Question

The variation of lengths of two metal rods $A$ and $B$ with change in temperature is shown in Fig. The coefficients of linear expansion ${\alpha }_A$ for the metal $A$ and the temperature $T$ will be (given ${\alpha }_B$ = 9$\times$${10}^{-6}$/ ${}^{\circ }C$)

• A. α_A=3x10^-6/°C, 500°C
• B. α_A=3x10^-6/°C, 222.22°C
• C. α_A=27x10^-6/°C, 500°C
• D. α_A=27x10^-6/°C, 222.22°C

Answer 1

The correct option is D

α_A=27x10^-6/°C, 222.22°C

Slope of line $A$ = $\frac{(1006-1000)mm}{T^{\circ }C}$ = $\frac{\Delta L}{\Delta T}$ = $L{\alpha }_A$

i.e., $\frac{6}{T}mm{/}^{\circ }C$ = (1000 mm)${\alpha }_A$ ...............(i)

Similarly, for line B,

$\frac{2}{T}mm{/}^{\circ }C$ = (1002 mm)${\alpha }_B$ ...............(ii)

Dividing Eq. (i) by Eq. (ii),

3 = $\frac{1000{\alpha }_A}{1002{\alpha }_B}$ $\simeq$ ${\alpha }_A$ = 3 ${\alpha }_B$ ...............(iii)

From Eq. (iii), ${\alpha }_A$ = $3\times 9\times {10}^{-6}$ = $27\times {10}^{-6}{/}^{\circ }C$

From Eq. (i), $T$ = $\frac{6}{1000{\alpha }_A}$ = $\frac{6\times {10}^6}{1000\times 27}$

= ${222.22}^{\circ }C$