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Question
A thin rod having length L0 at 0°C and coefficient of linear expansion α has its two ends maintained at temperatures θ1 and θ2, respectively. Find its new length.
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Solution
Consider the diagram
θ = `(θ_1 + θ_2)/2`
Let temperature varies linearly in the rod from one end to the other ends. Let θ be the temperature of the mid-point of the rod. At steady state.
Rate of flow of heat,
`((dQ)/dt) = (KA (θ_1 - θ))/(L_0/2) = (KA(θ - θ_2))/(L_0/2)`
Where K is the coefficient of thermal conductivity of the rod
or ⇒ `θ_1 - θ = θ - θ_2`
or ⇒ θ = `(θ_1 + θ_2)/2`
Using relation, `L = L_0 (1 + αθ)`
or `L = L_0 [1 + θ((θ_1 + θ_2)/2)]`
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