Understanding Stability of Floating and Submerged Bodies

The stability of a body, whether it is floating on the surface of a liquid or fully submerged within it, depends on the interplay between its weight and the buoyant force exerted by the liquid. Key points involved in determining stability are the locations of the body's centre of gravity ($\boldsymbol{G}$), the centre of buoyancy ($\boldsymbol{B}$), and for floating bodies, the metacenter ($\boldsymbol{M}$).

Let's define these important points:

• Centre of Gravity ($\boldsymbol{G}$): This is the point where the entire weight of the body is considered to act downwards. Its position depends on the distribution of mass within the body.
• Centre of Buoyancy ($\boldsymbol{B}$): This is the centroid of the displaced volume of liquid. The buoyant force, which is equal to the weight of the displaced liquid, acts upwards through this point. Its position depends on the shape of the submerged part of the body.
• Metacenter ($\boldsymbol{M}$): This point is relevant only for floating bodies. When a floating body is tilted by a small angle, the shape of the submerged volume changes, causing the centre of buoyancy to shift to a new position ($\boldsymbol{B'}$). The metacenter ($\boldsymbol{M}$) is the point where the vertical line passing through the new centre of buoyancy ($\boldsymbol{B'}$) intersects the original vertical line passing through the initial centre of buoyancy ($\boldsymbol{B}$) and centre of gravity ($\boldsymbol{G}$). For small angles of tilt, $\boldsymbol{M}$ is considered a fixed point relative to the body.

Stability Conditions for Submerged Bodies

A body that is completely submerged in a fluid has a constant volume of displaced fluid, regardless of its orientation (assuming the fluid is homogeneous and incompressible). Therefore, the position of the centre of buoyancy ($\boldsymbol{B}$) for a submerged body is fixed relative to the body's geometry and is the centroid of the body's entire volume. The position of the centre of gravity ($\boldsymbol{G}$) is also fixed relative to the body.

The stability of a submerged body depends solely on the relative positions of $\boldsymbol{G}$ and $\boldsymbol{B}$:

• Stable Equilibrium: If the centre of gravity ($\boldsymbol{G}$) is below the centre of buoyancy ($\boldsymbol{B}$), the body is in stable equilibrium. A slight tilt will cause a restoring moment that brings the body back to its original position. The buoyant force acts upwards through $\boldsymbol{B}$ and the weight acts downwards through $\boldsymbol{G}$. If $\boldsymbol{G}$ is below $\boldsymbol{B}$, these forces create a couple that rights the body.
• Unstable Equilibrium: If the centre of gravity ($\boldsymbol{G}$) is above the centre of buoyancy ($\boldsymbol{B}$), the body is in unstable equilibrium. A slight tilt will cause an overturning moment that causes the body to tilt further away from its original position.
• Neutral Equilibrium: If the centre of gravity ($\boldsymbol{G}$) coincides with the centre of buoyancy ($\boldsymbol{B}$), the body is in neutral equilibrium. The body will remain in any position to which it is displaced, as there is no moment created by the weight and buoyant forces.

Stability Conditions for Floating Bodies

For a floating body, the volume of displaced liquid changes when the body is tilted. This causes the centre of buoyancy ($\boldsymbol{B}$) to shift to a new position ($\boldsymbol{B'}$). The stability of a floating body is determined by the relative positions of the centre of gravity ($\boldsymbol{G}$) and the metacenter ($\boldsymbol{M}$).

The stability conditions for a floating body are:

• Stable Equilibrium: If the metacenter ($\boldsymbol{M}$) is above the centre of gravity ($\boldsymbol{G}$), the body is in stable equilibrium. A slight tilt creates a restoring moment. This happens when the line of action of the buoyant force in the tilted position passes above $\boldsymbol{G}$.
• Unstable Equilibrium: If the metacenter ($\boldsymbol{M}$) is below the centre of gravity ($\boldsymbol{G}$), the body is in unstable equilibrium. A slight tilt creates an overturning moment. This happens when the line of action of the buoyant force in the tilted position passes below $\boldsymbol{G}$.
• Neutral Equilibrium: If the metacenter ($\boldsymbol{M}$) coincides with the centre of gravity ($\boldsymbol{G}$), the body is in neutral equilibrium. However, for real floating bodies, $\boldsymbol{M}$ and $\boldsymbol{G}$ rarely coincide precisely.

