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Thermal Expansion

Two rods of l...

Question

Two rods of length $L_{1}$ and $L_{2}$ are welded together to make a composite rod of length ( $(L_{1} + L_{2})$ ). If the coefficient of linear expansion of the materials of the rods are $α_{1}$ and $α_{2}$ respectively, the effective coefficient of linear expansion of the composite rod is:

A

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

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B

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

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C

\sqrt{α_{1} α_{2}}

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D

\frac{α_{1} + α_{2}}{2}

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Solution

The correct option is B $\frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}}$
The new length of rod 1 will be $L_{1}^{'} = L_{1} (1 + α_{1} Δ T)$ and that of rod 2 will be $L_{2}^{'} = L_{2} (1 + α_{2} Δ T)$

Hence, for the combined rod,

$L_{1}^{'} + {L^{'}}_{2} = {(L}_{1} + L_{2}) (1 + α Δ T) = {(L}_{1} + L_{2}) + (L_{1} α_{1} + L_{2} α_{2}) Δ T$

Hence, $α = \frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}}$

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