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The temperature of a metal strip having coefficient of linear expansion α is increased from T₁ to T₂, resulting in an increase of its length by Δ L₁. The temperature is further increased from T₂ to T₃ such that the increase in its length is Δ L₂. Given T₃+T₁=2T₂ and T₂-T₁=Δ T, the value of Δ L₂ is ______.

Asked in JEE Main 4th April 2nd Shift 2026 · Linear expansion and composite rods

• (1) Δ L₁[1+2α²(Δ T)²]
• (2) Δ L₁[1+α²(Δ T)²]
• (3) Δ L₁[1+2αΔ T]
• (4) Δ L₁[1+αΔ T]

Step-by-step solution

Why the other options are wrong

• (1) comes from squaring the correction, as if the second stage started from L₁(1+αΔ T)², and then keeping the cross term 2α²(Δ T)² instead of the first-order αΔ T.
• (2) comes from keeps only the second-order piece of (1+αΔ T)²=1+2αΔ T+α²(Δ T)² and drops the term that actually matters; the growth in base length is first order in αΔ T.
• (3) comes from using the whole span T₃-T₁=2Δ T as the base-length correction, writing L₂=L₁(1+2αΔ T). The base length for the second stage is the length at T₂, which is only Δ T above T₁.

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