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. A)7.8°C B)6.3°C C)9.2°C D)15°C E)5.7°C

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Quizplus · Accepted Answer

The aluminum rod and copper tube will fit together when their change in radius due to temperature change is such that the aluminum rod's radius decreases to the original radius of the copper tube. The change in length (or radius, in this case) due to temperature change can be calculated using the formula $\Delta L = \alpha L_0 \Delta T$, where $\Delta L$ is the change in length, $\alpha$ is the coefficient of thermal expansion, $L_0$ is the original length, and $\Delta T$ is the change in temperature.For the aluminum rod to just fit into the copper tube, the final radius of the aluminum rod must equal the original radius of the copper tube, which is 1.000100 cm. The difference in radius between the aluminum rod and the copper tube at 20°C is $1.000200 \, \text{cm} - 1.000100 \, \text{cm} = 0.000100 \, \text{cm}$.Let $T_f$ be the final temperature, and $\Delta T = 20°C - T_f$. The change in radius for aluminum ($\Delta R_{Al}$) and copper ($\Delta R_{Cu}$) can be expressed as:$$\Delta R_{Al} = \alpha_{Al} R_{Al_0} \Delta T$$$$\Delta R_{Cu} = \alpha_{Cu} R_{Cu_0} \Delta T$$Since the aluminum rod needs to shrink more than the copper tube to fit, we are interested in the point where the difference in their original radii equals the difference in their changes in radius due to cooling. Thus:$$0.000100 \, \text{cm} = (\alpha_{Al} R_{Al_0} - \alpha_{Cu} R_{Cu_0}) \Delta T$$Given $\alpha_{Al} = 2.4 \times 10^{-5} \, \text{K}^{-1}$ and $\alpha_{Cu} = 1.7 \times 10^{-5} \, \text{K}^{-1}$, and considering the small difference in radii, we can approximate $R_{Al_0} \approx R_{Cu_0} \approx 1.000150 \, \text{cm}$ for the purpose of this calculation. Solving for $\Delta T$ and subsequently for $T_f$ gives us the temperature at which the aluminum rod just fits into the copper tube. Given the specific coefficients of thermal expansion and the need to solve for the exact temperature, the correct answer is determined through calculation rather than directly through the provided information. The calculation involves solving for the temperature that equalizes the expanded/contracted sizes of the aluminum and copper, taking into account their different rates of thermal expansion. The correct answer, after performing the necessary calculations, is found to be 5.7°C, which corresponds to choice E.

You Want to Insert an Aluminum Rod,which at 20°C Has