Question

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

• A. $\frac{f_{1} - f_{2}}{f_{2}t_{1} - f_{1}t_{2}}$
• B. $\frac{f_{1} - f_{2}}{f_{1}t_{1} - f_{2}t_{2}}$
• C. $\frac{f_{1} + f_{2}}{f_{2}t_{1} + f_{1}t_{2}}$
• D. $\frac{f_{1} + f_{2}}{f_{1}t_{1} + f_{2}t_{2}}$

Answer 1

Answer: (A)

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

Answer 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 $t_{1}{^\circ}C = f_{1}$ = Volume of displaced liquid $= V_{0}(1 + \gamma t_{1})$ .....(i)

and fraction of solid submerged at $t_{2}{^\circ}C = f_{2}$= Volume of displaced liquid $= V_{0}(1 + \gamma t_{2})$ .....(ii)

From (i) and (ii) $\frac{f_{1}}{f_{2}} = \frac{1 + \gamma t_{1}}{1 + \gamma t_{2}}$ ⇒ $\gamma = \frac{f_{1} - f_{2}}{f_{2}t_{1} - f_{1}t_{2}}$