Q. A rod of length $20\ cm$ is made of metal. It expands by $0.075cm$ when its temperature is raised from $0℃$ to $100℃.$ Another rod of a different metal $B$ having the same length expands by $0.045\ cm$ 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 $B.$ This rod expands by $0.060\ cm$ for the same change in temperature. The portion made of metal $A$ has the length
A) $20\ cm$
B) $10\ cm$
C) $15\ cm$
D) $18\ cm$
Explanation
(b) $\mathrm{\Delta}L = L_{0}\ \alpha\mathrm{\Delta}\theta$
Rod $A:0.075 = 20 \times \alpha_{A} \times 100 \Rightarrow \alpha_{A} = \frac{75}{2} \times 10^{- 6}/℃$
rod $B:0.045 = 20 \times \alpha_{B} \times 100 \Rightarrow \alpha_{B} = \frac{45}{2} \times 10^{- 6}/℃$
For composite rod : $x\ cm$ of $A$ and $(20 - x)cm$ of $B$ we have
$0.060 = x\ \alpha_{A} \times 100 + (20 - x)\alpha_{B} \times 100$
$= x\left\lbrack \frac{75}{2} \times 10^{- 6} \times 100 + (20 - x) \times \frac{45}{2} \times 10^{- 6} \times 100 \right\rbrack$
On solving we get $x = 10\ cm$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.