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  4. Two small spherical metal balls, having equal masses, are made from materials of densities ρ1 and ρ2(ρ1 = 8π2) and have radii of 1mm and 2mm, respectively. They are made to fall vertically (from rest) in a viscous medium whose coefficient of viscosity equals η and whose density is 0.1 ρ2. The ratio of their terminal velocities would be :

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Q. Two small spherical metal balls, having equal masses, are made from materials of densities $\rho_{1}$ and $\rho_{2}\left(\rho_{1} = 8\pi_{2}\right)$ and have radii of 1mm and 2mm, respectively. They are made to fall vertically (from rest) in a viscous medium whose coefficient of viscosity equals $\eta$ and whose density is $0.1 \,\rho_{2}$. The ratio of their terminal velocities would be :

NEETNEET 2019Mechanical Properties of Fluids

Solution:

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$v_r = \frac{2r^2(\sigma - \rho)g}{9\eta}$
$\frac{v_{1}}{v_{2}} = \left(\frac{r_{1}}{r_{2}}\right)^{2} \frac{\left(\sigma_{1} - \rho\right)}{\left(\sigma_{2} - \rho\right)}$
$= \left(\frac{1}{2}\right)^{2} \left(\frac{8\rho_{2} - 0.1 \rho_{2}}{\rho_{2} - 0.1 \rho_{2}}\right) = \frac{79}{36}$