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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 coefficients of linear expansion of the rod are $α_{1}$ and $α_{2}$ respectively, the effective coefficient of linear expansion of the composite rod will be

A

\frac{L_{1} α_{1} + L_{2} α_{2}}{6 (L_{1} + L_{2})}

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B

\frac{L_{1} α_{1} + L_{2} α_{2}}{4 (L_{1} + L_{2})}

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C

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

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D

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

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Solution

The correct option is C $\frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}}$
Given:
Lengths of the two rods are $L_{1}$ and $L_{2}$
Length of composite rod $= L_{1} + L_{2}$
Coefficients of linear expansion $= α_{1}$ and $α_{2}$ respectively
To find:
Effective coefficient of linear expansion of composite rod, $α = ?$
Let us assume:
Change in temperature $= Δ T$
We know that, for end to end connection of rod,
effective expansion $=$ expansion in rod $1$ $+$ expansion in rod $2$
$\Rightarrow (L_{1} + L_{2}) α Δ T = L_{1} α_{1} Δ T + L_{2} α_{2} Δ T$
$\Rightarrow α = \frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}}$
is the effective coefficient of linear expansion.

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Thermal Expansion

Standard XII Physics

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