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Chapter 13: Problem 6

A 2.4 -m length of copper pipe extends directly from a hot-water heater in a basement to a faucet on the first floor of a house. If the faucet isn't fixed in place, how much will it rise when the pipe is heated from $20.0^{\circ} \mathrm{C}\( to \)90.0^{\circ} \mathrm{C} .$ Ignore any increase in the size of the faucet itself or of the water heater.

Short Answer

Expert verified
Answer: The height increase of the copper pipe is approximately 2.87 mm.

Step by step solution

01

Identify the given values

The given values in this problem are: - The initial length of the copper pipe, \(L_0 = 2.4\,\text{m}\) - The initial temperature, \(T_0 = 20.0^{\circ} \text{C}\) - The final temperature, \(T_f = 90.0^{\circ} \text{C}\) - The coefficient of linear expansion for copper, \(\alpha = 17 \times 10^{-6}\,\text{C}^{-1}\)
02

Calculate the change in temperature

The change in temperature can be calculated using the formula: $$ \Delta T = T_f - T_0 $$ Substituting the given values, we get: $$ \Delta T = 90.0^{\circ} \text{C} - 20.0^{\circ} \text{C} = 70.0^{\circ} \text{C} $$
03

Find the change in length

Now, we can use the linear expansion formula to find the change in length of the copper pipe. The formula is: $$ \Delta L = \alpha L_0 \Delta T $$ Substitute the given values: $$ \Delta L = (17 \times 10^{-6}\,\text{C}^{-1})(2.4\,\text{m})(70.0^{\circ} \text{C}) $$ Calculate the change in length: $$ \Delta L \approx 2.87 \times 10^{-3}\,\text{m} $$
04

Find the height increase

The pipe will expand in all directions, but we are interested in the upward (vertical) expansion. Since we want to find the height increase, we can use the fact that the pipe is in a straight line to assume that the vertical and horizontal expansions are equal. Thus, the height increase is equal to the change in length: $$ \Delta H = \Delta L \approx 2.87 \times 10^{-3}\,\text{m} $$ So, the faucet will rise by approximately \(2.87\,\text{mm}\) when the pipe is heated from \(20.0^{\circ} \text{C}\) to \(90.0^{\circ} \text{C}\).

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