Two small balls of same density σ and radii r and 3 r are dropped into two different tubes of length L , filled with liquids of densities 2 σ and 3 σ and coefficients of viscosity η 1 and η 2

Question

Two small balls of same density σ and radii r and 3 r ​ are dropped into two different tubes of length L , filled with liquids of densities 2 σ ​ and 3 σ ​ and coefficients of viscosity η 1 ​ and η 2 ​ respectively. If both balls reach the bottom of the tubes simultaneously, having moved at their respective terminal velocities, and the ratio η 2 ​ η 1 ​ ​ = 4 x ​ , the value of x is

TestHub · Accepted Answer

The terminal velocity of a spherical ball of radius $R$ and density $ρ_{b o d y}$ falling through a liquid of density $ρ_{l i q u i d}$ and viscosity $η$ is given by:

$v = \frac{2}{9} \frac{R ^{2} ( ρ _{b o d y} - ρ _{l i q u i d} ) g}{η}$

For the first ball, substituting $R = r$ and $ρ_{l i q u i d} = \frac{σ}{2}$ :

$v_{1} = \frac{2}{9} \frac{r ^{2} ( σ - \frac{σ}{2} ) g}{η _{1}} = \frac{r ^{2} σ g}{9 η _{1}}$

For the second ball, substituting $R = \frac{r}{3}$ and $ρ_{l i q u i d} = \frac{σ}{3}$ :

$v_{2} = \frac{2}{9} \frac{( \frac{r}{3} ) ^{2} ( σ - \frac{σ}{3} ) g}{η _{2}} = \frac{2}{9} \frac{\frac{r ^{2}}{9} ( \frac{2 σ}{3} ) g}{η _{2}} = \frac{4 r ^{2} σ g}{243 η _{2}}$

Since both balls travel the same distance $L$ in the same time, their terminal velocities must be equal, $v_{1} = v_{2}$ :

$\frac{r ^{2} σ g}{9 η _{1}} = \frac{4 r ^{2} σ g}{243 η _{2}}$

$\frac{η _{1}}{η _{2}} = \frac{243}{36} = \frac{27}{4}$

Comparing this with $\frac{η _{1}}{η _{2}} = \frac{x}{4}$ , we get $x = 27$ .

Answer: $27$

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