Q1: A floating body oscillates about its __________ when it is displaced slightly.

ANS:B - metacentre

The metacentre (M) is a critical concept in the stability analysis of floating bodies, particularly ships, boats, and other floating structures. It plays a fundamental role in determining the stability characteristics of a floating body when it is subjected to external forces or disturbances. Below is a detailed explanation of the metacentre:

1. Definition: The metacentre is a theoretical point located above the centre of gravity (G) of a floating body. It is the point at which the vertical line passing through the buoyant force intersects the original vertical line passing through the body's centre of gravity before any displacement occurs.
2. Importance: The metacentre is crucial for assessing the stability of a floating body. It determines how the body behaves when subjected to external forces such as wind, waves, or changes in weight distribution. Understanding the position of the metacentre relative to the centre of gravity is essential for ensuring the safety and stability of floating structures.
3. Stability Analysis: The stability of a floating body depends on the relationship between its centre of gravity (G) and the metacentre (M). Specifically:
   • If the metacentre (M) is above the centre of gravity (G), the body is said to be in a stable equilibrium. This means that when the body is tilted or disturbed from its equilibrium position, a restoring moment is generated that tends to return it to its original position.
   • If the metacentre (M) is below the centre of gravity (G), the body is in an unstable equilibrium. In this case, any tilt or disturbance causes the body to move further away from its equilibrium position, potentially leading to capsizing or loss of stability.
4. Calculation: The position of the metacentre (M) can be determined through mathematical analysis based on the geometry and characteristics of the floating body. For simple shapes and uniform density distributions, the metacentre can be calculated using principles of geometry and fluid mechanics. For more complex structures, computational methods or physical experiments may be required to determine the metacentric height and position.
5. Design Considerations: Engineers and naval architects consider the position of the metacentre during the design of ships and other floating structures. By ensuring that the metacentre is located above the centre of gravity, they can design vessels that are inherently stable and resistant to capsizing under normal operating conditions.

In summary, the metacentre is a critical parameter in the stability analysis and design of floating bodies. Its position relative to the centre of gravity determines the stability characteristics of the body and influences its behavior when subjected to external forces. Understanding and controlling the position of the metacentre is essential for ensuring the safety, stability, and seaworthiness of ships, boats, and other floating structures.