Question

Two metal rods have coefficients of linear expansion \[1.1 \times {10^{ - 5}}{/^ \circ }C\] and \[1.65 \times {10^{ - 5}}{/^ \circ }C\] respectively. The difference in lengths is \[10cm\] at all temperatures. Their initial lengths must be respectively:
$(A)$ \[40cm\]and \[50cm\]
$(B)$ \[40cm\]and \[30cm\]
$(C)$ \[50cm\] and \[60cm\]
$(D)$ \[30cm\] and \[20cm\]

Answer 1

Hint: The concept of thermal expansion is applied to solving this type of problem. When heat is applied to a solid, especially a metal it results in expansion. The expansion depends on the coefficient of thermal expansion given by \[\alpha \] and the temperature difference. The coefficient of thermal expansion varies from solid to solid.
Formula used:
The increase in length of a metallic rod on the application of temperature is given by:
\[\Delta l = l \times \alpha \times \Delta T\]
Where, \[\Delta l\] is the change in length of the material, \[l\] is the original length of the material, \[\alpha \] is the linear coefficient of expansion, and \[\Delta T\] is the temperature change.

Complete step-by-step solution:
Given:
\[{\alpha _1} = 1.1 \times {10^{ - 5}}{/^ \circ }C\]
\[{\alpha _2} = 1.65 \times {10^{ - 5}}{/^ \circ }C\]
\[{l_1} - {l_2} = 10cm\]
We have already given the information that both rods are expanding by the same amount at the same temperature. So, we can equate a change in the length of each rod.
\[{l_1}{\alpha _1}\Delta T = {l_2}{\alpha _2}\Delta T\]
Since the temperature is the same, we can rewrite this equation as,
${l_1}{\alpha _1} = {l_2}{\alpha _2}$
$\Rightarrow \dfrac{{{l_1}}}{{{l_2}}} = \dfrac{{{\alpha _2}}}{{{\alpha _1}}} $
$\Rightarrow \dfrac{{{l_1}}}{{{l_2}}} = \dfrac{{1.65 \times {{10}^{ - 5}}}}{{1.1 \times {{10}^{ - 5}}}} $
$\Rightarrow \dfrac{{{l_1}}}{{{l_2}}} = 1.5 $
$\Rightarrow{l_1} = 1.5{l_2}$
We know that, the difference in lengths\[{l_1} - {l_2} = 10cm\].
\[{l_1} - {l_2} = 10cm\]
\[ \Rightarrow 1.5{l_2} - {l_2} = 10cm\]
\[ \Rightarrow 0.5{l_2} = 10cm\]
\[ \Rightarrow {l_2} = \dfrac{{10}}{{0.5}} = 20cm\]
And,
\[{l_1} - 20 = 10\]
\[ \Rightarrow {l_1} = 10 + 20 = 30cm\]
Their initial lengths must be \[30cm\] and \[20cm\]. Hence, option D is correct.

Note: We must keep in mind that the coefficient of thermal expansion changes with material changes. The change in dimension of the material can easily be calculated if we know the original dimension of the material and the change in temperature along with the coefficient of thermal expansion. In this case, the coefficient of linear expansion (\[\alpha \]) is considered. Likewise, for area and volume expansions as a result of heat, the coefficient of area expansion (\[\beta \]) and volume expansion \[\left( \gamma \right)\]are used.