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Question: A 2 meter long aluminum pipe at \(27^0C\) is heated until it is 2.0024 m at \(77^0C\). The coefficie...

A 2 meter long aluminum pipe at 270C27^0C is heated until it is 2.0024 m at 770C77^0C. The coefficient of linear expansion of aluminum is:
(A) 12×10−6/0C12 \times {10^{ - 6}}{/^0}C
(B) 24×10−6/0C24 \times {10^{ - 6}}{/^0}C
(C) 6×10−6/0C6 \times {10^{ - 6}}{/^0}C
(D) 20×10−2/0C20 \times {10^{ - 2}}{/^0}C

Explanation

Solution

Hint
Here, we use the formula for linear expansion given by,α=dldT×l\alpha = \dfrac{{dl}}{{dT \times l}}, where dl is the change in length, dT is the change in temperature and l is the actual length of the pipe.

Complete step by step answer
Given that, the actual length of the aluminum pipe, l = 2 m;
Change in length, dl = (2 – 2.0024) m = 0.0024 m.
⇒T1=270C,T2=770C\Rightarrow T_1 = 27^0C, T_2 = 77^0C. So, change in temperature, dT=T2–T1=(77−27)0C=500CdT = T_2 – T_1 = (77-27) ^0C = 50 ^0C.
Substitute the given values in the linear expansion formula given by,
⇒α=dldT×l\Rightarrow \alpha = \dfrac{{dl}}{{dT \times l}}
⇒α=0.0024m500C×2m ⇒α=24×10−6/0C\Rightarrow \alpha = \dfrac{{0.0024m}}{{{{50}^0}C \times 2m}}\\\ \Rightarrow \alpha = 24 \times {10^{ - 6}}{/^0}C
Thus, the coefficient of linear expansion of aluminum is, α=24×10−6/0C\alpha = 24 \times {10^{ - 6}}{/^0}C.
The correct answer is option (B).

Note
Coefficient of linear expansion is the rate of change of unit length per unit degree change in temperature. It is given by,
⇒α=dldT×l\Rightarrow \alpha = \dfrac{{dl}}{{dT \times l}}
Where dll\dfrac{{dl}}{l} is the change of unit length of the given material, and dT is the change in temperature.
One should convert the given value of temperature in 0C scale to Kelvin scale, if the answer is asked in degree Kelvin.