Question

Two rods, one of aluminum and other 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 of aluminium and steel are ${\alpha _A}$ and ${\alpha _S}$ respectively. It the length of each rod increases by same amount when the temperature is raised by ${t^0}C$ then find the relation $\dfrac{{{l_1}}}{{{l_1} + {l_2}}}$ .
(A) $\dfrac{{{\alpha _A}}}{{{\alpha _A} + {\alpha _S}}}$
(B) $\dfrac{{{\alpha _S}}}{{{\alpha _A}}}$
(C) $\dfrac{{{\alpha _A}}}{{{\alpha _S}}}$
(D) $\dfrac{{{\alpha _S}}}{{{\alpha _A} + {\alpha _S}}}$

Answer 1

Hint : Aluminum and steel are the alloys and can be used to make utensils or rods. The rod has initial lengths and coefficient of linear expansion. These both are inversely proportional to each other. The length can be increased by the increase in temperature.

Complete Step By Step Answer:
Given that two rods are there. One is made of aluminum and another rod is made up of steel. The aluminum rod has the initial length of ${l_1}$ and the steel rod has the initial length of ${l_2}$ . The aluminium rod has the coefficient of linear expansion of ${\alpha _A}$ and the steel rod has the coefficient of linear expansion of ${\alpha _S}$ .
The lengths and coefficient of linear expansion.
Thus, the relation will be
$\dfrac{{{l_2}}}{{{l_1}}} = \dfrac{{{\alpha _A}}}{{{\alpha _S}}}$
Addition of $1$ on both sides, we will get
$1 + \dfrac{{{l_2}}}{{{l_1}}} = 1 + \dfrac{{{\alpha _A}}}{{{\alpha _S}}}$
By simplification, the relation will be
$\dfrac{{{l_1} + {l_2}}}{{{l_1}}} = \dfrac{{{\alpha _A} + {\alpha _S}}}{{{\alpha _S}}}$
The inverse of the above relation is
$\dfrac{{{l_1}}}{{{l_1} + {l_2}}} = \dfrac{{{\alpha _S}}}{{{\alpha _A} + {\alpha _S}}}$
Thus, $\dfrac{{{l_1}}}{{{l_1} + {l_2}}}$ is equal to the $\dfrac{{{\alpha _S}}}{{{\alpha _A} + {\alpha _S}}}$ .
Option D is the correct one.

Note :
Alloy is a mixture of two or more chemical elements prepared mainly for the resistivity of corrosion. Steel is an alloy of iron and carbon. Stainless steel has a different composition than steel. The rods of alloy or metals expand in only direction along the length can be called as linear expansion.