Question

A rod of length $20\text{cm}$ is made of metal. It expands by $0.075\text{cm}$ when its temperature is raised from $0^{\circ }\text{C}$ to ${100}^{\circ }\text{C}$. Another rod of a different metal B having the same length expands by $0.045\text{cm}$ for the same change in temperature. A third rod of the same length is composed of two parts one of metal A and the other of metal B. Now, the rod expands by $0.06\text{cm}$ for the same change in temperature. The portion made of metal A has the length (in $\text{cm}$)

Answer 1

For rod 1 :-
$L_1=20\text{cm}$
$\Delta L_1=0.075\text{cm}$
$\Delta L_1=L_1{\alpha }_1\Delta T$
${\alpha }_1=\left[\frac{0.075}{20\times 100}\right]$
${\alpha }_1=37.5\times {10}^{-6}{\text{/}}^{\circ }\text{C}$
For rod 2 :-
$\Delta L_2=L_2{\alpha }_2\Delta T$
$0.045=\left(20\right){\alpha }_2\left[100\right]$
${\alpha }_2=\left[\frac{0.045}{20\times 100}\right]$
${\alpha }_2=22.5\times {10}^{-6}{\text{/}}^{\circ }\text{C}$
Now for third rod :-

$\Delta L=\Delta L_1+\Delta L_2$
$0.06=L[37.5\times {10}^{-6}]\times 100+(20-L)\times 22.5\times {10}^{-6}\times 100$
$6=0.375L+4.5-0.225L$
$1.5=0.15L$
$L=10\text{cm}$