Question

A particle attains its terminal settling velocity when ______

• A. buoyancy force + drag force = gravity force
• B. drag force = buoyancy force
• C. gravity force + drag force = buoyancy force
• D. gravity force - drag force = buoyancy force

Hint:

To quickly write the force balance for a falling body, remember that gravity pulls it down, while fluid displacement (buoyancy) and friction (drag) push it up.
At steady state (terminal velocity), upward forces must balance downward forces: \( F_b + F_d = F_g \).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the balance of forces acting on a solid particle falling through a fluid when it reaches its terminal settling velocity.
This is a core concept in fluid-particle dynamics and mechanical operations.

Step 2: Key Formula or Approach:
For a particle falling through a fluid, three main forces act on it:
1. Gravitational force (\( F_g \)) acting downwards.
2. Buoyancy force (\( F_b \)) acting upwards.
3. Drag force (\( F_d \)) acting upwards (opposing the direction of motion).
The equation of motion is given by Newton's second law:
\[ m \cdot \frac{dv}{dt} = F_g - F_b - F_d \]

Step 3: Detailed Explanation:

• Initial Phase: When a particle is released into a fluid, it initially accelerates because the downward gravitational force is greater than the sum of the upward buoyancy and drag forces.

• Acceleration Effect: As the particle's velocity increases, the fluid drag force (which is a function of velocity) also increases.

• Terminal State: Eventually, the upward forces (buoyancy force + drag force) increase to a point where they perfectly balance the downward gravitational force.
At this point, the net force acting on the particle is zero:
\[ F_g - F_b - F_d = 0 \quad \implies \quad F_b + F_d = F_g \]
Because the net force is zero, the acceleration \( \frac{dv}{dt} \) becomes zero, and the particle continues to fall at a constant, maximum velocity known as the terminal settling velocity.


Step 4: Final Answer:
The particle attains its terminal settling velocity when the sum of the buoyancy force and the drag force equals the gravitational force.