Question

A solid body of uniform specific gravity floats in a deep liquid pool. Take \( B \), \( G \), and \( M \) as the centre of buoyancy, centre of gravity, and metacentre of the body, respectively. Which one of the following options is correct for the stable floatation of the body in the pool when the body is given a small tilt angle?

• A. \( MG \) is the metacentric height and \( G \) should lie below \( M \)
• B. \( MG \) is the metacentric height and \( G \) should lie above \( M \)
• C. \( MB \) is the metacentric height and \( B \) should lie below \( M \)
• D. \( MB \) is the metacentric height and \( G \) should lie above \( M \)

Hint:

For stable floatation, the metacentre must lie above the center of gravity. If \( G \) is below \( M \), the body will be stable when tilted. If \( G \) lies above \( M \), the body will be unstable.

Answer 1

The Correct Option is A

In this problem, we are dealing with the concept of flotation stability. The stability of a floating body depends on the relationship between the center of gravity (\( G \)), the center of buoyancy (\( B \)), and the metacentre (\( M \)). Let’s break down the important concepts:

1. Metacentre (\( M \)):
The metacentre is a point where the buoyant force (the upward force exerted by the fluid on the body) acts when the body is slightly tilted. The metacentre is the point of intersection of the vertical line through the center of buoyancy \( B \) when the body is tilted, with the axis of symmetry of the body in its equilibrium position.

2. Centre of Gravity (\( G \)):
The center of gravity is the point where the total weight of the body can be considered to act. It is the point where the force of gravity is effectively concentrated.

3. Centre of Buoyancy (\( B \)):
The center of buoyancy is the point where the buoyant force (or upward force) acts. This point is the centroid of the displaced fluid volume. It depends on the shape and size of the object submerged in the fluid.

Condition for Stability:
- For the body to be stable, the metacentre \( M \) must lie above the center of gravity \( G \). This condition ensures that when the body is tilted slightly, the buoyant force will create a restoring moment to return the body to its original position.

- The metacentric height \( MG \) is the distance between the metacentre \( M \) and the center of gravity \( G \). A higher metacentric height results in greater stability.

- Restoring Moment: When the body is tilted, the buoyant force must create a moment (a rotational force) that acts to restore the body to its upright position. This happens when the metacentre \( M \) lies above the center of gravity \( G \).

Conclusion:
For a floating body to return to its equilibrium position after being tilted, the metacentre \( M \) must be above the center of gravity \( G \), and the restoring moment must oppose the tilting force. Therefore, the center of gravity \( G \) should lie below the metacentre \( M \).

Thus, the correct option is:

\[ \boxed{(A) \, MG \, \text{is the metacentric height and} \, G \, \text{should lie below} \, M} \]