A body floats when its weight equals upthrust, so Vρg = Vinσg. Hence a body will float in a liquid if and only if ρ ≤ σ.

• If ρ > σ, the body sinks because its weight exceeds upthrust.
• If ρ = σ, the body floats fully submerged in neutral equilibrium.
• The trap is to treat every floating body as partially immersed, which is false when densities are equal.

After a small tilt, the new vertical through the shifted centre of buoyancy meets the original vertical at the metacentre M. Stable equilibrium requires M above the centre of gravity G, neutral equilibrium requires M at G, and unstable equilibrium occurs when M lies below G.

• A floating body can satisfy translatory equilibrium and still be rotationally unstable.
• Raising the centre of gravity makes toppling easier.
• Students often confuse centre of buoyancy with metacentre; they are not the same point.

From Vρ = Vinσ, the fraction of volume outside the liquid is fout = 1 - ρ/σ. For a floating platform, an added mass m causing extra depression y obeys mg = Aσyg.

• The same body floats deeper in a less dense liquid because Vin must increase.
• Comparing submerged fractions directly compares body and fluid densities.
• The trap is to use total platform height instead of the additional depression caused by the added load.