A copper rod (α=1.70 ×10^-5 ^∘ C^-1) is 20 cm longer than...

A copper rod $\left(\alpha=1.70 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)$ is $20 \mathrm{~cm}$ longer than an aluminum $\operatorname{rod}\left(\alpha=2.20 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)$ How long should the copper rod be if the difference in their lengths is to be independent of temperature? For their difference in lengths not to change with temperature, $\Delta L$ must be the same for both rods under the same temperature change. That is, or $$ \begin{array}{c} \left(\alpha L_{0} \Delta T\right)_{\text {copper }}=\left(\alpha L_{0} \Delta T\right)_{\text {abminum }} \\ \left(1.70 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right) L_{0} \Delta T=\left(2.20 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)\left(L_{0}-0.20 \mathrm{~m}\right) \Delta T \end{array} $$ where $L_{0}$ is the length of the copper rod, and $\Delta T$ is the same for both rods. Solving for the original length yields $L_{0}=0.88 \mathrm{~m}$.

A copper rod $\left(\alpha=1.70 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)$ is $20 \mathrm{~cm}$ longer than an aluminum $\operatorname{rod}\left(\alpha=2.20 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)$
How long should the copper rod be if the difference in their lengths is to be independent of temperature?
For their difference in lengths not to change with temperature, $\Delta L$ must be the same for both rods under the same temperature change. That is,
or
$$
\begin{array}{c}
\left(\alpha L_{0} \Delta T\right)_{\text {copper }}=\left(\alpha L_{0} \Delta T\right)_{\text {abminum }} \\
\left(1.70 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right) L_{0} \Delta T=\left(2.20 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)\left(L_{0}-0.20 \mathrm{~m}\right) \Delta T
\end{array}
$$
where $L_{0}$ is the length of the copper rod, and $\Delta T$ is the same for both rods. Solving for the original length yields $L_{0}=0.88 \mathrm{~m}$.

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Key Concepts

Linear Thermal Expansion

This concept explains that most solid objects expand in length when the temperature increases. The amount by which an object expands is directly proportional to its original length, the temperature change, and a material-specific constant, making it a linear relationship that can be used to predict dimensional changes under thermal variations.

Coefficient of Linear Expansion

This material property quantifies how much a material expands per unit length for a unit change in temperature. Each material has its own coefficient, which plays a crucial role in analyzing thermal effects in components made from different substances and becomes central when comparing or designing systems with variable thermal responses.

Temperature Independence in Differential Length

This concept pertains to ensuring that the difference in lengths between two objects remains constant regardless of temperature changes. It involves balancing the thermal expansions of different materials so that their relative dimensional differences do not vary with temperature, a key consideration in precision engineering and composite material design.

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