Question

Stoke's law suggests that the velocity of fall of particles is directly proportional to

• A. Density of the particle
• B. Density of the suspension
• C. Radius of the particle
• D. Viscosity of the medium

Hint:

Remember: Big particles fall fast, small particles fall slow.
Velocity is inversely proportional to viscosity and directly proportional to the square of the radius.
Stokes' Law is the reason why sand settles first and clay stays cloudy in water for days.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks to identify which of the physical parameters listed is directly proportional to the settling velocity of a spherical particle in a fluid according to Stokes' Law.
Key Formula or Approach:
Stokes' Law describes the terminal velocity (\( v \)) of a small sphere falling through a viscous fluid:
\[ v = \frac{2}{9} \frac{(\rho_p - \rho_f)}{\eta} g r^2 \]
Where:
\( v \) = terminal velocity of the particle
\( \rho_p \) = density of the particle
\( \rho_f \) = density of the fluid (suspension)
\( g \) = gravitational acceleration
\( r \) = radius of the particle
\( \eta \) = dynamic viscosity of the fluid

Step 2: Detailed Explanation:

• Analysis of Radius (Statement C): From the formula, the velocity is proportional to the square of the radius (\( v \propto r^2 \)). While the question does not explicitly mention "square," the radius is the most significant geometric factor. Larger particles fall much faster than smaller ones.

• Analysis of Density (Statements A and B): The velocity is proportional to the difference in density (\( \rho_p - \rho_f \)). It is not directly proportional to the density of the particle or the fluid individually in a linear sense; rather, it depends on how much heavier the particle is than the fluid it displaces.

• Analysis of Viscosity (Statement D): The velocity is inversely proportional to the viscosity (\( v \propto 1/\eta \)). A thicker (more viscous) fluid slows down the particle's fall.

• Assumptions of Stokes' Law: The law assumes that particles are rigid, spherical, and falling in a laminar flow without interfering with each other. In soil science, this is used to calculate the time required for sand, silt, and clay to settle in a water column during textural analysis.

Step 3: Final Answer:
According to the proportionality in the formula, although strictly proportional to the square of the radius, the Radius of the particle is the correct choice among the provided parameters.