The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is $20 \mathrm{~cm} / \ma

Question

The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is $20 \mathrm{~cm} / \mathrm{s}$. The terminal velocity of another ball of same material and having radius 3 mm in the same fluid will be $\_\_\_\_$ $\mathrm{cm} / \mathrm{s}$.

Professor PC Thomas Classes · Answer

According to Stokes' law, the terminal velocity of a falling sphere is $v_t = \frac{2}{9} \frac{r^2 (ho - \sigma) g}{\eta}$. Since the materials and the fluid are identical, $v_t \propto r^2$. Taking the ratio for the two balls: $\frac{v_{t2}}{v_{t1}} = (\frac{r_2}{r_1})^2 = (\frac{3	ext{ mm}}{6	ext{ mm}})^2 = (\frac{1}{2})^2 = \frac{1}{4}$. Thus, $v_{t2} = \frac{v_{t1}}{4} = \frac{20}{4} = 5	ext{ cm/s}$.