Question

The lengths of steel and copper rods are so that the length of the steel rod is 5 cm longer than that of the copper rod at all temperatures, then length of each rod, are ( $\alpha$ for copper = $1.7\times {{10}^{-5}}$ per $^{\circ }C$ and $\alpha$ for steel=$1.1\times {{10}^{-5}}$ per ${}^{{}^{\circ }}C$)
(A) 9.17 cm; 14.17 cm
(B) 9.02 cm; 14.20 cm
(C) 9.08 cm; 14.08 cm
(D) 9.50 cm; 14.50 cm

Answer 1

The concept of thermal expansion is applied for solving this problem. When heat is applied on 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:
$\begin{aligned} & \Delta l=l\times \alpha \times \Delta T \\\ & \\\ \end{aligned}$

Complete answer:
Given,
${{\alpha }_{Cu}}=1.7\times {{10}^{-5}}$ Per ${}^{\circ }C$
${{\alpha }_{steel}}=1.1\times {{10}^{-5}}$ Per ${}^{\circ }C$
${{l}_{s}}-{{l}_{cu}}=5$ ………….. (1)
$\Delta {{l}_{s}}=\Delta {{l}_{Cu}}$
$\begin{aligned} & \Rightarrow {{l}_{s}}\times {{\alpha }_{s}}\times \Delta T={{l}_{Cu}}\times \alpha \times \Delta T \\\ & \Rightarrow {{l}_{s}}\times 1.1\times {{10}^{-5}}={{l}_{Cu}}\times 1.7\times {{10}^{-5}} \\\ & \Rightarrow \dfrac{{{l}_{s}}}{{{l}_{Cu}}}=\dfrac{1.7}{1.1} \\\ & \Rightarrow \dfrac{{{l}_{s}}}{{{l}_{Cu}}}=1.545 \\\ \end{aligned}$ …….. (2)
Solving equation (1) and (2), we have
${{l}_{s}}=14.17cm$
$\begin{aligned} & {{l}_{Cu}}=14.17-5 \\\ & \Rightarrow {{l}_{Cu}}=9.17cm \\\ \end{aligned}$
Thus the values of length of steel and copper rods are 14.17 and 9.17 cm respectively.

So, the correct answer is “Option A”.

Additional Information:
The coefficient of thermal expansion tells us how the size an object changes with changes in temperature. In other words, it gives us the value of fractional change in the size of an object per degree change in temperature.Gases and liquids expand substantially on heating but the behaviour shown by solids is also very important to observe since it plays a great role in designing the structure of machines and buildings. The larger the coefficient, the more it will expand per degree change in temperature. Thermal expansion takes place in one dimension, two dimensional and three dimensions.

Note:
We must keep in mind that the coefficient of thermal expansion changes with change in material. 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 ($\gamma$ ) are used.