What lengths of brass and iron rods at $0°$C must be used if the difference between their lengths is always $0.2$ m? The coefficients of linear expansion of brass and iron are $18 \times 10^{-6}\ °\text{C}^{-1}$ and $12 \times 10^{-6}\ °\text{C}^{-1}$ respectively.
Answer: (A) brass $0.4$ m, iron $0.6$ m
Concept. The difference in length stays constant only if both rods expand by the same amount: $l_b\alpha_b = l_i\alpha_i$.
Step 1. Ratio of lengths.
$$\frac{l_i}{l_b} = \frac{\alpha_b}{\alpha_i} = \frac{18}{12} = \frac32$$
The iron rod is the longer one.
Step 2. Use the difference. $l_i - l_b = 0.2$ m with $l_i = 1.5l_b$ gives $0.5l_b = 0.2$, so $l_b = 0.4$ m and $l_i = 0.6$ m.
Answer. Brass $0.4$ m, iron $0.6$ m (Option A).
Solution by Sreeraj P, M.Sc Physics