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# A metal rod that is 30.0 cm long expands by 0.0650 cm when its temperature is raised from $0.0^{\circ} C$ to $100.0^{\circ} \mathrm{C}$. A rod of a different metal and of the same length expands by 0.0350 cm for the same rise in temperature. A third rod, also 30.0 cm long, is made up of pieces of each of the above metals placed end to end and expands 0.0580 cm between $0.0^{\circ} C$ and $100.0^{\circ} \mathrm{C}$. Find the length of each portion of the composite rod.

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#### Given

$\text{\textcolor{#4257b2}{Metal One}}$

$l_{o}= 30 \times 10^{-2} \,\,\mathrm{m}$ ,,,, $\delta l = 0.065 \times 10^{-2} \,\,\mathrm{m} ,,,, T_{1} = 0 \,\,\mathrm{C^{o}} ,,,, T_{2} = 100 \,\,\mathrm{C^{o}}$

$\text{\textcolor{#c34632}{Metal Two}}$

$\delta T = 0.035 \times 10^{-2} \,\,\mathrm{m} ,,,, l_{o} = 30 \times 10^{-2} \,\,\mathrm{m}$

Composite Rod

$l_{o} = 30 \times 10^{-2} \,\,\mathrm{m} ,,,, \delta T = 0.058 \times 10^{-2} \,\,\mathrm{m}$

#### Solution

For $\text{\textcolor{#4257b2}{Metal One}}$

$\delta T = \alpha \times l_{o} \delta T$

The missed parameter here is is $\alpha$ so by solving for $\alpha$

\begin{align*} \alpha_{1} & = \dfrac{\delta l}{l_{o} \times \delta T}\\ &= \dfrac{0.065 \times 10^{-2} \,\,\mathrm{m}}{30 \times 10^{-2} \,\,\mathrm{m} [100 - 0]}\\ &= 2.167 \times 10^{-5} \,\,\mathrm{(C^{o})^{-1}} \end{align*}

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