Floating Body Stable Equilibrium Conditions

The stability of a floating body, such as a ship or a pontoon, is a crucial concept in fluid mechanics. It relates to the body's tendency to return to its original position after being tilted slightly by an external force.

Understanding Key Points

To understand the condition for stable equilibrium, we need to consider two important points related to a floating body:

• Center of Gravity (G): This is the point where the entire weight of the body acts vertically downwards.
• Metacenter (M): When the body is tilted, the center of buoyancy (the center of gravity of the displaced fluid) shifts. The point where the vertical line passing through the new center of buoyancy intersects the original vertical centerline of the body is called the metacenter (M).

Condition for Stable Equilibrium

A floating body is in stable equilibrium when, after being tilted, it experiences a restoring moment that brings it back to its original position. This occurs under a specific condition related to the positions of the metacenter (M) and the center of gravity (G).

The condition for stable equilibrium is met when the metacenter point (M) is above the center of gravity (G).

This distance between the metacenter and the center of gravity is known as the metacentric height, denoted as $GM$. For stable equilibrium, the metacentric height must be positive ($GM > 0$).

Comparing Equilibrium States

The relative positions of M and G determine the type of equilibrium:

• Stable Equilibrium: Occurs when the metacenter (M) is above the center of gravity (G). Any tilt creates a restoring couple that returns the body to its upright position. ($GM > 0$)
• Neutral Equilibrium: Occurs when the metacenter (M) coincides with the center of gravity (G). The body remains in its new position after being tilted. ($GM = 0$)
• Unstable Equilibrium: Occurs when the metacenter (M) is below the center of gravity (G). Any tilt creates an overturning couple that moves the body further away from its original position. ($GM < 0$)

Therefore, for a floating body to achieve stable equilibrium, the metacenter point must be situated above the center of gravity.