Question

Review. Inside the wall of a house, an L-shaped section of hot-water pipe consists of three parts: a straight, horizontal piece h = 28.0 cm long; an elbow; and a straight, vertical piece = 134 cm long. A stud and a second story floorboard hold the ends of this section of copper pipe stationary. Find the magnitude and direction of the displacement of the pipe elbow when the water flow is turned on, raising the temperature of the pipe from $$18.0^0C $$ to $$46.5^0C$$.

Answer 1

The horizontal section expands according to $$\Delta L =\alpha L_i\Delta T$$.
$$\Delta x =[17 \times 10^{-6} (^0C)^{-1}](28.0 cm)( 46.5^0C - 18.0^0C)$$
$$=1.36 \times 10^{-2} cm$$
The vertical section expands similarly by
$$\Delta y =[17 \times 10^{-6}(^0C)^{-1}](134 cm)(28.5^0C) =6.49 \times 10^{-2} cm $$
The vector displacement of the pipe elbow has magnitude
$$\Delta r=\sqrt{\Delta x^2 +\Delta y^2}=\sqrt{(0.136 mm)^2 +(0.649 mm)^2} =0.663 mm $$
and is directed to the right below the horizontal at angle
$$\theta =tan^{-1} (\dfrac{\Delta y}{\Delta x}) =tan^{-1}( \dfrac{0.649 mm}{0.136 mm}) =78.2^0$$
$$\Delta r =0.663 mm$$ to the right at $$78.2^0$$ below the horizontal