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} $