Question

In an experiment to verify Stokes law, a small spherical ball of radius $r$ and density $\rho$ falls under gravity through a distance $h$ in air before entering a tank of water. If the terminal velocity of the ball inside water is same as its velocity just before entering the water surface, then the value of $h$ is proportional to
(ignore viscosity of air)

• A. $r^4$
• B. $r$
• C. $r^3$
• D. $r^2$

Answer 1

The correct option is A $r^4$

Using,
$v^2-u^2=2gh$
$\Rightarrow v^2-0^2=2gh\Rightarrow v=\sqrt{2gh}$

Termianl velocity,

$V_T=\frac{2}{9}\frac{r^2(\rho -\sigma )g}{\eta }$

After falling through $h$ the velocity should be equal to terminal velocity

$\therefore \sqrt{2gh}=\frac{2}{9}\frac{r^2(\rho -\sigma )g}{\eta }$
$\Rightarrow 2gh=\frac{4}{81}\frac{r^4{g}^2(\rho -\sigma {)}^2}{{\eta }^2}$
$\Rightarrow h=\frac{2r^{4}g(\rho -\sigma {)}^2}{81{\eta }^2}\Rightarrow h\propto r^4$