Question

If two rods of length L and 2L having coefficients of linear expansion α and 2α 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

To find the linear coefficient of expansion for the rod made from the two rods, we first need to know the linear expansions in each rod, where they’re raised to the same temperature difference. We can then add these to get the linear expansion in the composition rod. From this, we can find its coefficient of the linear coefficient.
Formula used:
$\Delta L = L\alpha \Delta T$

Complete answer:
The linear expansion of any metal is given by
$\Delta L = L\alpha \Delta T$
Where,
$\Delta L$ is the expansion in the length of the metal
$L$ is the original length
$\alpha$ is the coefficient of linear expansion
$\Delta T$ is the change in temperature
In the question, they’ve given two rods of lengths ‘$L$’ and ‘$2L$’ with coefficients of linear expansion as ‘$\alpha$’ and ‘$2\alpha$’.
So, if we consider the first rod, the linear expansion for the change in temperature ‘$\Delta T$’ can be written as,
$\Delta {L_1} = L\alpha \Delta T$
Similarly, for the second rod, the linear expansion can be written as,
$\Delta {L_2} = \left( {2L} \right)\left( {2\alpha } \right)\Delta T = 4L\alpha \Delta T$
So, if we consider a rod made from the two, then the total length will be ‘$3L$’. Let us say that the coefficient of linear expansion for the new rod be ‘$\alpha '$’. When the temperature is raised by ‘$\Delta T$’, the linear expansion will be given by
$\Delta {L_3} = 3L\alpha '\Delta T$
And the expansion will be equal to the expansion in each rod individually. So,
\eqalign{ & \Delta {L_3} = \Delta {L_1} + \Delta {L_2} \cr & \Rightarrow 3L\alpha '\Delta T = L\alpha \Delta T + 4L\alpha \Delta T \cr & \Rightarrow 3L\alpha '\Delta T = 5L\alpha \Delta T \cr & \therefore \alpha ' = \dfrac{{5\alpha }}{3} \cr}
Thus, if the two rods are combined the new coefficient of linear expansion will be $\dfrac{{5\alpha }}{3}$.

So, the correct answer is “Option C”.

Note:
You need to understand that, the linear expansion of the rod is the increase in the length of the rod. For a given material, the coefficient of linear expansion is constant. Similar to linear expansion there is the areal expansion and volumetric expansion.