You need a 50 cm long metal rod that has a linear expansion coefficient of 20 x
ID: 1772506 • Letter: Y
Question
You need a 50 cm long metal rod that has a linear expansion coefficient of 20 x 10-6 / °C for a device you are building, but you don’t have access to any metals that have this linear expansion coefficient. However, you do have a Copper rod ( = 15 x 10-6 / °C) and an Aluminum rod ( = 23 x 10-6 / °C) laying around. You realize that you can make a 50 cm long rod that has = 20 x 10-6 / °C by connecting the Copper and Aluminum rods together lengthwise. How long a piece of each Copper and Aluminum rod must you use?
Explanation / Answer
change in the length of the 50 cm rod that has a linear expansion coefficient of 20*10^-6/oc =
the change in length of the copper - the change in length of the aluminium
the change in length of the copper = Lc*alpha_c*dT
the change in length of the aluminiumr = Lal*alpha_al*dT
change in the length of the 50 cm rod = L*alpha*dT
given L = Lc + Lal
L*alpha*dT = Lc*alpha_c*dT - Lal*alpha_al*dT
L*alpha = Lc*alpha_c - Lal*alpha*al
L*alpha = Lac*alpha_c - (L-Lc)*alpha_al
50*20*10^-7 = (Lc*15*10^-6) - ((50-Lc)*23*10^-6)
length of aluminium Lc = 33 cm
length of aluminium Lal = 50 - 33 = 17 cm <<<----ANSWER
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