Question

Two uniform rods $AB$ and $BC$ have Young's modulii $1.2\times {10}^{11}{\text{N/m}}^2$ and $1.5\times {10}^{11}{\text{N/m}}^2$ respectively. If coefficient of linear expansion of $AB$ is $1.5\times {10}^{-5}/{}^{\circ }\text{C}$ and both have equal area of cross section, then coefficient of linear expansion of $BC$, for which there is no shift of the junction at all temperatures, is

• A. $1.5\times {10}^{-5}/{}^{\circ }\text{C}$
• B. $1.2\times {10}^{-5}/{}^{\circ }\text{C}$
• C. $0.6\times {10}^{-5}/{}^{\circ }\text{C}$
• D. $0.8\times {10}^{-5}/{}^{\circ }\text{C}$

Answer 1

The correct option is B $1.2\times {10}^{-5}/{}^{\circ }\text{C}$

Given:
For rod $AB$,
Coefficient of thermal expansion, ${\alpha }_{AB}=1.5\times {10}^{-5}/{}^{\circ }\text{C}$
Young's modulus, $Y_{AB}=1.2\times {10}^{11}{\text{N/m}}^2$
For rod $BC$,
Young's modulus $Y_{BC}=1.5\times {10}^{11}{\text{N/m}}^2$
Coefficient of thermal expansion, ${\alpha }_{BC}=?$


For no shift of the junction at all temperatures, thermal strain should be countered by the compressive strain in the rods.
$\frac{\Delta L}{L}\left(\text{thermal}\right)=\frac{\Delta L}{L}\left(\text{compressive}\right)$
$\Rightarrow \frac{\alpha L\Delta T}{L}=\frac{FL}{AYL}$
$\Rightarrow \alpha \Delta T=\frac{F}{AY}$
$\Rightarrow \alpha \propto \frac{1}{Y}$
[ Temperature difference, compressive force and area of cross section are constant and equal for both rods]
So, $\frac{{\alpha }_{AB}}{{\alpha }_{BC}}=\frac{Y_{BC}}{Y_{AB}}$
Substituting values, $\frac{1.5\times {10}^{-5}}{{\alpha }_{BC}}=\frac{1.5\times {10}^{11}}{1.2\times {10}^{11}}$
$\Rightarrow {\alpha }_{BC}=1.2\times {10}^{-5}/{}^{\circ }\text{C}$