A metal strip has coefficient of linear expansion α. Its temperature is first increased from T₁ to T₂, causing its length to increase by ΔL₁. It is then heated from T₂ to T₃, causing an additional increase ΔL₂. Given T₃+T₁ = 2T₂ and T₂-T₁ = ΔT, which expression gives ΔL₂ according to the finite-step expansion treatment used in the source solution?
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Question

A metal strip has coefficient of linear expansion α. Its temperature is first increased from T₁ to T₂, causing its length to increase by ΔL₁. It is then heated from T₂ to T₃, causing an additional increase ΔL₂. Given T₃+T₁ = 2T₂ and T₂-T₁ = ΔT, which expression gives ΔL₂ according to the finite-step expansion treatment used in the source solution?

Options

G1 Physics · Accepted Answer

From T₃+T₁ = 2T₂, we get T₃-T₂ = T₂-T₁ = ΔT. Let the original length at T₁ be L₀. The first increase is ΔL₁ = L₀αΔT. Therefore the length at T₂ is L(T₂) = L₀+ΔL₁. Using the finite-step treatment stated in the provided solution, the second increase is calculated from the length present at T₂: ΔL₂ = [L₀+ΔL₁]αΔT. Expand: ΔL₂ = L₀αΔT+ΔL₁αΔT. Since L₀αΔT = ΔL₁, we obtain ΔL₂ = ΔL₁[1+αΔT]. Correct option: D.