Submerged Body Equilibrium in Fluids

When a body is completely submerged in a fluid, its stability depends on the relative positions of two important points: the center of gravity (G) and the center of buoyancy (O).

Understanding Center of Gravity (G) and Center of Buoyancy (O)

• Center of Gravity (G): This is the point where the entire weight of the body is considered to act downwards. It is determined by the mass distribution of the body itself.
• Center of Buoyancy (O): This is the centroid of the volume of the fluid displaced by the submerged body. The buoyant force, which acts upwards, is considered to pass through this point. For a completely submerged body, the volume of displaced fluid is equal to the body's own volume.

Equilibrium Conditions for Subpletely Submerged Bodies

The stability of a completely submerged body is determined by how it responds to a small angular disturbance. There are three main types of equilibrium:

• Stable Equilibrium: If, upon a small tilt, the body tends to return to its original position.
• Unstable Equilibrium: If, upon a small tilt, the body tends to move further away from its original position, leading to overturning.
• Neutral Equilibrium: If, upon a small tilt, the body remains in the new tilted position.

Stable Equilibrium for a Completely Submerged Body

For a body that is completely submerged in a fluid, the condition for stable equilibrium is specifically related to the positions of its center of gravity (G) and center of buoyancy (O). Unlike floating bodies where the metacenter plays a role, for fully submerged bodies, stability is simpler:

• A completely submerged body is in stable equilibrium if its center of buoyancy (O) lies above its center of gravity (G).

Explanation: When the body is tilted slightly, the weight of the body acts downwards through G, and the buoyant force acts upwards through O. If O is above G, these two forces will form a "restoring couple" that tends to rotate the body back to its original, upright position. This restoring action ensures the body's stability.

Analyzing Other Equilibrium Conditions for Submerged Bodies

• Unstable Equilibrium: If the center of buoyancy (O) lies below the center of gravity (G). In this case, a small tilt will create an "overturning couple" that further increases the tilt, leading to the body overturning.
• Neutral Equilibrium: If the center of buoyancy (O) coincides with the center of gravity (G). When tilted, the lines of action of the weight and buoyant force remain aligned, and no restoring or overturning couple is produced. The body will simply stay in the new tilted position.

Evaluating the Given Options

Let's examine each option based on the principles of equilibrium for completely submerged bodies:

• Option 1: O does not coincide with the centre of mass of the displaced fluid
  This statement is incorrect. The center of buoyancy (O) is, by definition, the center of mass (or centroid) of the displaced fluid volume. Therefore, O always coincides with the center of mass of the displaced fluid.
• Option 2: G coincides with the centre of mass of the displaced fluid
  This statement is also incorrect. G is the center of gravity of the body itself, while O is the center of mass of the displaced fluid. For neutral equilibrium, O and G coincide, but G is not generally the center of mass of the displaced fluid.
• Option 3: O lies below G
  This condition leads to unstable equilibrium. As explained, if O is below G, any disturbance will cause an overturning moment.
• Option 4: O lies above G
  This is the correct condition for a completely submerged body to be in stable equilibrium. When O is above G, a restoring moment is created upon disturbance, bringing the body back to its original orientation.

Therefore, for a body completely submerged in a fluid to be in stable equilibrium, the center of buoyancy (O) must lie above the center of gravity (G).