Two rods of length ℓ1 and ℓ2 are made of materials whose coefficient of linear expansion are α1 and α2. If difference between two lengths is independent of temperature

Engineering

Physics

Thermal Expansion

Question

Two rods of length ℓ₁ and ℓ₂ are made of materials whose coefficient of linear expansion are α₁ and α₂. If difference between two lengths is independent of temperature

$\frac{l_{1}}{l_{2}} = \frac{α_{2}}{α_{1}}$

$\frac{α_{1}^{2}}{l_{1}} = \frac{α_{2}^{2}}{l_{2}}$

$l_{2}^{2} α_{1} = l_{1}^{2} α_{2}$

$\frac{l}{l_{2}} = \frac{α_{1}}{α_{2}}$

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The difference in lengths (ℓ₁ - ℓ₂) must remain constant at any temperature. The change in length for a rod is given by Δℓ = ℓ α ΔT. For the difference to be temperature-independent, the thermal expansion of both rods must be equal: Δℓ₁ = Δℓ₂.

This gives: ℓ₁α₁ΔT = ℓ₂α₂ΔT. Canceling ΔT from both sides yields the condition:

$\frac{l_{1}}{l_{2}} = \frac{α_{2}}{α_{1}}$

Final Answer: ℓ₁/ℓ₂ = α₂/α₁