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Two rods, one made of aluminium and the other made of steel, having initial lengths $l_1$ and $l_2$ respectively are connected together to form a single rod of length $(l_1 + l_2)$. The coefficients of linear expansion for aluminium and steel of $\alpha_1$ and $\alpha_2$ respectively. If the length of each rod increases by the same amount when their temperature is raised by $t^oC$, then the ratio $l_1/(l_1 + l_2)$ :-
$\frac{\alpha_1}{\alpha_2}$
$\frac{\alpha_2}{\alpha_1}$
$\frac{\alpha_2}{(\alpha_1+\alpha_2)}$
$\frac{\alpha_1}{(\alpha_1+\alpha_2)}$
Solution
$\mathrm{L}_{\mathrm{t}}=\mathrm{L}_{0}(1+\alpha \mathrm{t})$
$\Delta \mathrm{L}=\mathrm{L}_{\mathrm{t}}-\mathrm{L}_{0}=\mathrm{L}_{0} \alpha \mathrm{t}$
If the length of each rod increases by same amount
on raising the temperature to $t,$ then
$l_{1} \alpha_{1} t=l_{2} \alpha_{2} t$
or $\quad l_{1} \alpha_{1}=l_{2} \alpha_{2}$
or $\quad \frac{l_{2}}{l_{1}}=\frac{\alpha_{1}}{\alpha_{2}}$
$1+\frac{l_{2}}{l_{1}}=1+\frac{\alpha_{1}}{\alpha_{2}}$
$\frac{l_{1}+l_{2}}{l_{1}}=\frac{\alpha_{1}+\alpha_{2}}{\alpha_{2}}$ or $\frac{l_{1}}{l_{1}+l_{2}}=\frac{\alpha_{2}}{\alpha_{1}+\alpha_{2}}$
