Floatation and sinking

Drop a steel nail into a bucket of water and it sinks straight to the bottom in an instant. Yet a steel ship, thousands of times heavier, sails across an ocean without a worry. Both objects are made of the same dense metal, pulled down by the same gravity, so what decides which one floats and which one sinks? The answer is a quiet upward push every liquid gives to anything placed inside it, called buoyancy, combined with a simple balance called the law of floatation. This article builds on the density and relative density you already know and shows exactly how that balance decides floating or sinking, why a hollow ship beats a solid block, and how the same idea explains icebergs, submarines and hydrometers.

The core insight

A body floats when the upward buoyant force from the liquid it displaces exactly balances its own weight, which happens whenever the body's average density is less than or equal to the liquid's density.

The core idea

Buoyancy is a push, not a preference. Every object placed in a liquid feels an upward force from the liquid surrounding it, called buoyancy or upthrust. This is why a bucket full of water feels noticeably lighter to lift while it is still submerged than the instant it clears the surface — the water was pushing up on it the whole time.

The law of floatation is a statement of balance. In words: a body floats in a liquid when the weight of the liquid it displaces equals the weight of the body itself. Floating is therefore not a special property of light objects — it is what happens whenever the downward pull of gravity on a body is exactly cancelled by the upward buoyant force from the liquid it pushes aside.

Density comparison predicts the outcome. Whether an object floats, sinks, or hovers depends on comparing its own overall (average) density with the liquid's density.

ρbody<ρliquid  ⟹  floats (partly submerged)\rho_{\text{body}} < \rho_{\text{liquid}} \implies \text{floats (partly submerged)}
ρbody=ρliquid  ⟹  floats fully submerged, neither rising nor sinking\rho_{\text{body}} = \rho_{\text{liquid}} \implies \text{floats fully submerged, neither rising nor sinking}
ρbody>ρliquid  ⟹  sinks\rho_{\text{body}} > \rho_{\text{liquid}} \implies \text{sinks}

where: ρbody\rho_{\text{body}} is the mass of the object divided by its total volume (its average density, including any hollow space), and ρliquid\rho_{\text{liquid}} is the density of the liquid it is placed in.

Shape changes average density without changing the material. A solid lump of steel is far denser than water and sinks. A steel ship made of the same metal floats, because its hull encloses a large volume of air. Spread the same mass of steel over a much bigger hollow volume and the average density of the whole ship — steel plus enclosed air — drops below that of water, even though every plate of steel in it is still far denser than water on its own.

Figure
A boat hull floats at a waterline with the submerged part shaded and labelled displaced water while a solid block of the same material rests sunk at the bottom
Same steel, different shape, different fate waterline hollow steel hull floats shaded region = water displaced weight of that water = weight of hull solid steel block sinks no enclosed air, so its average density stays above water's

Applications extend the same balance. A submarine changes its own average density on purpose, filling ballast tanks with water to sink and emptying them with compressed air to rise, so it can float, sink, or hover at will. An iceberg floats because ice is slightly less dense than liquid water, hiding most of its bulk underwater. A hydrometer is a weighted, sealed tube that floats higher in denser liquids, letting its calibrated stem read off relative density directly. A hot-air or gas balloon floats in air for the identical reason a ship floats in water: the gas inside is less dense than the surrounding air, so the balloon's average density is less than air's.

Derivation AWhat fraction of a floating body stays underwater

The goal is to turn the law of floatation — the weight of displaced liquid equals the weight of the body — into a working ratio that tells you exactly what fraction of a floating body's volume lies below the surface, using only the definition of density and basic algebra, with no calculus and no Archimedes' Principle beyond the plain word statement already given.

Step 1. Write the law of floatation as an equation of weights. A floating body of total volume VtotalV_{\text{total}} and average density ρbody\rho_{\text{body}} has weight ρbodyVtotalg\rho_{\text{body}} V_{\text{total}} g. It sinks in just enough to displace a volume VsubV_{\text{sub}} of liquid of density ρliquid\rho_{\text{liquid}}, and that displaced liquid has weight ρliquidVsubg\rho_{\text{liquid}} V_{\text{sub}} g. The law of floatation says these two weights are equal:

ρbody Vtotal g=ρliquid Vsub g\rho_{\text{body}}\, V_{\text{total}}\, g = \rho_{\text{liquid}}\, V_{\text{sub}}\, g

Step 2. Cancel gg from both sides, since it appears identically on each side and the balance holds regardless of its value. This leaves a purely density-and-volume statement:

ρbody Vtotal=ρliquid Vsub\rho_{\text{body}}\, V_{\text{total}} = \rho_{\text{liquid}}\, V_{\text{sub}}

Step 3. Rearrange to isolate the fraction of the body that is submerged, Vsub/VtotalV_{\text{sub}}/V_{\text{total}}, by dividing both sides by ρliquidVtotal\rho_{\text{liquid}} V_{\text{total}}:

VsubVtotal=ρbodyρliquid\frac{V_{\text{sub}}}{V_{\text{total}}} = \frac{\rho_{\text{body}}}{\rho_{\text{liquid}}}

Step 4. Read the result. The fraction of a floating body's volume that stays below the surface equals the ratio of its own density to the liquid's density. If ρbody\rho_{\text{body}} is much smaller than ρliquid\rho_{\text{liquid}}, only a small fraction is submerged and the body rides high; as ρbody\rho_{\text{body}} approaches ρliquid\rho_{\text{liquid}}, the fraction approaches 1 and the body floats almost fully submerged, matching the boundary case where equal densities give suspension anywhere in the liquid.

