First slide

Viscosity

Question

A metal ball of radius 'r' and density 'd' travels with a terminal velocity 'v' in a liquid of density d/4. The terminal velocity of another ball of radius '2r' and density '3d' in the same liquid is

Moderate

A

$\large \frac {44}{3}V$
B

$\large \frac {22}{3}V$
C

$\large \frac {11}{3}V$
D

$\large \frac {3}{44}V$

Solution

$\large r_1=r;\;r_2=2r$ ; $\large {\rho _m} = \frac{d}{4}$ ; $\large {\rho _{{b_1}}} = d$ ; $\large {\rho _{{b_2}}} = 3d$ ; $\large {V_{{t_1}}} = v$ ; $\large {V_{{t_2}}} = ?$
$\large {V_t} = \frac{2}{9}{r^2}\frac{g}{\eta }\left( {{\rho _b} - {\rho _m}} \right) \Rightarrow {V_t}{\alpha ^2}\left( {{\rho _b} - {\rho _m}} \right)$
$\large \frac {V_{t_1}}{V_{t_2}}=\left ( \frac {r_1}{r_2} \right )^2\left ( \frac { \rho _{b_1}-\rho_{m_1}}{ \rho _{b_2}-\rho_{m_2}} \right )$
$\large \left ( \frac {r}{2r} \right )^2\left ( \frac {d-\frac d4}{3d-\frac d4} \right )=\frac {3}{44}\Rightarrow V_{t_2}=\frac {44}{3}V$

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