Determine the ratio of the lengths of an iron rod and an aluminium rod such that the difference in their lengths remains independent of temperature. The coefficients of linear expansion for iron and aluminium are 12 10⁻⁶ ^ C ⁻¹ and 23 10⁻⁶ ^ C ⁻¹ respectively.

Step-by-step solution

Let L₁ and L₂ be the lengths of the iron and aluminium rods respectively. Let ₁ and ₂ be their coefficients of linear expansion. The change in length for a temperature change T is given by L = L T . For the difference in lengths to remain independent of temperature, the change in length of both rods must be equal. L₁ = L₂ L₁ ₁ T = L₂ ₂ T L₁ ₁ = L₂ ₂ L₁ L₂ = ₂ ₁ Substituting the given values, ₁ = 12 10⁻⁶ ^ C ⁻¹ and ₂ = 23 10⁻⁶ ^ C ⁻¹ : L₁ L₂ = 23 10⁻⁶ 12 10⁻⁶ = 23 12 The ratio of their lengths is 23 : 12 .