Consider two solid spheres P and Q each of density 8 gm cm\(^{-3}\) and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm cm\(^{-3}\) and viscosity \(\eta=3\) poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm cm\(^{-3}\) and viscosity \(\eta=2\) poiseulles. The ratio of the terminal velocities of P and Q is

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Fluid Mechanics

Consider two solid spheres P and Q each of density 8 gm cm

^{- 3}

and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm cm

^{- 3}

and viscosity

η = 3

poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm cm

^{- 3}

and viscosity

η = 2

poiseulles. The ratio of the terminal velocities of P and Q is

Answer: 3

Terminal velocity is given by:

v_{T} = \frac{2}{9} \frac{r ^{2}}{η} (ρ - σ) g

where

ρ

is sphere density and

σ

is liquid density.

For sphere P:

r_{P} = 0.5

cm,

ρ = 8

gm/cm³,

σ = 0.8

gm/cm³,

η = 3

poise

v_{P} \propto \frac{( 0.5 ) ^{2}}{3} (8 - 0.8) = \frac{0.25 \times 7.2}{3} = \frac{1.8}{3}

For sphere Q:

r_{Q} = 0.25

cm,

ρ = 8

gm/cm³,

σ = 1.6

gm/cm³,

η = 2

poise

v_{Q} \propto \frac{( 0.25 ) ^{2}}{2} (8 - 1.6) = \frac{0.0625 \times 6.4}{2} = \frac{0.4}{2}

\frac{v _{P}}{v _{Q}} = \frac{1.8/3}{0.4/2} = \frac{1.8 \times 2}{3 \times 0.4} = \frac{3.6}{1.2} = 3

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Integer answer