Question

If two rods of length \[L\] and \[2L\] having coefficients of linear expansion \[\alpha \] and \[2\alpha \] respectively are connected so that total length becomes \[3L\], the average coefficient of linear expansion of the composition rod equals:

A. \[\dfrac{3}{2}\alpha \]
B. \[\dfrac{5}{2}\alpha \]
C. \[\dfrac{5}{3}\alpha \]
D. None of these

Answer 1

Hint: Here first we will find the linear thermal expansion for the 1st and 2nd rods. Then we will find the linear thermal expansion of the composition rod and hence obtains the coefficient of linear expansion of the composition rod.
Formula used: \[\Delta L = \alpha L\Delta T\]

Complete Step-by-Step solution:
Given length of 1st rod = \[L\]
Coefficient of linear expansion of 1st rod = \[\alpha \]
We know the mathematical expression for linear thermal expansion is \[\Delta L = \alpha L\Delta T\] where \[\Delta L\] is the change in length , \[L\] is the initial length of the rod, \[\alpha \] is the coefficient of linear expansion and \[\Delta T\] is the change in temperature.
So, linear thermal expansion for 1st rod is \[{\left( {\Delta L} \right)_1} = \alpha L\Delta T\]
Given length of 2nd rod = \[2L\]
Coefficient of linear expansion of 2nd rod = \[2\alpha \]
So, linear thermal expansion of 2nd rod is \[{\left( {\Delta L} \right)_2} = 2\alpha 2L\Delta T\]
Now these two rods are connected to form a rod of length \[3L\] and let \[{\alpha _{{\text{avg}}}}\] be the coefficient of linear expansion. So, the linear thermal expansion of the resulted rod is given by \[{\left( {\Delta L} \right)_{{\text{avg}}}} = {\alpha _{{\text{avg}}}}3L\Delta T\].
As the resulted rod is combined by 1st and 2nd rods we have \[{\left( {\Delta L} \right)_{{\text{avg}}}} = {\left( {\Delta L} \right)_1} + {\left( {\Delta L} \right)_2}\]
\[
   \Rightarrow {\left( {\Delta L} \right)_{{\text{avg}}}} = {\left( {\Delta L} \right)_1} + {\left( {\Delta L} \right)_2} \\
   \Rightarrow {\alpha _{{\text{avg}}}}3L\Delta T = \alpha L\Delta T + 2\alpha 2L\Delta T \\
   \Rightarrow 3{\alpha _{{\text{avg}}}}L\Delta T = \alpha L\Delta T + 4\alpha L\Delta T \\
   \Rightarrow 3{\alpha _{{\text{avg}}}}L\Delta T = 5\alpha L\Delta T \\
\]
Cancelling the common terms, we have
\[
   \Rightarrow 3{\alpha _{{\text{avg}}}} = 5\alpha \\
  \therefore {\alpha _{{\text{avg}}}} = \dfrac{{5\alpha }}{3} \\
\]
Hence the average coefficient of linear expansion of the composition rod equals to \[\dfrac{{5\alpha }}{3}\]
Thus, the correct option is C. \[\dfrac{5}{3}\alpha \]

Note: Thermal expansion is the tendency of matter to change its shape, area, and volume in response to a change in temperature. Linear expansion means change in one dimension (length) as opposed to change in volume (volumetric expansion).