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. $1:64$
• E. $30:1$

Answer 1

Answer: (C)

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 $v=\frac{v}{2}$