question_answer

A rod of length 20 cm is made of metal. It expands by 0.075cm when its temperature is raised from 0°C to 100°C. 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                   [CPMT 1991]

A)            20 cm                                      

B)            10 cm

C)            15 cm                                      

D)            18 cm

Correct Answer: B

Solution :

                   \[\Delta L={{L}_{0}}\,\alpha \Delta \theta \] Rod A : 0.075 = 20 ´ aA ´ 100 Þ \[{{\alpha }_{A}}=\frac{75}{2}\times {{10}^{-6}}/{}^\circ C\] rod B : 0.045 = 20 ´ aB ´ 100 Þ \[{{\alpha }_{B}}=\frac{45}{2}\times {{10}^{-6}}/{}^\circ C\] For composite rod : x cm of A and (20 ? x) cm of B we have 0.060 = x aA ´ 100 + (20 ? x) aB ´ 100           \[=x\,\left[ \frac{75}{2}\times {{10}^{-6}}\times 100+(20-x)\times \frac{45}{2}\times {{10}^{-6}}\times 100 \right]\] On solving we get x = 10 cm.