Floating and sinking

Drop a pebble into a bucket of water and it sinks straight to the bottom. Drop in a piece of wood of roughly the same size and it floats, bobbing on the surface. Size alone cannot explain the difference — a large log floats just as easily as a small twig, and a tiny pebble sinks just as surely as a big rock. The real answer lies in comparing how tightly mass is packed into a given volume, a quantity called density, for the object and for the liquid it sits in. This article builds a quantitative feel for that comparison — using the everyday formula density equals mass divided by volume — while also showing, through the classic steel-ship puzzle, that density is never the whole story: shape matters too.

The core insight

An object floats in a liquid when its overall density (mass packed into its volume, including any hollow or trapped-air space) is less than the liquid's density, and sinks when it is greater, which is why solid steel sinks but a steel ship, shaped hollow, floats.

The core idea

Density is simply how much mass is squeezed into a given volume. For any object or liquid,

\rho = \frac{m}{V}

where: $\rho$ (the Greek letter rho) is density, usually in grams per cubic centimetre (g/cm³) or kilograms per cubic metre (kg/m³); $m$ is mass; and $V$ is volume. Water has a density of about $1\ \text{g/cm}^3$ , and this is the everyday benchmark against which most floating-and-sinking questions are judged. The floating rule follows directly from comparing two densities:

If the object's density is less than the liquid's density, the object floats.
If the object's density is greater than the liquid's density, the object sinks.
If the two densities are (nearly) equal, the object hovers, neither rising nor sinking noticeably.

Wood (about $0.6\ \text{g/cm}^3$ ) is less dense than water, so it floats; a rock or pebble (typically $2$ to $3\ \text{g/cm}^3$ ) is denser than water, so it sinks. This matches the qualitative rule you may already know — "lighter objects float, heavier ones sink" — except that "lighter" and "heavier" must always be read as density, not total mass. A huge log has far more mass than a small pebble, yet the log floats and the pebble sinks, because what matters is mass per unit volume, not mass alone. But density of the solid material is not the only factor. Shape changes the story. A block of steel (density about $7.8\ \text{g/cm}^3$ ) sinks in water every time. Yet ships built from that very same steel float, carrying tonnes of cargo. The trick is that a ship's hull is hollow — spread wide and enclosing a large volume of air. When you ask "what is the overall density of the ship," you must divide the ship's total mass by its total volume, including all that enclosed air, not just the volume of the steel plates themselves. That overall (or effective) density can easily drop below $1\ \text{g/cm}^3$ , even though the steel itself never changes.

Figure

Two identical beakers of water one holding a compact solid block that sinks and the other holding the same mass reshaped into a wide hollow hull that floats

Derivation AComparing an object's density with the liquid's density to predict floating or sinking

The goal of this derivation is to show, step by step, exactly how the single number called density is compared with the density of a liquid to decide whether an object floats or sinks, and why reshaping an object without changing its mass can flip the answer from sinking to floating by changing the volume used in that comparison.

Step 1. Write density as mass divided by volume for both the object and the liquid: $\rho_{\text{object}} = m_{\text{object}}/V_{\text{object}}$ and $\rho_{\text{liquid}} = m_{\text{liquid}}/V_{\text{liquid}}$ . Since we only ever compare the two densities (not the two masses or the two volumes directly), the sizes of the object and the sample of liquid do not need to match — density is a property of the material, independent of how much of it you have.

Step 2. State the comparison rule: the object floats if $\rho_{\text{object}} < \rho_{\text{liquid}}$ , sinks if $\rho_{\text{object}} > \rho_{\text{liquid}}$ , and hovers in near-neutral balance if $\rho_{\text{object}} \approx \rho_{\text{liquid}}$ . For water, $\rho_{\text{liquid}} \approx 1\ \text{g/cm}^3$ , so this becomes a simple test: is the object's density above or below $1\ \text{g/cm}^3$ ?

Step 3. Recognise that $V_{\text{object}}$ in Step 1 must be the total volume the object occupies, including any hollow or air-filled space, not just the volume of solid material in it. This is the step students most often skip. A solid steel block has $V_{\text{object}}$ equal to the volume of steel alone, giving a high density (about $7.8\ \text{g/cm}^3$ ) that sinks. Reshape the same mass of steel into a wide, hollow hull, and $V_{\text{object}}$ now includes the large volume of air trapped inside — the mass in the numerator has not changed, but the volume in the denominator has grown substantially.

Step 4. Conclude that a larger $V_{\text{object}}$ for the same $m_{\text{object}}$ makes $\rho_{\text{object}} = m_{\text{object}}/V_{\text{object}}$ smaller. If the hull's shape spreads the mass over enough enclosed volume, the overall density drops below the liquid's density, and the same material that sank as a solid block now floats as a hull. This is the full explanation, in density language, of why shape — not just the density of the raw material — decides floating and sinking.

