Question 234: Properties of Matter

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
$\_\_\_\_$ $\mathrm{cm} / \mathrm{s}$.

Accepted Answer

When a spherical ball falls through a viscous fluid, it eventually reaches a constant speed called terminal velocity. This occurs when the net force acting on the ball is zero.
The forces acting on the ball are :
Weight (W) which acts downwards. $\mathrm{W}=\frac{4}{3} \pi \mathrm{r}^3 \sigma \mathrm{~g}$ (where $\sigma$ is the density of the ball).
Up-thrust $\left(\mathrm{F}_{\mathrm{B}}\right)$ is Acting upwards. $F_B=\frac{4}{3} \pi r^3 \rho g$ (where $\rho$ is the density of the fluid).
Viscous drag $\left(\mathrm{F}_{\mathrm{v}}\right)$ is acting upwards. According to Stokes' Law, $\mathrm{F}_{\mathrm{v}}=6 \pi \eta \mathrm{r} \mathrm{v}$, where $\eta$ is the viscosity of fluid and v is the speed of the ball.
At equilibrium :
$$ \mathrm{W}=\mathrm{F}_{\mathrm{B}}+\mathrm{F}_{\mathrm{v}} $$
$$\Rightarrow $$ $\frac{4}{3} \pi r^3 \sigma g=\frac{4}{3} \pi r^3 \rho g+6 \pi \eta r v$
$$\Rightarrow $$ $6 \pi \eta r v=\frac{4}{3} \pi r^3 g(\sigma-\rho)$
$$\Rightarrow $$ $$ \mathrm{v}=\frac{2 \mathrm{r}^2 \mathrm{~g}(\sigma-\rho)}{9 \eta} $$
For balls of the same material in the same fluid :
$$ \mathrm{v} \propto \mathrm{r}^2 $$
Therefore :
$$ \frac{\mathrm{v}_2}{\mathrm{v}_1}=\left(\frac{\mathrm{r}_2}{\mathrm{r}_1}\right)^2 $$
Initial radius of the ball is $r_1=6 \mathrm{~mm}$ and the initial terminal velocity is $v_1=20 \mathrm{~cm} / \mathrm{s}$
Final radius of the ball is $r_2=3 \mathrm{~mm}$
Substituting the values :
$\frac{v_2}{20}=\left(\frac{3}{6}\right)^2$
$$\Rightarrow $$ $$ \frac{v_2}{20}=\frac{1}{4} \Rightarrow v_2=\frac{20}{4}=5 \mathrm{~cm} / \mathrm{s} $$
Therefore, the terminal velocity of the second ball will be $5 \mathrm{~cm} / \mathrm{s}$.
Hence, the correct answer is 5.

Properties of Matter

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}.

Integer / Numeric Answer Type