Question

Two rods of length $L_1$ and $L_2$ are welded together to make a composite rod of length $(L_1+L_2)$. If the coefficients of linear expansion of the rod are ${\alpha }_1$ and ${\alpha }_2$ respectively, the effective coefficient of linear expansion of the composite rod will be

• A. $\frac{L_1{\alpha }_1+L_2{\alpha }_2}{6(L_1+L_2)}$
• B. $\frac{L_1{\alpha }_1+L_2{\alpha }_2}{4(L_1+L_2)}$
• C. $\frac{L_1{\alpha }_1+L_2{\alpha }_2}{L_1+L_2}$
• D. $\frac{L_1{\alpha }_1+L_2{\alpha }_2}{2(L_1+L_2)}$

Answer 1

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

Given:
Lengths of the two rods are $L_1$ and $L_2$
Length of composite rod $=L_1+L_2$
Coefficients of linear expansion $={\alpha }_1$ and ${\alpha }_2$ respectively
To find:
Effective coefficient of linear expansion of composite rod, $\alpha =?$
Let us assume:
Change in temperature $=\Delta T$
We know that, for end to end connection of rod,
effective expansion $=$ expansion in rod $1$ $+$ expansion in rod $2$
$\Rightarrow (L_1+L_2)\alpha \Delta T=L_1{\alpha }_1\Delta T+L_2{\alpha }_2\Delta T$
$\Rightarrow \alpha =\frac{L_1{\alpha }_1+L_2{\alpha }_2}{L_1+L_2}$
is the effective coefficient of linear expansion.