Iron has coefficient of linear expansion of 12×10−6/𝐶 ∘ and copper has coefficient of linear expansion of 18×10−6/𝐶 ∘ An iron rod and a copper rod differ in lengths by 5 cm at all temperatures then, their lengths expressed in cm must respectively be
Question
Iron has coefficient of linear expansion of 12×10−6/𝐶 ∘ and copper has coefficient of linear expansion of 18×10−6/𝐶 ∘ An iron rod and a copper rod differ in lengths by 5 cm at all temperatures then, their lengths expressed in cm must respectively be
Solution
The coefficient of linear expansion (α) is the change in length per unit length per degree change in temperature.
Given: α_iron = 12×10−6/°C α_copper = 18×10−6/°C ΔL (difference in lengths) = 5 cm
We know that the change in length (ΔL) is given by the formula:
ΔL = α * L * ΔT
Where: ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, ΔT is the change in temperature.
In this case, the change in length is the same for both the iron and copper rod (5 cm), and we are looking for their original lengths (L_iron and L_copper).
We can set up the following equation:
ΔL_iron = ΔL_copper α_iron * L_iron * ΔT = α_copper * L_copper * ΔT
Since the change in temperature is the same for both rods, we can cancel out ΔT:
α_iron * L_iron = α_copper * L_copper
We can rearrange this equation to solve for L_iron:
L_iron = (α_copper / α_iron) * L_copper
Substituting the given values:
L_iron = (18×10−6/°C / 12×10−6/°C) * L_copper L_iron = 1.5 * L_copper
We also know that the difference in lengths is 5 cm, so:
L_copper - L_iron = 5 cm L_copper - 1.5 * L_copper = 5 cm -0.5 * L_copper = 5 cm L_copper = -5 cm / -0.5 = 10 cm
Substituting L_copper back into the equation for L_iron:
L_iron = 1.5 * 10 cm = 15 cm
So, the lengths of the iron and copper rods must be 15 cm and 10 cm respectively.
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