Question

An iron rod and a brass rod, both at $27{}^{\circ }\text{C}$ differ in length by ${10}^{-3}\text{m}$. The coefficient of linear expansion for iron is $1.1\times {10}^{-5}{}^{\circ }{\text{C}}^{-1}$ and for brass is $1.9\times {10}^{-5}{}^{\circ }{\text{C}}^{-1}$. The temperature at which both of these rods will have the same length is:

• A. $0{}^{\circ }\text{C}$
• B. $152{}^{\circ }\text{C}$
• C. $175{}^{\circ }\text{C}$
• D. Data is insufficient

Answer 1

The correct option is A $0{}^{\circ }\text{C}$

The given data is normally insufficient to determine the result. Either the length of one of the rods at $0{}^{\circ }\text{C}$ must be known or one has to assume that both the rods had the same length at $0{}^{\circ }\text{C}$ which is not possible if they have the same length again at another temperature. Suppose the two rods had the same length $L$ at $0{}^{\circ }\text{C}$. Then by the given problem,$L[1+1.9\times {10}^{-5}\times 27]$

$-L[1+1.1\times {10}^{-5}\times 27]$
= ${10}^{-3}$

$L\times 0.8\times 27\times {10}^{-5}$
= ${10}^{-3}$

Or $L$ =
$\frac{{10}^{-3}}{0.8\times 27\times {10}^{-5}}\text{m}$

= $4.63\text{m}$

Hence $0{}^{\circ }\text{C}$ is a possible choice and the rods had equal length of $4.63\text{m}$ at $0{}^{\circ }\text{C}$.