Question

A $30.0\, cm$ long metal rod expands by $0.0650\, cm$ when its temperature is raised from $0^{\circ} C$ to $100^{\circ} C$. A second rod of different metal and of the same length expands by $0.0350\, cm$ for the same rise in temperature. A third composite rod, also $30.0\, cm$ long, is made up of pieces of each of the above metals placed end to end and expands by $0.0580\, cm$ when temperature is increased from $0^{\circ} C$ to $100^{\circ} C$. Find the length (in $cm$ ) of the smaller portion of the composite rod at $0^{\circ} C$.

Answer 1

Using $\alpha=\frac{\Delta L}{L_{0} \cdot \Delta T}$, we have
$\alpha_{1}=\frac{0.0650}{30 \times 100},$
$\alpha_{2}=\frac{0.0350}{30 \times 100}$
$L_{01} \cdot \alpha_{1} \cdot \Delta T+L_{02} \cdot \alpha_{2} \cdot \Delta T=0.0580$
and $L_{01}+L_{02}=30$
Solving, $L_{01}=23\, cm$ and $L_{02}=7\, cm$.