Question

A uniform solid ball of density $d$ and of radius $r$ is moving vertically downward inside a viscous liquid $(density=\frac{d}{6}$ & coefficient of viscosity$=\eta )$ with an acceleration of $g/2$ at an instant. Speed of the ball at this instant is

• A. $\frac{1}{6}\frac{r^{2}gd}{\eta }$
• B. $\frac{5}{27}\frac{r^{2}gd}{\eta }$
• C. $\frac{2}{27}\frac{r^{2}gd}{\eta }$
• D. $\frac{2}{9}\frac{r^{2}gd}{\eta }$

Answer 1

The correct option is C $\frac{2}{27}\frac{r^{2}gd}{\eta }$

Net force acting on the ball is

$F=\frac{4}{3}\pi r^3(d-\frac{d}{6})-6\pi \eta rv=\frac{4}{3}\pi r^{3}d\frac{g}{2}$

$\therefore \frac{4}{3}{r}^{3}dg(1-\frac{1}{6}-\frac{1}{2})=6\pi \eta rv$

$\therefore v=\frac{2}{27}\frac{r^{2}dg}{\eta }$