Vertical Stability and Rotational Stability

When analyzing the stability of a floating body, there are two key aspects to consider: vertical stability and rotational stability.

Vertical Stability:
Vertical stability refers to the ability of the body to maintain its equilibrium in the vertical direction. In other words, it determines whether the body will return to its original position or not when it is displaced vertically.

Rotational Stability:
Rotational stability, on the other hand, pertains to the ability of the body to maintain its equilibrium when it is rotated about a vertical axis. It determines whether the body will tend to return to its original position or not when it is rotated.

Centre of Gravity (G):
The center of gravity of a body is the point through which the entire weight of the body acts vertically downwards. It is the point where the force of gravity can be considered to act.

Centre of Buoyancy (B):
The center of buoyancy is the point through which the buoyant force (a vertical upward force) due to the surrounding fluid acts on the body. It is the point where the upward force of buoyancy can be considered to act.

Metacenter (M):
The metacenter is a point that determines the rotational stability of a floating body. It is the point of intersection of the lines passing through the center of buoyancy in two different equilibrium positions of the body.

Analysis of the Floating Body:
In the given figure, the body is floating with its center of gravity (G) below its center of buoyancy (B). This configuration indicates that the body is in an unstable equilibrium.

When the body is slightly displaced vertically, the center of buoyancy shifts accordingly, creating a moment about the center of gravity (G). This moment causes the body to rotate further in the direction of the displacement, increasing the displacement even more. Therefore, the body is vertically unstable.

Additionally, the metacenter (M) is also located above the center of gravity (G), further confirming the rotational instability of the body. When the body is rotated slightly about a vertical axis, the metacenter creates a moment that reinforces the rotation, causing the body to rotate even more.

Therefore, based on the relative positions of the center of gravity (G), center of buoyancy (B), and metacenter (M), the body is both vertically unstable and rotationally unstable.