Question

Two rods, one of aluminum and the other made of steel, having initial lengths $l_1$ and $l_2$ are connected together to form a single rod of length $l_1+l_2$. The co-efficients of linear expansion for aluminum and steel are ${\alpha }_a$ and ${\alpha }_s$ respectively. If the length of each rod increases by the same amount when their temperature are raised by $t^{\circ }C$, then find the ratio $\frac{l_1}{(l_1+l_2)}$

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

By co-efficient of linear expainsion,
$\alpha =\frac{\Delta L}{L\times \Delta T},Where\Delta L=changeinlength\therefore \Delta L=\alpha L\Delta T\therefore \Delta L_a={\alpha }_a{l}_1\Delta Tand\Delta L_s={\alpha }_s{l}_2\Delta T\because \Delta L_a=\Delta L_{s}or{\alpha }_a{l}_1\Delta T={\alpha }_s{l}_2\Delta Tor\frac{l_2}{l_1}=\frac{{\alpha }_a}{{\alpha }_s}\Rightarrow \frac{l_2}{l_1}+1=\frac{{\alpha }_a}{{\alpha }_s}+1or\frac{l_2+l_1}{l_1}=\frac{{\alpha }_a+{\alpha }_s}{{\alpha }_s}\therefore \frac{l_1}{l_1+l_2}=\frac{{\alpha }_a}{{\alpha }_a+{\alpha }_s}$