Determine the lengths of an iron rod and a copper ruler at 0 ° C if the difference in their lengths at 50 ° C and 450 ° C is the same and is equal to 2 cm . The coefficient of linear expansion of iron = 12 × 10 - 6 K - 1 and that of copper = 17 × 10 - 6 K - 1 .

Step-by-step solution

Let x be the length of the iron rod at 0   ° C , y that of the copper rod at 0   ° C , and I the difference in lengths at t 1 and t 2 o C. Then l = x 1 + α 1 t 1 - y 1 + α 2 t 1 ...(i) and ± l = x 1 + α 1 t 2 - y 1 + α 2 t 2 ...(ii) Taking + 1 l = x 1 + α 1 t 2 - y 1 + α 2 t 2 ...(iii) from Eqs. (i) and (iii), we get x α 1 = y α 2 ...(iv) from Eqs. (i) and (iv), we get y = l ⁡ α 1 α 2 - α 1 = 2 × 1 2 × 1 0 - 6 1 7 - 1 2