Whether an unsupported object rises, sinks, or remains suspended in a fluid is determined by the competition between its weight and the buoyant force.

For a rigid body of mass $M$ and total volume $V$ completely immersed in a fluid of density $\rho_f$,

$$F_B=\rho_f gV,$$

while its weight is

$$W=Mg.$$

Define the object's average density by

$$\bar\rho=\frac{M}{V}.$$

Then the net vertical force while fully immersed is

$$F_B-W=(\rho_f-\bar\rho)gV.$$

Therefore, ignoring other forces such as drag or contact forces:

• if $\bar\rho>\rho_f$, the net force is downward and the object tends to sink;
• if $\bar\rho<\rho_f$, the net force is upward and the object tends to rise;
• if $\bar\rho=\rho_f$, the object is neutrally buoyant and can remain fully submerged without a net buoyant acceleration.

Floating at a free surface

An object less dense than the fluid rises until only enough of it remains submerged for buoyancy to balance its weight. At floating equilibrium,

$$Mg=\rho_fgV_{\rm disp}.$$

Since $M=\bar\rho V$,

$$\bar\rho Vg=\rho_fV_{\rm disp}g,$$

so

$$\boxed{\frac{V_{\rm disp}}{V}=\frac{\bar\rho}{\rho_f}}.$$

The fraction of the object's volume below the surface is therefore set by the ratio of its average density to the fluid density.

Example: an iceberg

If ice has average density

$$\rho_{\rm ice}=920,\mathrm{kg/m^3}$$

and seawater is approximated here as

$$\rho_f=1000,\mathrm{kg/m^3},$$

then

$$\frac{V_{\rm disp}}{V}=\frac{920}{1000}=0.92.$$

About $92%$ of the iceberg's volume is submerged in this simplified model. Only the remaining $8%$ lies above the waterline.

Why a heavy boat can float

Floating is controlled by average density, not by total mass alone. A steel hull contains a large volume of air, so the mass of the entire boat divided by the volume enclosed by its outer shape can be less than the density of water. The boat then reaches equilibrium after displacing a volume of water whose weight equals the boat's total weight.

Adding cargo increases $M$. The boat must then displace more water, so it settles deeper until the larger buoyant force again balances its weight. If the required displaced volume would exceed the hull's available buoyant volume, the boat can no longer maintain that floating equilibrium.

Buoyancy in gases

Archimedes' principle applies to gases as well as liquids. A balloon of external volume $V$ in air experiences approximately

$$F_B=\rho_{\rm air}gV.$$

To rise from rest, this buoyant force must exceed the total weight of the gas, envelope, payload, and everything else carried by the balloon:

$$\rho_{\rm air}V>M_{\rm total}.$$

Using a gas less dense than air helps, but low gas density alone is not sufficient: the mass of the envelope and payload also matters.

Floating, sinking, and neutral buoyancy are therefore different outcomes of the same force balance. Archimedes' principle supplies the buoyant force; the object's average density and constraints determine the resulting motion or equilibrium.