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  4. A marble of mass x and diameter 2 r is gently released in a tall cylinder containing honey. If the marble displaces mass y( < x) of the liquid, then the terminal velocity is proportional to

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Q. A marble of mass $x$ and diameter $2 r$ is gently released in a tall cylinder containing honey. If the marble displaces mass $y( < x)$ of the liquid, then the terminal velocity is proportional to

Mechanical Properties of Fluids

Solution:

$v_{0}=\frac{2}{9} \frac{r^{2}\left(\rho-\rho^{\prime}\right) g}{\eta}$
Now, $x=\frac{4}{3} \pi r^{3} \rho$ or $\rho \propto \frac{x}{r^{3}}$
Similarly, $\rho^{\prime} \propto \frac{y}{r^{3}}$
$\therefore v_{0} \propto \frac{x-y}{r}$