Floating stability – condition for neutral equilibrium using the metacentre A floating body is said to be in neutral equilibrium when which of the following conditions about the metacentre (M) relative to the centre of gravity (G) holds?

Introduction / Context:
Stability of floating bodies is assessed by the relative positions of the metacentre (M), the centre of buoyancy (B), and the centre of gravity (G). Depending on these positions, the body may be stable, unstable, or in neutral equilibrium for small angles of heel.

Given Data / Assumptions:

• Small angular displacements (initial stability analysis).
• Homogeneous fluid; hydrostatic conditions.
• The definitions of M, B, and G are standard.

Concept / Approach:
Initial stability is determined by metacentric height GM = distance between G and M. If GM > 0 (M above G), the body is stable. If GM < 0 (M below G), it is unstable. Neutral equilibrium corresponds to GM = 0, meaning M and G coincide so that the righting and overturning moments vanish for small inclinations.

Step-by-Step Solution:
1) Consider a small heel angle: buoyancy shifts, locating a new B and defining M as the intersection of buoyancy normals.2) Compute GM; sign indicates stability type.3) GM = 0 ⇔ M coincides with G.4) Therefore, neutral equilibrium condition is M and G coincident.

Verification / Alternative check:
In neutral equilibrium, no restoring couple acts for small heels; experimentally, a body neither rights itself nor continues to overturn when slightly disturbed, matching GM = 0.

Why Other Options Are Wrong:

• M above G: stable equilibrium (not neutral).
• M below G: unstable equilibrium.
• M between B and G (not coincident): implies nonzero GM, yielding stability or instability depending on ordering.
• M at water surface: incorrect generalization; M depends on geometry and heel.

Common Pitfalls:
Confusing centre of buoyancy B with metacentre M; B always lies below the waterline, while M may lie above or below G depending on geometry.

Final Answer:
M coincides with G