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Thermal Expansion

Two rods, one...

Question

Two rods, one of aluminum and the other made of steel, having initial lengths $l_{1}$ and $l_{2}$ are connected together to form a single rod of length $l_{1} + l_{2}$ . The co-efficients of linear expansion for aluminum and steel are $α_{a}$ and $α_{s}$ respectively. If the length of each rod increases by the same amount when their temperature are raised by $t^{\circ} C$ , then find the ratio $\frac{l_{1}}{(l_{1} + l_{2})}$

A

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B

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C

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D

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Solution

The correct option is D
By co-efficient of linear expainsion,
$α = \frac{Δ L}{L \times Δ T}, W h e r e Δ L = c h a n g e i n l e n g t h ∴ Δ L = α L Δ T ∴ Δ L_{a} = α_{a} l_{1} Δ T a n d Δ L_{s} = α_{s} l_{2} Δ T ∵ Δ L_{a} = Δ L_{s} o r α_{a} l_{1} Δ T = α_{s} l_{2} Δ T o r \frac{l_{2}}{l_{1}} = \frac{α_{a}}{α_{s}} \Rightarrow \frac{l_{2}}{l_{1}} + 1 = \frac{α_{a}}{α_{s}} + 1 o r \frac{l_{2} + l_{1}}{l_{1}} = \frac{α_{a} + α_{s}}{α_{s}} ∴ \frac{l_{1}}{l_{1} + l_{2}} = \frac{α_{a}}{α_{a} + α_{s}}$

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