Question

Two rods, one of aluminium and the other made of steel, having initial lengths ${\text{l}}_1$ and ${\text{l}}_2$ are connected together to form a single rod of length $3\text{cm}$. If the length of each rod increases by the same amount when the temperature is raised by a certain value, and if $22\times {10}^{-6}{}^{\circ }{\text{C}}^{-1}$ and $11\times {10}^{-6}{}^{\circ }{\text{C}}^{-1}$ are the coefficients of linear expansion of aluminium and steel, then:

• A. ${\text{l}}_1=2\text{cm},{\text{l}}_2=1\text{cm}$
• B. ${\text{l}}_1=0.5\text{cm},{\text{l}}_2=2.5\text{cm}$
• C. ${\text{l}}_1=1\text{cm},{\text{l}}_2=2\text{cm}$
• D. ${\text{l}}_1=2.5\text{cm},{\text{l}}_2=0.5\text{cm}$

Answer 1

The correct option is C ${\text{l}}_1=1\text{cm},{\text{l}}_2=2\text{cm}$

Given:
The coefficient of linear expansion of aluminium, ${\alpha }_{\text{A}}=22\times {10}^{-6}{}^{\circ }{\text{C}}^{-1}$
The coefficient of linear expansion of steel, ${\alpha }_{\text{S}}=11\times {10}^{-6}{}^{\circ }{\text{C}}^{-1}$
The total length, ${\text{l}}_1+{\text{l}}_2=3\text{cm}$
when the temperature is raised by a certain value, say $\text{t}{}^{\circ }\text{C}$, and if ${\text{l}}_1^{\text{*}}$ and ${\text{l}}_2^{\text{*}}$ are the new lengths of the rods are respectively, then:
${\text{l}}_1^{\text{*}}={\text{l}}_1(1+{\alpha }_{\text{A}}\text{t})---\left(1\right)$
${\text{l}}_2^{\text{*}}={\text{l}}_2(1+{\alpha }_{\text{S}}\text{t})---\left(2\right)$
Also, both the rods increase by the same amount.
${\text{l}}_1^{\text{*}}-{\text{l}}_1={\text{l}}_2^{\text{*}}-{\text{l}}_2---\left(3\right)$
From (1), (2) and (3), we get ${\text{l}}_1{\alpha }_{\text{A}}={\text{l}}_2{\alpha }_{\text{S}}$
$⟹\frac{{\text{l}}_1}{{\text{l}}_2}=\frac{{\alpha }_{\text{S}}}{{\alpha }_{\text{A}}}=\frac{11\times {10}^{-6}}{22\times {10}^{-6}}=0.5$
$⟹{\text{l}}_1=0.5{\text{l}}_2$
We know that ${\text{l}}_1+{\text{l}}_2=3\text{cm}$
$⟹1.5{\text{l}}_2=3$
$⟹{\text{l}}_2=\frac{3}{1.5}=2\text{cm}$
and ${\text{l}}_1=1\text{cm}$