Question

The table below lists the original length, the coefficient of linear expansion, and the original temperature for five different metal bars. Heat is added to all the bars so that they change temperature by the same amount.
What is the correct order for the amount of increase in length for the bars, greatest first?

Metal Rod Number >	$1$	$2$	$3$	$4$	$5$
Original Length	$5.0m$	$2.0m$	$3.0m$	$4.0m$	$1.0m$
Coefficient of Linear Expansion	$15\times {10}^{-6}{K}^{-1}$	$35\times {10}^{-6}{K}^{-1}$	$20\times {10}^{-6}{K}^{-1}$	$20\times {10}^{-6}{K}^{-1}$	$65\times {10}^{-6}{K}^{-1}$
Original Temperature	${25}^{\circ }C$	${35}^{\circ }C$	${15}^{\circ }C$	${20}^{\circ }C$	${10}^{\circ }C$

• A. $1,4,3,2,5$
• B. $5,2,3$ and $4$ tie, $1$
• C. $2,1,4,3,5$
• D. $4,1,2,5,3$
• E. $1,4,2,3,5$

Answer 1

The correct option is D $4,1,2,5,3$

Using $L^{\prime }=L(1+\alpha \Delta T)$ $⟹L^{\prime }-L=L\alpha \Delta T$

$⟹$ Increase in length $\Delta L=L\alpha \Delta T$

(1) : $\Delta L_1=5.0\times 15\times {10}^{-6}\times \Delta T=75\Delta T$

(2) : $\Delta L_2=2.0\times 35\times {10}^{-6}\times \Delta T=70\Delta T$

(3) : $\Delta L_3=3.0\times 20\times {10}^{-6}\times \Delta T=60\Delta T$

(4) : $\Delta L_4=4.0\times 20\times {10}^{-6}\times \Delta T=80\Delta T$

(5) : $\Delta L_5=1.0\times 65\times {10}^{-6}\times \Delta T=65\Delta T$ $⟹\Delta L_4>\Delta L_1>\Delta L_2>\Delta L_5>\Delta L_3$