Predict floating fraction and neutral buoyancy

Find the volume that balances weight

A fully submerged cube of density 500 kg/m³ in liquid of density 1000 kg/m³ has 10 N buoyancy and only 5 N weight. Predict what must change for a possible floating equilibrium. Weight does not disappear when an object floats, and buoyancy need not remain at its fully submerged value.

As some of the cube emerges, it displaces less liquid. Its buoyancy decreases. Vertical equilibrium requires ρ_liquid gV_submerged = mg. For a rigid object of average density ρ_object and total volume V_object, mass is ρ_object V_object. Canceling g and dividing by total volume gives V_submerged/V_object = ρ_object/ρ_liquid.

This dimensionless ratio is the submerged volume fraction. The equation is an equilibrium requirement. Check that its result is physically possible before treating it as a floating configuration. The OpenStax buoyancy reference connects the displaced-fluid condition to floating fractions.

Calculate the default floating geometry

At object density 500 kg/m³ and liquid density 1000 kg/m³, the required fraction is 500/1000 = 0.5, or 50%. The one-litre cube displaces 0.5 L, equivalent to 0.0005 m³. Buoyancy at that configuration is 1000 × 10 × 0.0005 = 5 N, equal to the 5 N weight.

Because this cube has a constant horizontal cross-section and a fixed orientation, its submerged height fraction equals its submerged volume fraction. Half of its 0.1 m height is below the surface. For an irregular shape, a 50% submerged volume need not mean 50% of its height.

The diagram shows this possible vertical equilibrium. The separate force bars still show the fully submerged test, where buoyancy was 10 N. Compare the ledger’s two buoyancy entries to avoid mistaking different configurations for inconsistent calculations. No animation or timing claims to show how the cube settles from one to the other.

Change the surrounding liquid

Keep object density at 500 kg/m³ and increase liquid density to 1500 kg/m³. The required fraction is 1/3, or about 33.3%. The object displaces about 0.333 L and its equilibrium buoyancy is still 5 N because its weight has not changed. The denser liquid supports that weight with less displaced volume.

The fully submerged buoyancy would be 15 N, which is a different question. A floating object’s buoyancy does not increase beyond its weight merely because the liquid becomes denser; its equilibrium displaced volume changes instead. Capture both configurations and explain which quantity remains fixed.

Average density includes all mass and the sealed external volume. A hollow vessel can have low average density even when its wall material is dense. The present model changes average density directly; it does not calculate hull structure, flooding or loading capacity. For an open or leaking object, the volume that excludes liquid needs a separate analysis.

Treat equal density as neutral suspension

Set object and liquid densities both to 1000 kg/m³. The one-litre cube weighs 10 N and fully submerged buoyancy is also 10 N. The required fraction is 100%, and the vertical hydrostatic force difference is zero.

This is neutral buoyancy while fully submerged. The model does not select a unique depth: moving the unchanged cube to another fully submerged depth in the same uniform liquid leaves the balance unchanged. The drawn depth is illustrative. Zero net static force is not evidence of a restoring force toward that particular depth, nor a claim that an already moving object instantly stops.

If the cube partly emerges, it displaces less than its full volume, so buoyancy becomes less than weight under the neglected-air-buoyancy assumption. This is why equal density is distinct from the partially emerged floating configurations of lower-density objects.

Reject an impossible floating fraction

Set object density to 1500 kg/m³ and liquid density to 1000 kg/m³. The equilibrium equation demands a submerged fraction of 1.5, or a displaced volume of 1.5 L. But the sealed object has only 1 L available. It cannot displace 150% of itself in this liquid.

The model reports no unsupported floating equilibrium. When fully submerged, its 15 N weight exceeds the 10 N buoyancy. It has a downward tendency while still receiving buoyant support. The diagram is an initial submerged configuration, not an invented final resting depth. A floor could supply an additional normal force, but floor contact is outside this experiment.

Distinguish force balance from stability and settling

The floating equation tests vertical force balance for an assumed orientation. It does not establish whether the cube will tip after a disturbance. Rotational stability depends on the locations of the center of mass and the buoyant-force line of action as orientation changes. This model holds orientation fixed and makes no stability claim.

Likewise, a possible equilibrium is not a simulated final state. Settling depends on motion, fluid resistance and other effects not included here. A notebook should label its conclusion as a predicted equilibrium geometry and state the constant-volume, uniform-liquid assumptions.

Transfer to a measured fraction

A sealed object floats with 60% of its volume submerged in liquid of density 1250 kg/m³. Infer its average density. If moved to liquid of density 1000 kg/m³ without changing its mass or external volume, predict its new submerged fraction. State whether the first observation alone establishes its mass.

Check the floating fraction transfer

Average object density is 0.60 × 1250 = 750 kg/m³. In the second liquid, the required fraction is 750/1000 = 0.75, or 75%. The observation determines density under the stated assumptions, not mass by itself; the object's total volume is also needed to calculate mass.