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Two rods, one of aluminum 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 aluminum and steel are ${\alpha _a}$ and ${\alpha _s}$ 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 $\frac{{{l_1}}}{{({l_1} + {l_2})}}$
$\frac{{{\alpha _s}}}{{{\alpha _a}}}$
$\frac{{{\alpha _a}}}{{{\alpha _s}}}$
$\frac{{{\alpha _s}}}{{({\alpha _a} + {\alpha _s})}}$
$\frac{{{\alpha _a}}}{{({\alpha _a} + {\alpha _s})}}$
Solution
(c) Given $\Delta {l_1} = \Delta {l_2}$ or ${l_1}{\alpha _a}t = {l_2}{\alpha _s}t$
$\therefore $ $\frac{{{l_1}}}{{{l_2}}} = \frac{{{\alpha _s}}}{{{\alpha _a}}}$ or $\frac{{{l_1}}}{{{l_1} + {l_2}}} = \frac{{{\alpha _s}}}{{{\alpha _a} + {\alpha _s}}}$.
