Question

A metal rod A of length ${l_0}$expands by $\Delta l$ when its temperature is raised by $100\;^\circ C$. Another rod B of different metal of length $2{l_0}$ expands by $\Delta l/2$ for the same rise in temperature. A third rod C of length $3{l_0}$ is made up of pieces of rods A and B placed end to end expands by $2\Delta l$ on heating through 100K.The length of each portion of the composite rod is:
A) $\dfrac{5}{3}{l_0},\dfrac{4}{3}{l_0}$
B) ${l_0},\;2{l_0}$
C) $\dfrac{3}{2}{l_0},\dfrac{3}{2}{l_0}$
D) $\dfrac{2}{3}{l_0},\dfrac{7}{3}{l_0}$

Answer 1

Recall that since the rod is being heated up, it undergoes thermal stress. Use the relation(s) for thermal stress appropriately for each case. Note that in the first case the calculation is straightforward but in the second part where the ends of the rod yield by 1mm there is a change in length of the rod, which contributes to a linear strain caused by the thermal stress.

Complete step by step answer:
Therefore, do not forget to account for this change in length and its subsequent implications on the stress experienced by the rod in this case. While calculating the change in length do not forget to calculate the net change as the rod may not actually increase in length to the extent we calculate.
The extension in the rod A is:
$\Delta l = {l_0}{\alpha _1}\Delta T......(I)$
The extension in the rod B is:
$\dfrac{{\Delta l}}{2} = 2{l_0}{\alpha _1}\Delta T......(II)$
The extension in the third rod is:
$2\Delta l = x{\alpha _1}\Delta T + \left( {3{l_0} - x} \right){\alpha _1}\Delta T.......(III)$
We will now substitute eq(I) and eq(II) in eq(III) and the result becomes:
$\begin{array}{l} 2\Delta l = x\dfrac{{\Delta l}}{{{l_0}}} + \left( {3{l_0} - x} \right)\dfrac{{\Delta l}}{{4{l_0}}}\\\ x = \dfrac{{5{l_0}}}{3} \end{array}$
Therefore, $\left( {3{l_0} - x} \right)$ becomes, $\left( {3{l_0} - \dfrac{{5{l_0}}}{3}} \right) = \dfrac{{4{l_0}}}{3}$

Therefore, the correct option is (A).

Note: Note that we call the linear strain as the compressive strain through the rod ends up expanding. This is because the linear strain is caused by the thermal stress whereas the compressive strain is caused by the rod in response to the linear strain and is equivalent in magnitude. This is obvious because the rod exerts forces to prevent its expansion, by means of which it is in fact trying to compress the rod to hold on to expansion caused by the thermal stress incident on it.