Two rods, one of aluminium and the other made of steel, having initial length l1

Question

Two rods, one of aluminium and the other made of steel, having initial length

l_{1}

and

l_{2}

are connected together to form a single rod of length

l_{1} + l_{2} .

The coefficients of linear expansion for aluminium and steel are

α_{a}

and

α_{a}

and respectively. If the length of each rod increases by the same amount when their temperature are raised by

t^{o} C

, then find the ratio of length of aluminium rod to the whold rod.
(IIT JEE 2003)

Options:

A . αs/αa

B . αa/αs

C . αs/(αa+αs)

D . αa/(αa+αs)

Answer: Option C
:
C
The length of each rod increases by the same amount

∴ △ l_{a} = △ l_{s} \Rightarrow △ I_{1} α_{a} t = l_{2} α_{s} t

\Rightarrow \frac{l_{2}}{l_{1}} = \frac{α_{a}}{α_{s}} \Rightarrow \frac{l_{2}}{l_{1}} + 1 = \frac{α_{a}}{α_{s}} + 1

\Rightarrow \frac{l_{2} + l_{1}}{l_{1}} = \frac{α_{a} + α_{s}}{α_{s}} \Rightarrow \frac{l_{1}}{l_{1} + l_{2}} = \frac{α_{s}}{α_{a} + α_{s}}

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