A 30.0 cm long metal rod expands by 0.0650 cm when its temperature is raised from 0°C to 100°C. A second rod of different metal and of the same length expands by 0.0350 cm for the same rise in temperature. A third composite rod, also 30.0 cm long, is made-up of pieces of each of the above metals placed end to end and expands by 0.0580 cm when temperature is increased from 0°C to 100°C. Find the length of the shorter portion of the composite bar in cm at 0°C.
Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.
Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA
\[ \begin{aligned} \Delta T &= 100^\circ\text{C}, \quad L_0=30\ \text{cm} \\[4pt] \alpha_1 &= \frac{0.0650}{30\times100} =2.1667\times10^{-5}\ \text{^\circ C}^{-1},\\ \alpha_2 &= \frac{0.0350}{30\times100} =1.1667\times10^{-5}\ \text{^\circ C}^{-1} \\[6pt] \text{Let shorter length } &= x,\ \text{longer } = 30-x \\[4pt] 0.0580 &= \Delta T\big(\alpha_1 x+\alpha_2(30-x)\big) \\[4pt] 5.80\times10^{-4} &=2.1667\times10^{-5}x+3.5001\times10^{-4}-1.1667\times10^{-5}x \\[4pt] 2.30\times10^{-4} &= 1.00\times10^{-5}x \\[4pt] x &= 7.0\ \text{cm} \end{aligned} \]