Two small spherical metal balls are made from materials of densities $\rho_{1}$ and $\rho_{2}$ respe

Question

Two small spherical metal balls are made from materials of densities $\rho_{1}$ and $\rho_{2}$ respectively, where $\rho_{1}=4\rho_{2}$. Their radii are $1 \mathrm{~mm}$ and $2 \mathrm{~mm}$ respectively. They are dropped and reach terminal velocity in a viscous medium of density $0.5\rho_{2}$. The ratio of their terminal velocities ($v_1/v_2$) would be

Professor PC Thomas Classes · Answer

(d) : Terminal velocity is given by $v = \frac{2r^2(ho - \sigma)g}{9\eta}$, where $ho$ is the density of the ball and $\sigma$ is the density of the medium. The ratio of their velocities is $\frac{v_1}{v_2} = \left(\frac{r_1}{r_2}ight)^2 \frac{(ho_1 - \sigma)}{(ho_2 - \sigma)}$. Substituting the given values: $r_1=1$, $r_2=2$, $ho_1 = 4ho_2$, $ho_2 = ho_2$, and $\sigma = 0.5ho_2$. Thus, $\frac{v_1}{v_2} = \left(\frac{1}{2}ight)^2 \frac{(4ho_2 - 0.5ho_2)}{(ho_2 - 0.5ho_2)} = \frac{1}{4} 	imes \frac{3.5ho_2}{0.5ho_2} = \frac{1}{4} 	imes 7 = \frac{7}{4}$.