First slide

Fluid statics

Question

A solid cube of density p, floats in a liquid of density $ρ_{1}$ at 0°C as shown in fig. (2). If $γ$ and f be the coefficient of volume expansion of the liquid and fraction of the cube immersed in the liquid respectively, then the temperature at which the solid will be completely immersed will be

Moderate

A

$\frac{1 - f}{γf}$
B

$\frac{1 + f}{γf}$
C

$\frac{γf}{1 - f}$
D

$\frac{γf}{1 + f}$

Solution

Let V be the total volume of the solid and $∆ V$ be the volume of the solid immersed in the liquid. Then
$ρ_{s} Vg = ρ_{l} ΔVg or f = \frac{ΔV}{V} = \frac{ρ_{s}}{ρ_{l}}$
The density of the liquid decreases as the temperature is increased to T.
$\begin{array}{l} ρ_{l}^{'} = ρ_{l} / (1 + γT) \\ ∴ f^{'} = \frac{ρ_{s}}{ρ_{l}^{'}} = \frac{ρ_{s}}{ρ_{l}} \times (1 + γT) = f (1 + γT) \end{array}$
The solid gets completely immersed in the liquid when f '= 1. So,
$1 = f (1 + γT) = f + fγT or T = \frac{(1 - f)}{γf}$

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