Question

A small spherical ball falling through a viscous medium of negligible density has terminal velocity $v$. Another ball of the same mass but of radius twice that of the earlier falling through the same viscous medium will have terminal velocity

• A. $v$
• B. $v/4$
• C. $1:4$
• D. $1:64$

Answer 1

Answer: (C)

$1:4$

Answer 2

Terminal velocity of the ball through a viscous medium
$v=\frac{2}{9}\times \frac{g}{\eta }(\rho -\sigma ){{r}^{2}}$
Or $v=\frac{2}{9}\times \frac{g}{\eta }(\rho ){{r}^{2}}$
Because for viscous medium of negligible density
$(\sigma =0)$ $\therefore$
$v=\frac{2}{9}\times \frac{g}{\eta }\times \frac{m}{\frac{4}{3}\pi {{r}^{3}}}\times {{r}^{2}}$
$\left[ \because \rho =\frac{m}{\frac{4}{3}\pi {{r}^{3}}} \right]$
Or $v=\frac{2}{9}\times \frac{g}{\eta }\times \frac{m}{\frac{4}{3}\pi r}$
$\Rightarrow$ $v\propto \frac{1}{r}$
For the second ball $v\propto \frac{1}{2r}$
$\therefore$ $\frac{v}{v}=\frac{2r}{r}$ Or $v=\frac{v}{2}$