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(C) The terminal velocity $v_T$ of a sphere of radius $r$ and density $\sigma$ falling in a liquid of density $\rho$ and viscosity $\eta$ is given by

Vedclass · Accepted Answer

$r_2: r_1$

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(C) The terminal velocity $v_T$ of a sphere of radius $r$ and density $\sigma$ falling in a liquid of density $ho$ and viscosity $\eta$ is given by $v_T = \frac{2r^2(\sigma - ho)g}{9\eta}$.Since the masses are equal, $m = \frac{4}{3}\pi r^3 \sigma$, which implies $\sigma \propto \frac{1}{r^3}$.Substituting $\sigma = \frac{m}{\frac{4}{3}\pi r^3}$ into the terminal velocity formula:$v_T = \frac{2r^2}{9\eta} \left( \frac{m}{\frac{4}{3}\pi r^3} - ho ight)g = \frac{2g}{9\eta} \left( \frac{3m}{4\pi r} - r^2ho ight)$.For a liquid of infinite column where the viscous force dominates or assuming the density of the sphere is much larger than the liquid $(\sigma \gg ho)$, we have $v_T \propto r^2 \sigma$.Since $r^3 \sigma = 	ext{constant}$, then $\sigma \propto r^{-3}$.Therefore, $v_T \propto r^2 \cdot r^{-3} = r^{-1} = \frac{1}{r}$.Thus, $\frac{v_1}{v_2} = \frac{r_2}{r_1}$.

Two spheres of equal masses but radii $r_1$ and $r_2$ are allowed to fall in a liquid of infinite column. The ratio of their terminal velocities is

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