Worked example

Question: A piece of wood has a mass of $60\ \text{g}$ and a volume of $100\ \text{cm}^3$ . A pebble of the same volume, $100\ \text{cm}^3$ , has a mass of $250\ \text{g}$ . Both are placed in a bucket of water, whose density is $1\ \text{g/cm}^3$ . Which floats and which sinks?

Step 1. Find the density of the wood: $\rho_{\text{wood}} = m/V = 60/100 = 0.6\ \text{g/cm}^3$ .

Step 2. Find the density of the pebble: $\rho_{\text{pebble}} = m/V = 250/100 = 2.5\ \text{g/cm}^3$ .

Step 3. Compare each with water's density of $1\ \text{g/cm}^3$ . The wood's density ( $0.6$ ) is less than water's, so the wood floats. The pebble's density ( $2.5$ ) is greater than water's, so the pebble sinks.

Answer: the wood ( $0.6\ \text{g/cm}^3$ ) floats and the pebble ( $2.5\ \text{g/cm}^3$ ) sinks, matching the general density rule and confirming the qualitative result you would observe by simply dropping both into the bucket.

Competitive stretch

Question: A block of steel has a mass of $2000\ \text{g}$ and occupies $250\ \text{cm}^3$ as a solid cube — it sinks. The same $2000\ \text{g}$ of steel is reshaped into a hollow hull that now encloses a total volume (steel plus trapped air) of $3000\ \text{cm}^3$ . Does the hull float in water ( $\rho = 1\ \text{g/cm}^3$ )?

Step 1. Confirm the solid cube sinks: $\rho_{\text{cube}} = 2000/250 = 8.0\ \text{g/cm}^3$ , far greater than water's $1\ \text{g/cm}^3$ , so it sinks — consistent with the known fact that solid steel sinks.

Step 2. Compute the hull's overall density using its total enclosed volume: $\rho_{\text{hull}} = 2000/3000 = 0.67\ \text{g/cm}^3$ .

Step 3. Compare with water: $0.67 < 1$ , so the hull floats. Answer: the reshaped hull floats, with an overall density of about $0.67\ \text{g/cm}^3$ , even though it is made of the exact same $2000\ \text{g}$ of steel that sank as a solid cube — the only thing that changed was the volume over which that mass is spread.

Common traps

Comparing an object's mass with the liquid's mass or volume, instead of comparing densities. A huge log has more mass than a small pebble, but it is the density comparison, not the mass, that predicts floating.
Computing an object's density using only the volume of solid material, and forgetting to include hollow or air-filled space when the object has been shaped (like a hull or a boat), which silently gives the wrong (too high) density.
Assuming a material's density is fixed no matter what — density of the raw material (steel is always about $7.8\ \text{g/cm}^3$ ) is different from the overall density of a shaped object (a hollow steel hull can be under $1\ \text{g/cm}^3$ ).
Forgetting the comparison is always with the specific liquid used — an object can float in water but sink in a less dense liquid, or the reverse, since the rule depends on both densities, not the object alone.

Ask the tutor: Want to work through why a coin sinks but a foil boat of the same metal floats, or try more density-comparison problems?

Why does the same amount of steel sink as a block but float as a ship?

Because floating depends on overall density, not on the density of the raw material alone. As a solid block, the steel's volume is just the steel itself, giving a high density of about $7.8\ \text{g/cm}^3$ that sinks in water. Shaped into a hollow hull, the same mass of steel now spreads over a much larger total volume, since the enclosed air counts too. Dividing the same mass by that larger volume gives a much smaller overall density, often well under $1\ \text{g/cm}^3$ , so the hull floats.

Is density the only thing that decides floating and sinking?

Not quite, and this chapter is careful about it. Density of the raw material is the main clue, but the shape of the object also matters, because shape changes the total volume over which the mass is spread — and it is that overall density, mass divided by total volume including any hollow space, that actually decides floating or sinking. Two objects made of the same material can behave differently in water if one is solid and the other is hollow.

How can I quickly check whether something will float in water without a formula?

If you can estimate or measure both mass and volume, compute density as mass divided by volume and compare it with water's density of $1\ \text{g/cm}^3$ : below $1$ it floats, above $1$ it sinks. Without measurements, a rough mental check works too — solid wood, ice, and most plastics are less dense than water and float; solid metals, rocks, and glass are denser and sink, unless shaped hollow (like a ship or a bowl) so that trapped air lowers their overall density.

Floating and sinking

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads