At 22.0°C an aluminum ring has an inner diameter of 7.000 cm,and a brass rod has a diameter of 7.090 cm.Keeping the brass rod at 22.0°C,which of the following temperatures of the ring will allow the ring to just slip over the brass rod? (α_Al = 2.4 × 10−⁵ /C°,α_brass = 1.9 × 10−⁵/C°)

Question

Quizplus · Accepted Answer

To solve this problem, we need to calculate the temperature at which the expansion of the aluminum ring allows it to just slip over the brass rod. The formula for linear expansion is $\Delta L = \alpha L_0 \Delta T$, where $\Delta L$ is the change in length (or diameter in this case), $\alpha$ is the coefficient of linear expansion, $L_0$ is the original length (or diameter), and $\Delta T$ is the change in temperature.Given:- Initial diameter of aluminum ring, $D_{Al, initial} = 7.000$ cm- Diameter of brass rod, $D_{brass} = 7.090$ cm- Coefficient of linear expansion for aluminum, $\alpha_{Al} = 2.4 \times 10^{-5} /°C$- Coefficient of linear expansion for brass, $\alpha_{brass} = 1.9 \times 10^{-5} /°C$- Initial temperature, $T_{initial} = 22.0°C$We need the aluminum ring to expand to at least 7.090 cm to slip over the brass rod. The brass rod's temperature and size remain constant, so we only consider the expansion of the aluminum ring.The change in diameter needed for the aluminum ring is $D_{change} = D_{brass} - D_{Al, initial} = 7.090 cm - 7.000 cm = 0.090 cm$.Using the linear expansion formula for the aluminum ring, we get:$$0.090 cm = (2.4 \times 10^{-5} /°C) \times 7.000 cm \times \Delta T$$Solving for $\Delta T$ (change in temperature from the initial 22.0°C), we find:$$\Delta T = \frac{0.090 cm}{(2.4 \times 10^{-5} /°C) \times 7.000 cm}$$$$\Delta T \approx 536°C$$However, this is the change in temperature from the initial 22.0°C. To find the final temperature at which the ring can slip over the rod, we add this change to the initial temperature:$$T_{final} = T_{initial} + \Delta T = 22°C + 536°C = 558°C$$Therefore, the correct answer is 558°C, which allows the aluminum ring to just slip over the brass rod.

At 220°C an Aluminum Ring Has an Inner Diameter of 7