Question

A metal rod $A$ of $25cm$ lengths expands by $0.050cm$. When its temperature is raised from $0^{\circ }C$ to ${100}^{\circ }C$. Another rod $B$ of a different metal of length $40cm$ expands by $0.040cm$ for the same rise in temperature. A third rod $C$ of $50cm$ length is made up of pieces of rods $A$ and $B$ placed end to end expands by $0.03cm$ on heating from $0^{\circ }C$ to ${50}^{\circ }C$. Find the lengths of each portion of the composite rod.

Answer 1

From the equation of linear thermal expansion,

$\Delta L=L\alpha \Delta T$

$0.05=25{\alpha }_1\left(100\right)$

${\alpha }_1=2X{10}^{-5}cm{/}^{0}C$

$0.04=40{\alpha }_2\left(100\right)$

${\alpha }_2={10}^{-5}cm{/}^{0}C$

Thus, for the combined rod,

$0.03=x\left(2X{10}^{-5}\right)\left(50\right)+(50-x)\left({10}^{-5}\right)\left(50\right)$

$50+x=60$

$x=10cm$, $50-x=40cm$

Answer is 10cm and 40cm.