Analyzing the Statements on Stability

Let's examine each given statement based on the principles of stability for floating and submerged bodies:

Statement 1: A submerged body is in unstable equilibrium if the centre of gravity is below the centre of buoyancy.

Based on our understanding of submerged bodies, unstable equilibrium occurs when the centre of gravity ($\boldsymbol{G}$) is above the centre of buoyancy ($\boldsymbol{B}$). Stable equilibrium occurs when $\boldsymbol{G}$ is below $\boldsymbol{B}$. Therefore, this statement is incorrect.

Statement 2: A floating body is in stable equilibrium if the centre of gravity is below the metacenter.

For a floating body, stable equilibrium occurs when the metacenter ($\boldsymbol{M}$) is above the centre of gravity ($\boldsymbol{G}$). This is the same as saying the centre of gravity ($\boldsymbol{G}$) is below the metacenter ($\boldsymbol{M}$). Therefore, this statement is correct.

Statement 3: A submerged body is in neutral equilibrium if the centre of gravity coincides with the centre of buoyancy.

As discussed for submerged bodies, neutral equilibrium occurs precisely when the centre of gravity ($\boldsymbol{G}$) and the centre of buoyancy ($\boldsymbol{B}$) occupy the same point. Therefore, this statement is correct.

Statement 4: A floating body is in unstable equilibrium if the centre of gravity is above the metacenter.

For a floating body, unstable equilibrium occurs when the metacenter ($\boldsymbol{M}$) is below the centre of gravity ($\boldsymbol{G}$). This is the same as saying the centre of gravity ($\boldsymbol{G}$) is above the metacenter ($\boldsymbol{M}$). Therefore, this statement is correct.

The question asks for the incorrect statement regarding the stability of floating and submerged bodies. Based on our analysis, Statement 1 is the incorrect statement.

Conclusion on Stability Statement

Comparing the statements with the summary table and detailed analysis, the statement "A submerged body is in unstable equilibrium if the centre of gravity is below the centre of buoyancy" is false. When the centre of gravity of a submerged body is below its centre of buoyancy, it is in stable equilibrium.

Revision Table: Key Stability Concepts

Additional Information on Fluid Stability

The concept of stability is crucial in naval architecture and engineering. For floating bodies, the distance between the centre of gravity ($\boldsymbol{G}$) and the metacenter ($\boldsymbol{M}$) is called the metacentric height ($\boldsymbol{GM}$).

• Metacentric Height ($\boldsymbol{GM}$): The vertical distance between the centre of gravity ($\boldsymbol{G}$) and the metacenter ($\boldsymbol{M}$).
• If $\boldsymbol{M}$ is above $\boldsymbol{G}$, $\boldsymbol{GM}$ is positive, and the floating body is stable.
• If $\boldsymbol{M}$ is below $\boldsymbol{G}$, $\boldsymbol{GM}$ is negative, and the floating body is unstable.
• If $\boldsymbol{M}$ coincides with $\boldsymbol{G}$, $\boldsymbol{GM}$ is zero, and the floating body is in neutral equilibrium.

The metacentric height is a direct measure of the initial static stability of a floating vessel. A larger positive metacentric height indicates greater initial stability. However, for large angles of tilt, the simple metacentric theory is not sufficient, and the righting moment curve analysis is required.

For submerged bodies, the concept of metacenter is not applicable in the same way because the centre of buoyancy $\boldsymbol{B}$ remains fixed relative to the body for any orientation, as the volume displaced is always equal to the body's volume.