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I have rubber as having a negative coefficient of thermal expansion when it is under tension and that value being -43 x 10^-6/deg K
I have rubber as having a negative coefficient of thermal expansion when it is under tension and that value being -43 x 10^-6/deg K. A rubberband will shrink when heated if it is under tension.
The problem has a large rubberband attached to a ceiling, a wire of aluminum of unknown cm. length attached to the hanging end of the rubberband. A mass of 100 grams is then hung from the lower end of the aluminum wire. The rubberband is stretched 10 cm. long at this point. How do I figure out the length of the aluminum wire such that the mass remains in the same position with respect to the floor of the room, no matter what the room temperature is?
Expert Solution
The positive expansion of aluminum must compensate for the contraction of the rubber.
Aluminum linear thermal expansion coefficient is bAl = 0.000023 /K (taken from page:
http://en.wikipedia.org/wiki/Coefficient_of_thermal_expansion).
If aluminum wire length is L_Al, it will expand by
bAl* L_Al
for every degree of K of temperature rise
The rubber under tension has length L_r = 10 cm, and its coefficient of thermal expansion is
b_r = -0.000043 /K, and the change of its length for every degree of K of temperature rise is
b_r * L_r
We want the sum of the changes of length of the rubber and the aluminum wire when temperature grows by 1K to be 0:
bAl* L_Al + b_r * L_r = 0
From this we find the needed length of the aluminum wire:
L_Al = - b_r * L_r / bAl = - (-0.000043 /K * 10 cm) / (0.000023 /K) = 18.7 cm
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