Question

The length of steel rod is \(5 \text{ cm}\) longer than the copper rod at all temperatures. The length of the steel and copper rod is respectively (\(\alpha_s = 1.1 \times 10^{-5}/^{\circ} \text{C}, \alpha_c = 1.7 \times 10^{-5}/^{\circ} \text{C}\))

• A. nearly 15 cm and 10 cm
• B. nearly 14 cm and 9 cm
• C. nearly 12 cm and 7 cm
• D. nearly 13 cm and 8 cm

Hint:

For the difference in length to remain independent of temperature, the rod with the smaller expansion coefficient must be longer.

Answer 1

The Correct Option is B

Step 1: Condition for Constant Difference
If the difference in lengths $(L_s - L_c)$ is constant at all temperatures, then their expansions must be equal: $\Delta L_s = \Delta L_c$.

Step 2: Expansion Formula
$L_s \alpha_s \Delta T = L_c \alpha_c \Delta T \implies L_s \alpha_s = L_c \alpha_c$.

Step 3: Calculation
$L_s (1.1) = L_c (1.7) \implies L_s = \frac{1.7}{1.1} L_c$.
Given $L_s - L_c = 5 \implies \frac{1.7}{1.1} L_c - L_c = 5 \implies \frac{0.6}{1.1} L_c = 5$.
$L_c = \frac{5.5}{0.6} \approx 9.16 \text{ cm}$ and $L_s \approx 14.16 \text{ cm}$.
Final Answer: (B)