Thermal expansion: You want to insert an aluminum rod, which at 20°C has a radius of 1.000200 cm into a copper tube which has a radius of 1.000100 cm at the same temperature. You decide to put both of them in the refrigerator. At what temperature will the rod just fit if both are cooled to the same temperature? The coefficient of thermal expansion for aluminum is 2.4 × 10^-5K^-1, and that of copper is 1.7 × 10^-5 K^-1.

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Quizplus · Accepted Answer

To solve this problem, we use the formula for linear expansion, $\Delta L = L_0 \alpha \Delta T$, where $\Delta L$ is the change in length (or radius, in this case), $L_0$ is the original length (or radius), $\alpha$ is the coefficient of thermal expansion, and $\Delta T$ is the change in temperature. We need the aluminum rod to shrink enough to fit into the copper tube, so their final radii must be equal.Given:- Initial radius of aluminum rod, $R_{Al,0} = 1.000200$ cm- Initial radius of copper tube, $R_{Cu,0} = 1.000100$ cm- Coefficient of thermal expansion for aluminum, $\alpha_{Al} = 2.4 \times 10^{-5} / ^\circ C$- Coefficient of thermal expansion for copper, $\alpha_{Cu} = 1.7 \times 10^{-5} / ^\circ C$- Initial temperature, $T_0 = 20^\circ C$We want the final radii to be equal, so:$$R_{Al,0} + \Delta R_{Al} = R_{Cu,0} + \Delta R_{Cu}$$Substituting the linear expansion formula for $\Delta R$:$$R_{Al,0} + R_{Al,0} \alpha_{Al} \Delta T = R_{Cu,0} + R_{Cu,0} \alpha_{Cu} \Delta T$$Given that $\Delta T = T_{final} - T_0$, we can solve for $T_{final}$ to find the temperature at which the rod just fits into the tube. Rearranging and solving this equation with the given values leads to the correct temperature, which is 5.7°C, making the correct choice E.

Thermal Expansion: You Want to Insert an Aluminum Rod, Which