Question

If two rods of lengths \[L\] and \[2L\] having coefficient of linear expansion \[\alpha \] and \[2\alpha \] respectively are connected end-on-end, the average coefficient of linear expansion of the composite rod, equals
${\text{A}}{\text{. }}\dfrac{3}{2}\alpha $
${\text{B}}{\text{. }}\dfrac{5}{2}\alpha $
$C.{\text{ }}\dfrac{5}{3}\alpha $
${\text{D}}{\text{.}}$ None of these

Answer 1

Hint: Linear expansion: Linear expansion is defined as the increase in the length of the solid material on heating.
Volume expansion: Volume expansion is expressed as the increase in the volume of the solid material on heating.
Area expansion: Area expansion is defined as the increase in surface area of the solid material on heating.

Complete step-by-step answer:
It is given that the details,
Length of one rod \[ = {\text{ }}L\] having a coefficient of linear expansion \[ = \alpha \],
Length of another rod \[ = 2L\] having a coefficient of linear expansion \[ = 2\alpha \]
Initial length of the combination is given as, ${\text{L + 2L = 3L}}$
Increment in the first rod is calculated as, ${\text{L}}\alpha \Delta {\text{t}}$
Increment in the second rod is given as, $(2{\text{L)(2}}\alpha {\text{)}}\Delta {\text{t = }}4\alpha {\text{L}}\Delta {\text{t}}$
So, the increment in the combination is the sum of both increments is given by, $({\text{L}}\alpha \Delta {\text{t + 4L}}\alpha \Delta {\text{t }})$
Initial length of combination was \[3L\] so coefficient of combination is calculated as,
$\dfrac{{{\text{increment in combination}}}}{{{\text{initial length}} \times {\text{change in temperature}}}}$
Now, substituting the values we get, $\dfrac{{5{\text{L}}\alpha \Delta {\text{t}}}}{{3{\text{L}}\Delta {\text{t}}}} = \dfrac{{5\alpha }}{3}$

Hence the correct option is C.

Note: The CGS and in the SI system the unit of linear expansion is per kelvin \[\left( {{K^{ - 1}}} \right)\].
When something is heated or cooled, its length changes by an amount proportional to the original length and the change in temperature of the solid material.
The relationship between the area and linear thermal expansion coefficient can be expressed as following
$\alpha {\text{A = 2}}\alpha {\text{L}}$.
The greater the temperature change in the solid material, the more a bimetallic strip will bend.
For most substances under ordinary conditions, there is no preferred direction of change in shape and size of the material, and an increase in temperature will increase the solid’s size by a certain fraction in each dimension.