First slide

Thermal expansion

Question

Two metal rods of lengths L₁ and L₂ and coefficients of linear expansion $α_{1}$ and $α_{2}$ respectively are welded together to make a composite rod of length (L₁ + L₂) at 0°C. Find the effective coefficient of linear expansion of the composite rod. [EAMCET 2015]

Moderate

A

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

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

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

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

Solution

Given, the lengths of rods are L₁ and L₂
and coefficients of linear expansions are $α_{1}$ and $α_{2}$ .
As, $∆ L_{1} = α_{1} L_{1} ∆ t$ and $∆ L_{2} = α_{2} L_{2} ∆ t$
and for combined rod,
$∆ L = α (L_{1} + L_{2}) ∆ t$
Now, $∆ L = ∆ L_{1} + ∆ L_{2}$
$\Rightarrow α (L_{1} + L_{2}) ∆ t = α_{1} L_{1} ∆ t + α_{2} L_{2} ∆ t$
$\Rightarrow α (L_{1} + L_{2}) = α_{1} L_{1} + α_{2} L_{2}$
$\Rightarrow$ $α = \frac{α_{1} L_{1} + α_{2} L_{2}}{L_{1} + L_{2}}$

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