A solid whose volume does not change with temperature floats in a liquid. For two different temperatures $t_{1}$ and $t_{2}$ of the liquid, fractions $f_{1}$ and $f_{2}$ of the volume of the solid remain submerged in the liquid. The coefficient of volume expansion of the liquid is equal to

\frac{f_{1} - f_{2}}{f_{2} t_{1} - f_{1} t_{2}}

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\frac{f_{1} - f_{2}}{f_{1} t_{1} - f_{2} t_{2}}

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\frac{f_{1} + f_{2}}{f_{2} t_{1} + f_{1} t_{2}}

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\frac{f_{1} + f_{2}}{f_{1} t_{1} + f_{2} t_{2}}

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Solution

The correct option is A $\frac{f_{1} - f_{2}}{f_{2} t_{1} - f_{1} t_{2}}$
As with the rise in temperature, the liquid undergoes volume expansion. Therefore the fraction of solid submerged in liquid increases.
Fraction of solid submerged at $t_{1}^{\circ} C = f_{1} \propto$ Volume of displaced liquid
$\propto V_{0} (1 + γ t_{1}) . . . (i)$
and fraction of solid submerged at $t_{2}^{\circ} C = f_{2} \propto$ Volume of displaced liquid
$\propto V_{0} (1 + γ t_{2}) . . . (i i)$
From Eqs. (i) and (ii), $\frac{f_{1}}{f_{2}} = \frac{1 + γ t_{1}}{1 + γ t_{2}}$
$\Rightarrow γ = \frac{f_{1} - f_{2}}{f_{2} t_{1} - f_{1} t_{2}}$

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