3.3.3.2. Floating and Sinking

Table of Contents

Why Objects Float or Sink

Floating and sinking are determined by a comparison between two effects acting on an object in a fluid. One is the object's weight, which pulls downward. The other is the buoyant force, which pushes upward. The buoyant force itself is introduced in the parent topic on Archimedes' principle, so here we focus on how that principle tells us whether an object rises, stays at a level, or sinks.

If the buoyant force is greater than the object's weight, the object rises. If the buoyant force is less than the object's weight, the object sinks. If the two are equal, the object is in vertical equilibrium.

An object in a fluid behaves as follows:
If $F_B > W$, it rises.
If $F_B < W$, it sinks.
If $F_B = W$, it floats in equilibrium or remains suspended.

Role of Density

The most useful idea for predicting floating and sinking is density. Density is mass per unit volume, written as

$$
\rho = \frac{m}{V}
$$

For an object completely submerged in a fluid, its weight is

$$
W = mg = \rho_{\text{object}} V g
$$

The buoyant force equals the weight of the displaced fluid,

$$
F_B = \rho_{\text{fluid}} V g
$$

when the whole object is under the fluid. Comparing these two gives a simple rule. If the object's density is greater than the fluid's density, it sinks. If the object's density is less, it rises. If the densities are equal, it can remain suspended.

For a completely submerged object:
$$
F_B = \rho_{\text{fluid}} V g
$$
$$
W = \rho_{\text{object}} V g
$$
Therefore,
If $\rho_{\text{object}} > \rho_{\text{fluid}}$, the object sinks.
If $\rho_{\text{object}} < \rho_{\text{fluid}}$, the object rises.
If $\rho_{\text{object}} = \rho_{\text{fluid}}$, the object is neutrally buoyant.

Floating at the Surface

Many objects do not stay completely submerged. A piece of wood placed in water usually rises until only part of it remains below the surface. At that point, the buoyant force exactly balances the weight.

For a floating object, the fluid displaced is only the submerged part of the object. If $V_{\text{sub}}$ is the submerged volume, then

$$
F_B = \rho_{\text{fluid}} V_{\text{sub}} g
$$

and equilibrium requires

$$
\rho_{\text{fluid}} V_{\text{sub}} g = mg
$$

If the total object volume is $V$, and the object density is $\rho_{\text{object}}$, then $m = \rho_{\text{object}} V$, so

$$
\rho_{\text{fluid}} V_{\text{sub}} g = \rho_{\text{object}} V g
$$

which gives

$$
\frac{V_{\text{sub}}}{V} = \frac{\rho_{\text{object}}}{\rho_{\text{fluid}}}
$$

This tells us what fraction of the object is below the surface.

For an object floating at the surface,
$$
\frac{V_{\text{sub}}}{V} = \frac{\rho_{\text{object}}}{\rho_{\text{fluid}}}
$$
The fraction submerged equals the ratio of object density to fluid density.

Examples

A block with density $800\ \text{kg/m}^3$ placed in water, with density about $1000\ \text{kg/m}^3$, will float because its density is smaller than the fluid density. The fraction submerged is

$$
\frac{V_{\text{sub}}}{V} = \frac{800}{1000} = 0.8
$$

So about 80 percent of the block is below the water surface.

A steel ball with density much greater than water sinks because its weight is larger than the maximum buoyant force that water can provide when the ball is fully submerged.

A fish can adjust its average density and become nearly neutrally buoyant, meaning it can stay underwater without strongly rising or sinking.

Average Density Matters

Some large objects made of dense materials can still float. A steel ship is the classic example. Steel itself is denser than water, but the ship contains a large volume of air. The total mass divided by the total outer volume gives an average density that can be less than the density of water.

This means the important quantity is often not the density of the material alone, but the average density of the whole object.

Object	Material density compared to water	Average density of whole object	Result
Solid wood block	Less	Less	Floats
Solid steel ball	Greater	Greater	Sinks
Steel ship	Greater	Less	Floats
Fish with neutral buoyancy	Varies	Equal	Suspended

Neutral Buoyancy

Neutral buoyancy occurs when the weight exactly equals the buoyant force while the object is fully submerged. In that case, there is no net vertical force. The object neither rises nor sinks, as long as conditions remain unchanged.

Submarines use this principle by adjusting how much water is in their ballast tanks. By changing their average density, they can sink, float upward, or remain suspended at a chosen depth.

Floating Higher or Lower in Different Fluids

An object may float in one fluid and sink in another. Salt water is denser than fresh water, so the same object floats higher in salt water. That happens because a smaller submerged volume is needed to displace enough fluid weight to balance the object.

If $\rho_{\text{fluid}}$ increases, then for the same floating object,

$$
\frac{V_{\text{sub}}}{V} = \frac{\rho_{\text{object}}}{\rho_{\text{fluid}}}
$$

becomes smaller. So less of the object needs to be underwater.

Visualizing Floating and Sinking

Rendering scientific sketch…

Floating and sinking in a fluid

Common Misunderstanding

A heavy object does not always sink just because it is heavy. What matters is whether the buoyant force can balance its weight. A very massive ship can float, while a small pebble can sink. The key comparison is density and displaced fluid, not weight alone.

Floating and sinking are not decided by mass alone.
They are decided by the balance between weight and buoyant force, or equivalently by comparing average density with fluid density.

Summary Relation

The whole chapter can be summarized by one central idea. A fluid can support an object if the object can displace enough fluid so that the weight of the displaced fluid matches the object's weight. If this can happen before the object is fully submerged, it floats. If even full submersion is not enough, it sinks. If balance happens while fully submerged, the object is neutrally buoyant.

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