Question

The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is 20 cm/s. The terminal velocity of another ball of same material and having radius 3 mm in the same fluid will be _________ cm/s.

Hint:

In ratio problems where all other factors are constant, identify the power of the variable ($r$ in this case) and apply it directly to the change factor ($1/2$ radius $\rightarrow 1/4$ velocity).

Answer 1

Correct Answer: 5

Step 1: Understanding the Concept:
Terminal velocity is the constant velocity attained by an object falling through a viscous fluid when the net force on it becomes zero.
This happens when the sum of the buoyant force and the viscous drag force equals the gravitational force acting on the object.

Step 2: Key Formula or Approach:
The terminal velocity ($v_t$) of a spherical ball of radius $r$ is given by Stokes' Law:
\[ v_t = \frac{2}{9} \frac{r^2 (\rho - \sigma) g}{\eta} \]
where:
$\rho$ = Density of the ball
$\sigma$ = Density of the fluid
$\eta$ = Coefficient of viscosity of the fluid
From this expression, we see that for a given material and fluid:
\[ v_t \propto r^2 \]

Step 3: Detailed Explanation:
Let $r_1 = 6$ mm and $v_1 = 20$ cm/s.
Let $r_2 = 3$ mm and $v_2$ be the required terminal velocity.
Using the proportionality relation:
\[ \frac{v_2}{v_1} = \left( \frac{r_2}{r_1} \right)^2 \]
Substitute the values:
\[ \frac{v_2}{20} = \left( \frac{3}{6} \right)^2 \]
\[ \frac{v_2}{20} = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \]
\[ v_2 = \frac{20}{4} = 5 \text{ cm/s} \]

Step 4: Final Answer:
The terminal velocity of the ball with a 3 mm radius is 5 cm/s.