The coefficient of linear expansion of iron is 0.000011 /k. An iron rod is 10 m long at 27°C. The length of the rod will be decreased by 1.1 mm when the temperature of the rod changes to ?
1 Answer
#T_f ~~ 17^@ "C"# to decrease the length by#0.011%# .
The coefficient of thermal expansion is defined as
#alpha = 1/V((delV)/(delT))_P#
For linear expansion, it is just:
#alpha_L = 1/L ((delL)/(delT))_P# where
#L# is the length and#T# is temperature.#((delL)/(delT))_P# is the change in length due to change in temperature at constant pressure.
For solids, we can neglect the pressure dependence, and also assume that if the temperature was increased linearly,
#alpha_L ~~ 1/L ((DeltaL)/(DeltaT))#
So now we seek the final temperature.
#DeltaT = (DeltaL)/(alpha_LL) = T_f - T_i#
#= (8.9 cancel"mm" - 10 cancel"mm")/("0.000011 K"^(-1) cdot 10000 cancel"mm")#
#= -"10 K"#
Therefore:
#color(blue)(T_f) = T_i + DeltaT#
#= 27^@ "C" + (-10^@ "C")#
#= color(blue)(17^@ "C")#
