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Difference in lengths of two rods of different metals 0°C is 1 mm and that at 100°C is 2 mm. If coefficient of linear expansions of the metals are 12x10^-6 /°C and 8 x 10^-6/°C, find the length of the longer rod at 0°C.

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0.75 m

3.125 m

0.5 m

1.25 m

answer is D.

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Detailed Solution

$At 0 ° C, l_{1} - l_{2} = x$

$At t ° C, l_{1}^{1} - l_{2}^{1} = 2 x$

$ ∴ (l_{1} + l_{2}) + (l_{1} α_{1} - l_{2} α_{2}) t = 2 x$

$ \Rightarrow l_{1} α_{1} - (l_{1} - x) α_{2} = \frac{x}{t}$

$ \Rightarrow l_{1} (α_{1} - α_{2}) = \frac{x}{t} - {xα}_{2} = x (\frac{1}{t} - α_{2})$

$ \Rightarrow l_{1} = \frac{x (\frac{1}{t} - α_{2})}{(α_{1} - α_{2})} = \frac{10^{- 3} (\frac{1}{100} - 8 x 10^{- 6})}{(12 - 8) x 10^{- 6}} m$

$ \Rightarrow l_{1} = \frac{2}{4} m = 0 .5 m$

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