In an experiment to verify Stokes' law,a smal | vedclass

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(B) The velocity of the ball after falling through a distance $h$ in air is given by $v = \sqrt{2gh}$.The terminal velocity $v_t$ of a spherical ball

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$r^{4}$

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(B) The velocity of the ball after falling through a distance $h$ in air is given by $v = \sqrt{2gh}$.The terminal velocity $v_t$ of a spherical ball of radius $r$ and density $\rho$ in a liquid of density $\rho_{\ell}$ and viscosity $\eta$ is given by Stokes' law:$v_t = \frac{2}{9} \frac{r^2 g}{\eta} (\rho - \rho_{\ell})$.According to the problem,the velocity just before entering the water is equal to the terminal velocity inside the water:$\sqrt{2gh} = \frac{2}{9} \frac{r^2 g}{\eta} (\rho - \rho_{\ell})$.Squaring both sides:$2gh = \left( \frac{2}{9} \frac{r^2 g}{\eta} (\rho - \rho_{\ell}) \right)^2$$2gh = \frac{4}{81} \frac{r^4 g^2}{\eta^2} (\rho - \rho_{\ell})^2$Solving for $h$:$h = \frac{2}{81} \frac{r^4 g}{\eta^2} (\rho - \rho_{\ell})^2$.Since $g$,$\eta$,$\rho$,and $\rho_{\ell}$ are constants for the given experiment,we have:$h \propto r^4$.

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