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  4. A solid sphere, of radius R acquires a terminal velocity v1 when falling (due to gravity) through a viscous fluid having a coefficient of viscosity h. The sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity, v2, when falling through the same fluid, the ratio (v1/v2) equals :

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Q. A solid sphere, of radius $R$ acquires a terminal velocity $v_1$ when falling (due to gravity) through a viscous fluid having a coefficient of viscosity $h$. The sphere is broken into $27$ identical solid spheres. If each of these spheres acquires a terminal velocity, $v_2$, when falling through the same fluid, the ratio ($v_1/v_2$) equals :

JEE MainJEE Main 2019Mechanical Properties of Fluids

Solution:

We have
$V_T \, = \, \frac{2}{9} \, \frac{r^2}{\eta} (\rho_0 - \rho_1)g \Rightarrow \, \, \, v_T \propto r^2$
since mass of the sphere will be same
$\rho \frac{4}{3} \pi R^3 \, = \, 27. \frac{4}{3} \, \pi r^3 \rho \, \, \Rightarrow = \frac{R}{3}$
$\therefore \, \, \, \, \frac{v_1}{v_2} \, = \frac{R^2}{r^2} = 9$