Short Answer
The static equilibrium of submerged and floating bodies occurs when the forces and moments acting on the body are balanced. This means the buoyant force acting upward must equal the gravitational force acting downward, ensuring the body remains at rest.
For submerged bodies, equilibrium is achieved when the weight of the body equals the buoyant force, and the center of gravity (G) and center of buoyancy (B) are aligned. For floating bodies, equilibrium occurs when the buoyant force balances the weight, and the metacentric height (GM) is positive, ensuring stability. These conditions are essential for ship design, dam structures, and underwater constructions.
Detailed Explanation
Static Equilibrium of Submerged Bodies
A submerged body is completely under the fluid surface. It is in static equilibrium when:
- Buoyant Force = Weight of the Body: The body neither rises nor sinks when the two forces are equal.
- Center of Gravity (G) and Center of Buoyancy (B) are Vertically Aligned: If not aligned, the body experiences a moment, causing rotation.
Mathematically, equilibrium occurs when:
Buoyant Force=Weight=ρgV\text{Buoyant Force} = \text{Weight} = \rho g VBuoyant Force=Weight=ρgV
where:
- ρ = Density of the fluid
- g = Acceleration due to gravity
- V = Volume of displaced fluid
If the body is disturbed, stability depends on whether it returns to its original position or overturns.
Static Equilibrium of Floating Bodies
A floating body is partially submerged, such as a ship or a raft. It is in equilibrium when:
- Buoyant Force = Weight of the Body: The body neither sinks nor rises.
- Metacentric Height (GM) is Positive: The metacenter (M) is the point where the buoyant force acts when the body tilts. If GM > 0, the body is stable; if GM < 0, the body capsizes.
For stability, the metacentric height is calculated as:
GM=BM−BGGM = BM – BGGM=BM−BG
where:
- BM = Distance between the center of buoyancy (B) and metacenter (M)
- BG = Distance between center of buoyancy (B) and center of gravity (G)
Applications in Engineering:
- Ship and Boat Design: Ensuring proper stability to prevent capsizing.
- Bridge Pontoons and Floating Structures: Used in floating bridges and offshore platforms.
- Submarine Operations: Maintaining equilibrium when diving and surfacing.
Conclusion
Static equilibrium of submerged and floating bodies occurs when the buoyant force balances the weight, and stability conditions are met. In submerged bodies, alignment of G and B is essential, while in floating bodies, positive metacentric height (GM) ensures stability. These principles are crucial in marine engineering, shipbuilding, and hydraulic structure designs to prevent instability and failure.