Summary of "Fluid Mechanics | Module 2 | Buoyancy (Lecture 17)"

This lecture covers the fundamental concepts of Buoyancy in Fluid Mechanics, including the principles, forces involved, equilibrium conditions, and practical problem-solving methods related to floating and submerged bodies.

Main Ideas and Concepts

Introduction to Buoyancy:
- Buoyancy is the net upward force exerted by a fluid on a submerged or floating body.
- It arises due to the difference in pressure at different depths in a fluid.
- Archimedes’ principle states that the buoyant force on a body immersed in a fluid is equal to the weight of the fluid displaced by the body.
Hydrostatic Pressure and Resultant Force:
- Hydrostatic Pressure varies with depth in a fluid.
- The resultant hydrostatic force on a submerged surface acts normal (perpendicular) to the surface.
- The buoyant force is the net upward force resulting from pressure differences on the submerged body.
Volume and Displacement:
- The volume of the displaced fluid is crucial in determining the buoyant force.
- For simple shapes like cylinders, the volume can be calculated and used to find buoyant force.
- The volume of the fluid displaced by the submerged part of the body determines the magnitude of the buoyant force.
Centers of Gravity and Buoyancy:
- The center of gravity (G) is the point where the weight of the body acts.
- The center of Buoyancy (B) is the centroid of the displaced fluid volume.
- The relative positions of G and B determine the stability of the floating or submerged body.
Equilibrium Conditions:
- Stable Equilibrium: When the body returns to its original position after a small disturbance (restoring moment exists).
- Unstable Equilibrium: When the body moves further away from its original position after disturbance.
- Neutral Equilibrium: When the body remains in its new position after disturbance (no restoring or overturning moment).
- These conditions depend on the relative positions of G and B and the metacentric height.
Metacenter and Stability:
- The Metacenter (M) is the point about which the body oscillates when tilted.
- Stability criteria involve the metacentric height (distance between G and M).
- If M is above G, the body is stable; if below, unstable.
Application Problems:
- Example involving a steel block partially submerged in mercury and water.
- Use of density differences to calculate buoyant forces and equilibrium positions.
- Calculation of Specific Gravity using volumes displaced in multiple fluids.
- Application of Archimedes’ principle to multi-fluid systems.

Methodology / Instructions Presented

To Calculate Buoyant Force:
- Identify the volume of fluid displaced by the submerged part of the body.
- Multiply the displaced fluid volume by the fluid density and gravitational acceleration: Buoyant Force = ρ_fluid × V_displaced × g
To Determine Stability:
- Locate the center of gravity (G) of the body.
- Locate the center of Buoyancy (B) of the displaced fluid volume.
- Determine the Metacenter (M) by analyzing the body’s rotational behavior.
- Compare positions of G and M:
  - If M is above G → stable equilibrium.
  - If M is below G → unstable equilibrium.
  - If M coincides with G → neutral equilibrium.
To Solve Multi-fluid Buoyancy Problems:
- Calculate the volume of the body submerged in each fluid.
- Use the density of each fluid to find the buoyant force contributed by each fluid.
- Sum the buoyant forces and equate to the weight of the body to find equilibrium conditions.
- Calculate Specific Gravity by comparing weights and volumes.
To Find Specific Gravity:
- Use the formula: Specific Gravity = (Weight of body in air) / (Weight of displaced fluid)
- In multi-fluid systems, consider the contribution of each fluid separately.

Key Terms Defined

Buoyant Force: Upward force exerted by fluid on a submerged or floating object.
Center of Gravity (G): Point where the weight of the body acts.
Center of Buoyancy (B): Centroid of the displaced fluid volume.
Metacenter (M): Point about which the body oscillates when tilted.
Stable, Unstable, Neutral Equilibrium: Conditions describing the body's response to disturbance.
Specific Gravity: Ratio of the density of a substance to the density of a reference substance (usually water).