What are conditions of equilibrium for floating bodies?

Short Answer:

The conditions of equilibrium for floating bodies are based on the positions of the center of gravity (G) and the metacenter (M). A floating body is said to be in equilibrium when the buoyant force acting upward equals the weight of the body acting downward.

There are three main conditions:

  1. If the metacenter (M) is above the center of gravity (G), the body is in stable equilibrium.
  2. If M and G coincide, the body is in neutral equilibrium.
  3. If M is below G, the body is in unstable equilibrium.
    These conditions determine whether the body will float upright, stay tilted, or overturn in a fluid.
Detailed Explanation :

Conditions of Equilibrium for Floating Bodies

In fluid mechanics, a floating body is one that is partially submerged in a fluid and is supported by the upward buoyant force acting on it. For a body to float, the weight of the body must be equal to the buoyant force acting on it, according to Archimedes’ principle.

However, even when this condition is satisfied, the body may not always remain in a stable position. The stability of the floating body depends on the relative positions of its center of gravity (G) and metacenter (M). When a floating body is tilted slightly, these two points determine whether the body will return to its original position, remain tilted, or overturn completely.

Explanation of Equilibrium Conditions

A body in a fluid can have three types of equilibrium — stable, neutral, or unstable. These depend on how the metacenter (M) is positioned relative to the center of gravity (G).

Let’s understand the concept step-by-step:

When a body is floating in equilibrium:

  • The weight (W) of the body acts vertically downward through its center of gravity (G).
  • The buoyant force (FB) acts vertically upward through the center of buoyancy (B), which is the centroid of the displaced fluid volume.
  • If the body is tilted, the center of buoyancy (B) shifts to a new position (B′), and the line of action of the new buoyant force passes through the metacenter (M).

The relative position of M and G determines the equilibrium condition of the floating body.

  1. Stable Equilibrium (M above G)

A floating body is said to be in stable equilibrium when the metacenter (M) lies above the center of gravity (G).

  • In this condition, the metacentric height (GM) is positive.
  • When the body is slightly tilted, the center of buoyancy (B) moves toward the side of immersion.
  • The line of action of the buoyant force now passes through M, producing a restoring moment (or couple) that tries to bring the body back to its original upright position.

Example:

  • A ship that returns to its upright position after being disturbed by waves.
  • A wooden block floating stably in water.

Conclusion:
In stable equilibrium, the floating body resists overturning and maintains its balance due to the restoring couple created by the upward buoyant force and downward weight.

  1. Neutral Equilibrium (M coincides with G)

A floating body is said to be in neutral equilibrium when the metacenter (M) coincides with the center of gravity (G).

  • Here, the metacentric height (GM) is zero.
  • When the body is slightly tilted, no restoring or overturning moment is produced.
  • The body remains in its new position and does not return to its original one, nor does it overturn further.

Example:

  • A perfectly balanced floating cylinder that stays in its new tilted position without returning or sinking.

Conclusion:
In neutral equilibrium, the floating body neither restores itself to its original position nor becomes unstable; it simply stays in its new tilted position.

  1. Unstable Equilibrium (M below G)

A floating body is said to be in unstable equilibrium when the metacenter (M) lies below the center of gravity (G).

  • In this case, the metacentric height (GM) is negative.
  • When the body tilts, the center of buoyancy (B) shifts, and the line of action of the buoyant force creates an overturning couple rather than a restoring one.
  • This couple increases the tilt instead of reducing it, causing the body to overturn completely.

Example:

  • A ship or boat that is top-heavy (center of gravity is high).
  • A narrow cylindrical vessel floating vertically in water that tips over when disturbed.

Conclusion:
In unstable equilibrium, the floating body becomes top-heavy and easily capsizes because the buoyant force cannot counteract the overturning moment.

Mathematical Representation

The stability of a floating body can be expressed mathematically using the metacentric height (GM):

Where,

  •  = Metacentric height
  •  = Metacentric radius (distance between B and M)
  •  = Distance between B and G

The conditions of equilibrium are as follows:

  • Stable equilibrium:
  • Neutral equilibrium:
  • Unstable equilibrium:

Hence, the sign and magnitude of the metacentric height (GM) directly determine the type of equilibrium of the floating body.

Factors Affecting Stability of Floating Bodies

  1. Shape of the Body:
    • Wider bases increase stability because they increase the metacentric height (M moves upward).
  2. Distribution of Weight:
    • Lowering the center of gravity (G) improves stability.
    • Adding weight at the top raises G, reducing stability.
  3. Density and Immersion:
    • The volume of displaced fluid affects the center of buoyancy and metacentric radius.
  4. Metacentric Height:
    • A large metacentric height ensures good stability but may cause excessive rolling motion (oscillations).

Applications of Equilibrium Conditions

  1. Ship and Submarine Design:
    • Engineers ensure that ships have a high metacenter above the center of gravity for stability in rough seas.
  2. Floating Structures:
    • Used in the design of floating bridges, oil rigs, and buoys.
  3. Hydraulic Equipment:
    • Ensures floating parts maintain stable operation.
  4. Marine Safety:
    • Helps prevent capsizing of boats and ships.
  5. Experimental Studies:
    • Used in fluid mechanics labs to analyze floating stability.
Conclusion

In conclusion, the conditions of equilibrium for floating bodies depend on the relative positions of the center of gravity (G) and the metacenter (M). When M lies above G, the body is in stable equilibrium and returns to its upright position when disturbed. When M coincides with G, the body is in neutral equilibrium and stays in its new position. When M lies below G, the body is in unstable equilibrium and tends to overturn. Thus, understanding these conditions is essential in designing stable floating structures in marine and mechanical engineering.