What is the difference between stable and unstable equilibrium?

Short Answer:

The difference between stable and unstable equilibrium depends on how a body reacts when slightly disturbed from its original position. In stable equilibrium, the body tends to return to its original position when disturbed, meaning it resists overturning. In contrast, in unstable equilibrium, the body moves further away from its original position when disturbed, meaning it easily topples or overturns.

In simple terms, a body with a low center of gravity and high metacenter is stable, like a ship that regains its upright position, while a body with a high center of gravity and low metacenter is unstable, like a tall, narrow vessel that tips over easily.

Detailed Explanation :

Difference between Stable and Unstable Equilibrium

In fluid mechanics and engineering, equilibrium refers to the condition of a body when the sum of all forces and moments acting on it is zero. A floating or submerged body is said to be in equilibrium when the weight of the body acting downward equals the buoyant force acting upward. However, equilibrium can be of different types based on the body’s behavior when slightly disturbed — mainly stable, unstable, and neutral equilibrium.

The terms stable equilibrium and unstable equilibrium are used to describe the tendency of the body to return to its original position or to move away from it after a disturbance. These conditions are very important in the design and stability analysis of ships, submarines, buoys, and other floating structures.

  1. Stable Equilibrium

A floating or submerged body is said to be in stable equilibrium when, after being slightly disturbed, it returns to its original position.

This happens when the metacenter (M) is located above the center of gravity (G). In this case, the metacentric height (GM) is positive.

Explanation:
When the body is tilted slightly, the center of buoyancy (B) shifts to a new position (B′) because the shape of the displaced fluid changes. The line of action of the buoyant force now passes through a point called the metacenter (M). The upward buoyant force and the downward weight create a restoring couple or moment, which brings the body back to its original position.

Characteristics of Stable Equilibrium:

  1. The metacenter (M) lies above the center of gravity (G).
  2. The metacentric height (GM) is positive.
  3. When disturbed, the body returns to its upright position.
  4. The body resists overturning.

Examples:

  • A ship that straightens after rocking due to waves.
  • A cone floating base down in water.
  • A wide, flat-bottomed vessel or raft.

Reason for Stability:
In stable equilibrium, the restoring moment helps balance the tilting moment. This happens because the upward buoyant force passes on the side of the tilt opposite to the center of gravity, producing a force that pushes the body back to equilibrium.

  1. Unstable Equilibrium

A floating or submerged body is said to be in unstable equilibrium when, after being slightly disturbed, it moves further away from its original position instead of returning.

This happens when the metacenter (M) is located below the center of gravity (G). In this case, the metacentric height (GM) is negative.

Explanation:
When such a body is tilted slightly, the center of buoyancy (B) shifts, but the buoyant force now acts through the metacenter (M), which lies below the center of gravity (G). The upward buoyant force and the downward weight produce an overturning couple, which increases the tilt rather than correcting it. Hence, the body becomes unstable and may overturn completely.

Characteristics of Unstable Equilibrium:

  1. The metacenter (M) lies below the center of gravity (G).
  2. The metacentric height (GM) is negative.
  3. When disturbed, the body continues to tilt or overturn.
  4. The body cannot regain its original position.

Examples:

  • A tall, narrow ship with heavy load above its deck.
  • A cone floating tip down in water.
  • A bottle standing upright in water.

Reason for Instability:
In unstable equilibrium, the line of action of the buoyant force and the weight of the body form an overturning couple, causing the body to topple instead of regaining balance.

Mathematical Explanation

The metacentric height (GM) determines whether a body is in stable or unstable equilibrium:

Where,

  •  = Metacentric height
  •  = Metacentric radius (distance between B and M)
  •  = Distance between center of buoyancy and center of gravity

For Stable Equilibrium:   (M above G)
For Unstable Equilibrium:   (M below G)

Physical Explanation

When a floating body is tilted:

  • The center of buoyancy moves sideways to the submerged side.
  • If M is above G, the buoyant force acts to restore equilibrium.
  • If M is below G, the buoyant force acts to increase the tilt.

The balance between these two opposing forces determines whether the body returns to equilibrium or overturns.

Comparison between Stable and Unstable Equilibrium

Property Stable Equilibrium Unstable Equilibrium
Position of M Above G Below G
Metacentric height (GM) Positive Negative
Nature of moment Restoring moment Overturning moment
Reaction to disturbance Returns to original position Moves further from original position
Stability Highly stable Easily overturned
Example Ship or raft Floating cone (tip down)

Applications of Equilibrium Concept

  1. Ship Design:
    Ships are designed with a large positive metacentric height to ensure stability even in rough waters.
  2. Submarines:
    The weight distribution and buoyant chambers are adjusted so that the submarine remains stable underwater.
  3. Floating Structures:
    Used in offshore platforms, floating bridges, and buoys to maintain balance.
  4. Engineering of Watercraft:
    Helps in calculating how much load can be safely added without losing stability.
  5. Hydrostatic Studies:
    Used in studying the rolling and pitching motion of floating bodies.

Example for Understanding

A rectangular floating body has a width  , draft  , and the distance between its center of gravity and center of buoyancy  .

 

Since   is positive, the body is in stable equilibrium. If   were negative, the body would be in unstable equilibrium.

Conclusion

In conclusion, the difference between stable and unstable equilibrium lies in how a floating or submerged body responds to disturbance. A body with the metacenter (M) above the center of gravity (G) has a positive metacentric height and is in stable equilibrium, meaning it returns to its original position. Conversely, when M lies below G, the metacentric height becomes negative, resulting in unstable equilibrium, where the body overturns. Understanding these conditions is essential in the design and analysis of floating and marine structures to ensure safety and balance.