Worked example

A block of wood has a mass of 600 g600\ \text{g} and a volume of 800 cm3800\ \text{cm}^3, floating in water of density 1 g/cm31\ \text{g/cm}^3. Find what fraction, and what volume, of the block stays below the water surface.

Step 1. Find the average density of the wood block. \rho_{\text{wood}} = \frac{m}{V} = \frac{600\ \text{g}}{800\ \text{cm}^3} = 0.75\ \text{g/cm}^3 Since 0.75 g/cm30.75\ \text{g/cm}^3 is less than water's 1 g/cm31\ \text{g/cm}^3, the block floats, confirming the earlier result.

Step 2. Apply the submerged-fraction relation derived above. \frac{V_{\text{sub}}}{V_{\text{total}}} = \frac{\rho_{\text{wood}}}{\rho_{\text{water}}} = \frac{0.75}{1} = 0.75 So 75%75\% of the block's volume lies below the waterline.

Step 3. Convert that fraction into an actual submerged volume. V_{\text{sub}} = 0.75 \times 800\ \text{cm}^3 = 600\ \text{cm}^3 The wood block floats with 600 cm³ (75%) of its 800 cm³ submerged, and the remaining 200 cm³ stays above the surface.

Competitive stretch

Real icebergs float in seawater of density about 1.025 g/cm31.025\ \text{g/cm}^3, while glacial ice has a density of about 0.92 g/cm30.92\ \text{g/cm}^3. Using the same relation, Vsub/Vtotal=0.92/1.025≈0.90V_{\text{sub}}/V_{\text{total}} = 0.92/1.025 \approx 0.90. So roughly 90%90\% of an iceberg's bulk hides underwater, with only about a tenth showing above the surface — the origin of the phrase "just the tip of the iceberg."

Common traps

  • Thinking "heavy things always sink." What matters is the object's average density compared with the liquid's density, not its absolute weight — a huge steel ship floats, but a small steel pin sinks.
  • Forgetting a floating object displaces liquid equal to its own weight, not its own volume. A floating body usually displaces only part of its volume, exactly enough to match its weight, as the wood block example shows.
  • Believing ice is denser than water because it is a solid. Water is the exception: ice is less dense than liquid water, which is why icebergs float and ponds freeze from the top down.
  • Mixing up buoyancy with floatation. Buoyancy (upthrust) is the upward force every immersed object feels, whether it floats or sinks; floatation is the specific balanced condition where that upthrust equals the object's weight.

Ask the tutor: if a float-or-sink prediction, a submerged-fraction calculation, or the ship-versus-block reasoning still feels shaky, ask the AI tutor to walk through another worked density comparison with you.

Why does a submarine sink and rise on command if the law of floatation is fixed

The law of floatation is not fixed for a submarine, because a submarine can change its own average density. Its ballast tanks are flooded with water to raise the sub's average density above the surrounding seawater's, so it sinks; compressed air then blows the water back out, lowering the average density until it is less than seawater's again, so it rises. By adjusting exactly how much water sits in the tanks, a submarine can also balance its average density to equal seawater's and hover at a fixed depth, submerged but neither sinking nor rising.

Does a floating object displace a liquid volume equal to its own volume or its own weight

Its own weight, not its own volume, and this is the single most common slip on this topic. A floating body sinks in only until the weight of the liquid it has pushed aside equals its own weight; if it floated with less of itself underwater, the buoyant force would be too small to support it, and if it sank in more than needed, the buoyant force would exceed its weight and push it back up. A fully submerged sinking object, by contrast, does displace a volume of liquid equal to its own total volume, since all of it is underwater, but that displaced weight is still less than the object's own weight, which is precisely why it keeps sinking.

If ice is less dense than water, why does a full glass of ice water not overflow as the ice melts

Because the floating ice cube is already displacing a volume of water whose weight equals the ice cube's own weight, by the law of floatation. When the ice melts, it turns into a volume of liquid water with exactly that same weight, which by definition occupies exactly the volume of water that was already displaced. So the water level stays the same before and after melting. This logic is sometimes used to reassure people that melting sea ice, which is already floating, does not by itself raise sea levels the way melting land-based ice sheets do.

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