Question

An aluminium rod (length ${l_1}$ and coefficient of linear expansion ${\alpha _A}$) and steel rod (length ${l_2}$ and coefficient of linear expansion ${\alpha _B}$) are joined together. If the length of each rod increased by the same amount when their temperature is raised by ${t^ \circ }\,C$, then $\dfrac{{{l_1}}}{{\left( {{l_1} + {l_2}} \right)}}$ is:
(A) $\dfrac{{{\alpha _A}}}{{{\alpha _B}}}$
(B) $\dfrac{{{\alpha _B}}}{{{\alpha _A}}}$
(C) $\dfrac{{{\alpha _B}}}{{{\alpha _A} + {\alpha _B}}}$
(D) $\dfrac{{{\alpha _A}}}{{{\alpha _A} + {\alpha _B}}}$

Answer 1

In this problem it is given that the two rods of different materials undergo the thermal expansion process, then the length of the both rods of different materials are increased. The relation between their length and coefficient of linear expansion is given by using the linear expansion equation.

Complete step-by-step solution:
The two rods of different material are joined together and undergoes the thermal expansion process, if the length of each rod of the different material is increased by same length, then the relation is given as,
${l_1}{\alpha _A} = {l_2}{\alpha _B}\,............\left( 1 \right)$
By taking the length ratio in the above equation (1), then the above equation (1) is written as,
$\dfrac{{{l_2}}}{{{l_1}}} = \dfrac{{{\alpha _A}}}{{{\alpha _B}}}\,...............\left( 2 \right)$
By adding the term $1$ on both sides for further calculation, then the above equation is written as,
$\dfrac{{{l_2}}}{{{l_1}}} + 1 = \dfrac{{{\alpha _A}}}{{{\alpha _B}}} + 1$
By cross multiplying the terms on both sides, then the above equation is written as,
$\dfrac{{{l_2} + {l_1}}}{{{l_1}}} = \dfrac{{{\alpha _A} + {\alpha _B}}}{{{\alpha _B}}}$
Taking reciprocal on both sides, then the above equation is written as,
$\dfrac{{{l_1}}}{{{l_2} + {l_1}}} = \dfrac{{{\alpha _B}}}{{{\alpha _A} + {\alpha _B}}}$
Thus, the above equation shows the relation between the length and the coefficient of linear expansion when each rod is increased by the same length.
Hence, the option (C) is the correct answer.

Note:- After the equation (2), the extra term $1$ is added with the ratios of length and coefficient of linear expansions on both sides, it is added here for the further calculations. And after the cross-multiplication step, the reciprocal is done to give the exact solution.