First slide

Viscous flow/Laminar flow

Question

A ball of radius r and density $ρ$ falls freely under gravity through a distance h before entering water. Velocity of ball does not change even on entering water. If viscosity of water is $η$ , the value of h is given by

Moderate

A

$\frac{2}{9} r^{2} (\frac{1 - ρ}{η}) g$
B

$\frac{2}{81} r^{2} (\frac{ρ - 1}{η}) g$
C

$\frac{2}{81} r^{4} {(\frac{ρ - 1}{η})}^{2} g$
D

$\frac{2}{9} r^{4} {(\frac{ρ - 1}{η})}^{2} g$

Solution

Velocity of ball when it strikes the water surface
$v = \sqrt{2 g h} . . . . . . . (i)$
Terminal velocity of ball inside the water
$v = \frac{2}{9} r^{2} g \frac{(ρ - 1)}{η} . . . . . . . . (i i)$
Equating (i) and (ii), we get $\sqrt{2 g h} = \frac{2}{9} \frac{r^{2} g}{η} (ρ - 1)$
$\Rightarrow h = \frac{2}{81} r^{4} {(\frac{ρ - 1}{η})}^{2} g$

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