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

Two rods, one of aluminium 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 coefficients of linear expansion for aluminium and steel are ${α}_{a}$ and& ${α}_{s}$ respectively. If the length of each rod increases by the same amount when their temperature is raised by $t ° C$ , then find the ratio $\dfrac{l1}{{l}_{1} + {l}_{2}}$

Options

A $\dfrac{{\alpha}_{s}}{{\alpha}_{a}}$
B $\dfrac{{\alpha}_{a}}{{\alpha}_{s}}$
C $\dfrac{{α}_{s}}{{( \alpha}_{a} + {α}_{s} )}$
D $\dfrac{{\alpha}_{a}}{{( \alpha}_{a} + {\alpha}_{s} )}$

Correct answer

C. $\dfrac{{α}_{s}}{{( \alpha}_{a} + {α}_{s} )}$

Step-by-step solution

Given,

\Delta {l}_{1} = \Delta {l}_{2}

or,

{l}_{1}{α}_{a}t = {l}_{2}{α}_{s}t

∴ \dfrac{l1}{l2} = \dfrac{\alpha\text{s}}{\alphaa}

or, &

\dfrac{l1}{{l}_{1} + {l}_{2}} = \dfrac{\alpha\text{s}}{{α}_{a} + {α}_{s}}

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