Question

A iron rod of length 50 cm is joined at an end to an aluminum rod of length 100 cm. All measurements refer to ${20}^{\circ }C$. The coefficients of linear expansion of iron and aluminum are $12\times {10}^{-6}{/}^{\circ }C$ and $24\times {10}^{-6}{/}^{\circ }C$ respectively. The average coefficient of composite system is

• A. $36\times {10}^{-6}{/}^{\circ }C$
• B. $12\times {10}^{-6}{/}^{\circ }C$
• C. $20\times {10}^{-6}{/}^{\circ }C$
• D. $48\times {10}^{-6}{/}^{\circ }C$

Answer 1

The correct option is C $20\times {10}^{-6}{/}^{\circ }C$

Initially (at ${20}^{\circ }$C) length of composite system L = 50 + 100 = 150 cm
Length of iron rod at ${100}^{\circ }$C $=50[1+12\times {10}^{-6}\times (100-20\left)\right]=50.048cm$
Length of aluminum rod at 100${}^{\circ }$C $=100[1+24\times {10}^{-6}\times (100-20\left)\right]=100.192cm$
Finally (at ${100}^{\circ }$C) length of composite system L' = $50.048+100.192=150.24cm$
Change in length of the composite system $\Delta$L = L' - L = 150.24 -150 = 0.24 cm
$\therefore$ Average coefficient of expansion at ${100}^{\circ }$C $\alpha =\frac{\Delta L}{L\times \Delta T}$= $\frac{0.24}{150\times (100-20)}$ = $20\times {10}^{-6}{/}^{\circ }C$