Understanding the Centre of Buoyancy

The question asks about the definition of the centre of buoyancy. Understanding this concept is fundamental in fluid mechanics, particularly when dealing with floating or submerged bodies.

What is Buoyancy?

Buoyancy is the upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. This force arises because the pressure exerted by the fluid increases with depth. The force on the bottom surface of a submerged object is greater than the force on the top surface, resulting in a net upward force.

The Buoyant Force and Archimedes' Principle

Archimedes' principle states that the buoyant force on an object submerged or floating in a fluid is equal to the weight of the fluid displaced by the object. Mathematically, the buoyant force $F_B$ is given by:

$$F_B = \rho_f \cdot V_{disp} \cdot g$$

where:

• $\rho_f$ is the density of the fluid
• $V_{disp}$ is the volume of the displaced fluid
• $g$ is the acceleration due to gravity

This buoyant force acts vertically upwards through a specific point known as the centre of buoyancy.

Locating the Centre of Buoyancy

The centre of buoyancy (often denoted by B) is the point through which the buoyant force acts. Based on Archimedes' principle, this point corresponds to the centroid of the volume of the fluid that is displaced by the body. For a fluid of uniform density, the centroid of the displaced volume is also the centre of mass of the displaced fluid.

Analyzing the Options

Let's evaluate the given options in the context of the centre of buoyancy:

• Option 1: Centre of gravity of the body
  The centre of gravity (CG or G) is the point where the entire weight of the body is considered to act. This is different from the centre of buoyancy, which is related to the displaced fluid. The relative positions of the centre of gravity and the centre of buoyancy are crucial for the stability of a floating body, but they are generally not the same point.
• Option 2: Centre of mass of displaced fluid
  As discussed, the buoyant force is the weight of the displaced fluid acting upwards through the centroid of that displaced volume. For a fluid with uniform density, the centroid of the volume coincides with the centre of mass of that volume. Therefore, the centre of buoyancy is indeed the centre of mass of the displaced fluid.
• Option 3: Mid point between Centre of gravity and metacentre
  The metacentre is a concept related to the stability of floating bodies. It is the point of intersection of the line of action of the buoyant force when the body is slightly tilted and the original vertical line passing through the centre of buoyancy. The mid point between the centre of gravity and the metacentre does not represent the centre of buoyancy itself.
• Option 4: The point of intersection of buoyant force and centre line of the body
  The buoyant force acts through the centre of buoyancy. While the centre line might be relevant for some symmetrical bodies, the centre of buoyancy's location is determined solely by the geometry of the *displaced volume* and not necessarily by the overall centre line of the *body*, especially if the body is not symmetrical or is tilted. The definition is fundamentally tied to the displaced fluid volume.

Based on the definitions and principles of fluid mechanics, the correct description of the centre of buoyancy is the centre of mass of the displaced fluid.

Revision Table: Key Hydrostatic Points

Additional Information: Buoyancy and Stability

The position of the centre of buoyancy (B) relative to the centre of gravity (G) is critical for the stability of a floating body. When a body floats in equilibrium, the buoyant force $F_B$ is equal in magnitude to the weight of the body $W$, and both forces are collinear, acting through G and B respectively along the same vertical line.

If the body is tilted, the shape of the displaced volume changes, causing the centre of buoyancy B to shift its position. The buoyant force still acts vertically upwards through the new centre of buoyancy B'. The stability depends on whether this buoyant force creates a restoring moment (trying to return the body to its original position) or an overturning moment (trying to tilt it further).

For floating bodies, the concept of the metacentre (M) is used to determine stability. If the metacentre M is above the centre of gravity G, the body is stable. If M is below G, it is unstable. For submerged bodies, stability depends simply on whether G is below B; if G is below B, it's stable; if G is above B, it's unstable.