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: Understanding the Question:
If the difference in lengths remains constant at all temperatures, it means the change in length for a given change in temperature must be identical for both rods.
Step 2: Key Formula or Approach:
1) Change in length: \(\Delta L = L \alpha \Delta T\).
2) Condition for constant difference: \(\Delta L_{steel} = \Delta L_{copper} \implies L_s \alpha_s = L_c \alpha_c\).
Step 3: Detailed Explanation:
Given:
\(L_s - L_c = 5 \text{ cm} \dots (1)\)
\(\alpha_s = 1.1 \times 10^{-5}/^{\circ} \text{C}\)
\(\alpha_c = 1.7 \times 10^{-5}/^{\circ} \text{C}\)
From the constant difference condition:
\[ L_s \alpha_s = L_c \alpha_c \]
\[ L_s \times 1.1 \times 10^{-5} = L_c \times 1.7 \times 10^{-5} \]
\[ L_s = \frac{1.7}{1.1} L_c = \frac{17}{11} L_c \]
Substitute \(L_s\) in equation (1):
\[ \frac{17}{11} L_c - L_c = 5 \]
\[ \frac{17 - 11}{11} L_c = 5 \]
\[ \frac{6}{11} L_c = 5 \implies L_c = \frac{55}{6} \approx 9.17 \text{ cm} \]
Now find \(L_s\):
\[ L_s = L_c + 5 = 9.17 + 5 = 14.17 \text{ cm} \]
The closest values in the options are \(14 \text{ cm}\) and \(9 \text{ cm}\).
Step 4: Final Answer:
The lengths are nearly \(14 \text{ cm}\) and \(9 \text{ cm}\).