Question

Two metal rods of length, $L_1$ and $L_2$ and co-efficients of linear expansion ${\alpha }_1$ and ${\alpha }_2$ respectively are welded together to make a composite rod of length $(L_1+L_2)$ at $0^{\circ }C$. Find the effective co-efficient of linear expansion of the composite rod.

• A. $\frac{L_1{\alpha }_1^2-L_2{\alpha }_2^2}{L_1^2+L_2^2}$
• B. $\frac{L_1^2{\alpha }_1{L}_2^2{\alpha }_2}{L_1^2+L_2^2}$
• C. $\frac{L_1{\alpha }_1+L_2{\alpha }_2}{L_1-L_2}$
• D. $\frac{L_1{\alpha }_1+L_2{\alpha }_2}{L_1+L_2}$

Answer 1

The correct option is D $\frac{L_1{\alpha }_1+L_2{\alpha }_2}{L_1+L_2}$

Change in length of rod for a change of temperature by $\Delta T$ is given by :

$L_1{\alpha }_1\Delta T+L_2{\alpha }_2\Delta T=(L_1+L_2){\alpha }_{eff}\Delta T$

$⟹{\alpha }_{eff}=\frac{L_1{\alpha }_1+L_2{\alpha }_2}{L_1+L_2}$