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Chapter 13: Problem 21

A copper washer is to be fit in place over a steel bolt. Both pieces of metal are at \(20.0^{\circ} \mathrm{C} .\) If the diameter of the bolt is $1.0000 \mathrm{cm}\( and the inner diameter of the washer is \)0.9980 \mathrm{cm},$ to what temperature must the washer be raised so it will fit over the bolt? Only the copper washer is heated.

Short Answer

Expert verified
Answer: The copper washer must be raised to approximately \(192.3^{\circ} \mathrm{C}\) to fit over the steel bolt.

Step by step solution

01

Write down the formula for linear expansion

The formula for linear expansion is: \(\Delta L = L_0 \alpha \Delta T\) where: - \(\Delta L\) is the change in length (diameter) of the object, - \(L_0\) is the initial length (initial diameter) of the object, - \(\alpha\) is the coefficient of linear expansion for the material, - and \(\Delta T\) is the change in temperature.
02

Find the coefficients of thermal expansion for copper and steel

The coefficients of linear thermal expansion for copper and steel are: Copper: \(\alpha_{Cu} = 17 \times 10^{-6} \mathrm{K^{-1}}\) Steel: \(\alpha_{Steel} = 11 \times 10^{-6} \mathrm{K^{-1}}\)
03

Calculate the change in diameter required for the washer to fit over the bolt

From the problem statement, we know the inner diameter of the washer \(L_0 = 0.9980\ \mathrm{cm}\) and the diameter of the bolt \(D_{bolt} = 1.0000\ \mathrm{cm}\). To fit over the bolt, the washer needs to expand by: \(\Delta L_{req} = D_{bolt} - L_0 = 1.0000\ \mathrm{cm} - 0.9980\ \mathrm{cm} = 0.0020\ \mathrm{cm}\)
04

Calculate the change in temperature needed for the washer to expand enough to fit over the bolt

Using the formula for linear expansion, we now solve for the change in temperature \(\Delta T\): \(\Delta T = \frac{\Delta L_{req}}{L_0 \alpha_{Cu}} = \frac{0.0020\ \mathrm{cm}}{0.9980\ \mathrm{cm} \times 17 \times 10^{-6}\ \mathrm{K^{-1}}}\) \(\Delta T \approx 172.3 \ \mathrm{K}\)
05

Calculate the final temperature of the copper washer

Now, we add the change in temperature to the initial temperature to find the temperature at which the washer will fit over the bolt: \(T_{final} = T_{initial} + \Delta T = 20.0^{\circ} \mathrm{C} + 172.3 \ \mathrm{K}\) Convert temperature from Kelvin to Celsius: \(T_{final} \approx 192.3^{\circ} \mathrm{C}\) Therefore, the copper washer must be raised to approximately \(192.3^{\circ} \mathrm{C}\) to fit over the steel bolt.

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