The difference in length between two rods A and B is 60 cm at all temperatures. If $\alpha_A = 18 \times 10^{-6}/^\circ C$ and $\alpha_B = 27 \times 10^{-6}/^\circ C$, then the length of rod A and rod B at $0^\circ C$ is respectively ______.
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"Constant length difference" always means $L_1 \alpha_1 = L_2 \alpha_2$. The initial lengths are inversely proportional to their expansion coefficients. The rod with the smaller $\alpha$ must be much longer!
Step 1: Understanding the Concept:
If the difference in length ($l_A - l_B$) is constant at all temperatures, then the change in length for a given temperature change must be equal: $\Delta l_A = \Delta l_B$. Step 2: Formula Application:
$l_A \alpha_A \Delta T = l_B \alpha_B \Delta T \implies l_A \alpha_A = l_B \alpha_B$. Step 3: Explanation:
$l_A (18 \times 10^{-6}) = l_B (27 \times 10^{-6}) \implies \frac{l_A}{l_B} = \frac{27}{18} = \frac{3}{2} = 1.5$.
So, $l_A = 1.5 l_B$.
Given $l_A - l_B = 60$.
$1.5 l_B - l_B = 60 \implies 0.5 l_B = 60 \implies l_B = 120$ cm.
$l_A = 120 + 60 = 180$ cm. Step 4: Final Answer:
The lengths are $l_A = 180$ cm and $l_B = 120$ cm.