First slide

Thermal expansion

Question

Two rods of lengths L₁ and L₂ are welded together to make a composite rod of length (L₁+L₂). If the coefficient of linear expansion of the materials of the rods are α₁ and α₂ respectively, the effective coefficient of linear expansion of the composite rod is

Moderate

A

$\frac{L_{1} α_{1} - L_{2} α_{2}}{L_{1} + L_{2}}$
B

$\frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}}$
C

$\sqrt{α_{1} α_{2}}$
D

$\frac{α_{1} + α_{2}}{2}$

Solution

$\frac{Δ_{1}}{Δt} = L_{1} a_{1}; \frac{Δ_{2}}{Δt} = L_{2} a_{2}$
$∴ \frac{ΔL}{Δt} = \frac{Δ_{1}}{Δt} + \frac{Δ_{2}}{Δt}; (L_{1} + L_{2}) a = L_{1} a_{1} + L_{2} a_{2}$

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