Question:
In an experiment to verify Stokes law, a small spherical ball of radius $\mathrm{r}$ and density $\rho$ falls under gravity through a distance $\mathrm{h}$ 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 $\mathrm{h}$ is proportional to :
(ignore viscosity of air)
Correct Option: , 2
Solution:
After falling through $\mathrm{h}$, the velocity be equal to terminal velocity
$\sqrt{2 \mathrm{gh}}=\frac{2}{9} \frac{\mathrm{r}^{2} \mathrm{~g}}{\eta}\left(\rho_{\ell}-\rho\right)$
$\Rightarrow \mathrm{h}=\frac{2}{81} \frac{\mathrm{r}^{4} \mathrm{~g}\left(\rho_{\ell}-\rho\right)^{2}}{\eta^{2}}$
$\Rightarrow \mathrm{h} \propto \mathrm{r}^{4}$