Question

For stable equilibrium of floating bodies, the centre of gravity has to:

• A. Coincide with metacentre
• B. Be always above the metacentre
• C. Be always below the metacentre
• D. Be always below the centre of buoyancy

Hint:

Remember the mnemonic: "M above G for stability" in floating bodies. The metacentre (M) must be higher than the centre of gravity (G) for the equilibrium to be stable.

Answer 1

The Correct Option is C

Step 1: Understand the concepts of centre of gravity, centre of buoyancy, and metacentre.
Centre of Gravity (G): The point at which the entire weight of the body is assumed to act.
Centre of Buoyancy (B): The centre of gravity of the displaced fluid. The buoyant force acts vertically upwards through this point.
Metacentre (M): The point of intersection of the vertical line passing through the centre of buoyancy of a slightly displaced body and the original vertical line passing through the centre of gravity and the centre of buoyancy of the body in the equilibrium position. Step 2: Analyze the conditions for stable equilibrium of a floating body.
A floating body is said to be in stable equilibrium if, when given a small angular displacement, it tends to return to its original equilibrium position. This stability depends on the relative positions of the centre of gravity (G) and the metacentre (M). Step 3: Explain the restoring moment in stable equilibrium.
When a floating body is slightly tilted, the centre of buoyancy shifts to a new position (B'). For stable equilibrium, the buoyant force acting upwards through B' and the weight acting downwards through G must create a restoring moment that opposes the tilt and tends to bring the body back to its original position. This restoring moment occurs when the metacentre (M) is above the centre of gravity (G). In this case, the buoyant force and the weight form a couple that acts in the opposite direction to the displacement. Step 4: Explain the conditions for unstable and neutral equilibrium.
Unstable Equilibrium: If the metacentre (M) is below the centre of gravity (G), a small angular displacement will result in an overturning moment that further tilts the body away from its original position.
Neutral Equilibrium: If the metacentre (M) coincides with the centre of gravity (G), a small angular displacement will result in no restoring or overturning moment, and the body will remain in its new position.
Step 5: Evaluate the given options based on the conditions for stable equilibrium.
Option 1 (Coincide with metacentre): This corresponds to neutral equilibrium, not stable equilibrium.
Option 2 (Be always above the metacentre): For stable equilibrium, the metacentre must be above the centre of gravity.
Option 3 (Be always below the metacentre): This is the condition for stable equilibrium.
Option 4 (Be always below the centre of buoyancy): The relative position of the centre of gravity and the centre of buoyancy affects the initial equilibrium, but stability is determined by the position of the metacentre relative to the centre of gravity.
Step 6: Select the correct answer.
For stable equilibrium of floating bodies, the centre of gravity has to be always below the metacentre.