Define metacentric height.

Short Answer:

The metacentric height is defined as the distance between the center of gravity (G) of a floating body and its metacenter (M). It is an important measure used to determine the stability of floating bodies such as ships, boats, and pontoons.

In simple terms, the metacentric height shows how stable a floating body is in water. If the metacenter lies above the center of gravity (positive metacentric height), the body is stable; if it lies below, the body becomes unstable and may overturn. Therefore, metacentric height helps engineers design stable floating structures.

Detailed Explanation :

Metacentric Height

The metacentric height (GM) is a key parameter in fluid mechanics that indicates the stability of a floating body when it is tilted or disturbed. It is defined as the vertical distance between the center of gravity (G) of the body and the metacenter (M).

Mathematically,

Where,

  •  = Metacentric height
  •  = Distance between the center of buoyancy (B) and the metacenter (M), known as the metacentric radius
  •  = Distance between the center of buoyancy (B) and the center of gravity (G)

The metacentric height determines whether a body will return to its original position after being slightly tilted in a fluid. It is therefore a direct measure of stability.

Concept of Metacentric Height

When a floating body such as a ship or cylinder is in equilibrium, the center of buoyancy (B) and center of gravity (G) lie on the same vertical line. The buoyant force acts vertically upward through B, and the weight of the body acts vertically downward through G.

When the body is slightly tilted due to an external disturbance (like waves or wind):

  1. The shape of the displaced fluid volume changes, causing the center of buoyancy to shift to a new position, say  .
  2. The line of action of the buoyant force now passes through the new center of buoyancy  .
  3. The vertical line through the new buoyant force intersects the original vertical line through B at a point called the metacenter (M).

The distance between M and G, known as metacentric height, decides the floating body’s response to tilting.

Types of Equilibrium Based on Metacentric Height

The stability of a floating body depends on the position of its metacenter (M) with respect to its center of gravity (G). There are three possible cases:

  1. Stable Equilibrium (M above G)
    • If the metacenter lies above the center of gravity, the metacentric height (GM) is positive.
    • When the body is tilted, the buoyant force creates a restoring couple that brings the body back to its original upright position.
    • Example: A well-designed ship that returns upright after a wave passes.
  2. Neutral Equilibrium (M coincides with G)
    • If the metacenter coincides with the center of gravity, then  .
    • The body remains in its new position after tilting; it neither returns nor overturns.
    • Example: A perfectly balanced floating cylinder.
  3. Unstable Equilibrium (M below G)
    • If the metacenter lies below the center of gravity, the metacentric height (GM) is negative.
    • When tilted, the buoyant force produces an overturning couple, making the body unstable and likely to capsize.
    • Example: A top-heavy ship or an overloaded boat.

Mathematical Expression for Metacentric Height

The metacentric height can be calculated using the following formula:

Where,

  •  = Moment of inertia of the waterline area about the longitudinal axis
  •  = Volume of the fluid displaced by the floating body
  •  = Distance between the center of buoyancy and the center of gravity

The term   represents the metacentric radius (BM).

Thus,

Example:
For a rectangular floating body of width   and draft  :

This formula shows that the metacentric height increases with the width of the body and decreases with the depth of immersion.

Experimental Determination of Metacentric Height

The metacentric height can be found practically by the tilting method, which involves the following steps:

  1. Place the floating body (like a model of a ship) in water and ensure it is at rest.
  2. Measure its initial equilibrium position.
  3. Shift a known small weight ( ) through a known horizontal distance ( ) from the centerline of the body.
  4. This causes the body to tilt by a small angle  .
  5. The angle of tilt is measured using a plumb line or scale.

The metacentric height is then calculated by the formula:

Where,

  •  = Total weight of the floating body
  •  = Shifted weight
  •  = Distance moved by the weight
  •  = Angle of tilt

This experimental method is commonly used for ships, submarines, and floating models to test their stability.

Importance of Metacentric Height

  1. Stability of Floating Bodies:
    • Determines whether a ship, boat, or structure will return to its original position after being tilted.
  2. Ship Design:
    • Naval architects use metacentric height to design stable ships that resist capsizing even in rough seas.
  3. Marine and Offshore Engineering:
    • Used to ensure the stability of floating oil platforms, buoys, and docks.
  4. Load Distribution:
    • Helps determine the safe loading and unloading of ships to maintain proper stability.
  5. Hydrodynamics Studies:
    • Used in analyzing rolling and pitching motion of floating bodies.

Example Calculation

A floating body of width  , draft  , and  .
Find the metacentric height.

 

Hence, the metacentric height (GM) is 0.733 m, indicating that the body is stable.

Conclusion

In conclusion, the metacentric height is the vertical distance between the center of gravity (G) and the metacenter (M) of a floating body. It is the most important factor that determines the stability of floating objects. A large positive metacentric height ensures better stability, while a small or negative value indicates instability. Engineers and naval architects use this concept to design safe, stable, and efficient floating bodies such as ships, submarines, and floating platforms.