that $1.2 \times 10^{-2}\text{ m}$ matches option C.

Exam Tip: Always pay attention to the powers of ten. When multiplying numbers like $400 \times 3$, you get $1200$. Remember that $1200$ is the same as $1.2 \times 10^3$. When you multiply that by $10^{-5}$, you simply add the exponents ($3 + (-5) = -2$). This quick mental math saves time in the exam hall!