# Density and buoyancy

Canonical: https://duckyhelper.com/learn/physics/density-and-buoyancy/
Updated: 2026-10-01

Density is mass per unit volume, \(\rho = m/V\), usually in g/cm³ or kg/m³. Water's density is about 1.00 g/cm³. An object floats if its average density is less than the fluid's and sinks if it is greater. Archimedes' principle says the upward buoyant force on an object equals the weight of the fluid it pushes aside: \(F_b = \rho_{\text{fluid}} V_{\text{displaced}}\, g\).

## Key ideas

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

- \(\rho\) (rho) is density, in kg/m³ or g/cm³. \(1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3\).
- \(m\) is mass (kg or g) and \(V\) is volume (m³ or cm³). 1 mL = 1 cm³ and 1 L = 1000 cm³.

$$
F_b = \rho_{\text{fluid}}\, V_{\text{displaced}}\, g
$$

- \(F_b\) is the buoyant force, in newtons (N), pointing up.
- \(\rho_{\text{fluid}}\) is the density of the fluid, in kg/m³ (water: 1000 kg/m³).
- \(V_{\text{displaced}}\) is the volume of fluid pushed aside, in m³. For a fully submerged object, it is the object's whole volume.
- \(g = 9.8\ \text{m/s}^2\).

A floating object sinks until it displaces its own weight of fluid. That gives a quick rule: the fraction of a floating object under the surface equals its density divided by the fluid's density.

**Densities of some materials (OpenStax College Physics 2e, Table 11.1)**

| Material | Density (g/cm³) | In water |
| --- | --- | --- |
| Air (0 °C) | 0.00129 | Rises |
| Ice | 0.917 | Floats |
| Water (4 °C) | 1.000 | Neutral |
| Seawater | 1.025 | Neutral (in seawater) |
| Aluminum | 2.7 | Sinks |
| Iron or steel | 7.8 | Sinks |
| Gold | 19.32 | Sinks |

## Worked examples

**Example 1: identify a metal by its density**

Problem: A metal block has a mass of 54.0 g and a volume of 20.0 cm³. What is its density, and what metal could it be?

1. Divide mass by volume.

   $$
   \rho = \frac{54.0\ \text{g}}{20.0\ \text{cm}^3} = 2.70\ \text{g/cm}^3
   $$
2. Compare with the table: aluminum is 2.7 g/cm³.

Answer: 2.70 g/cm³, which matches aluminum

**Example 2: buoyant force on a sunken rock**

Problem: A rock with a volume of 0.0030 m³ is fully under water. What buoyant force acts on it?

1. Fully submerged, so the displaced volume is the rock's volume.

   $$
   F_b = (1000\ \text{kg/m}^3)(0.0030\ \text{m}^3)(9.8\ \text{m/s}^2)
   $$
2. Multiply.

   $$
   F_b = 29.4\ \text{N}
   $$

Answer: 29 N upward

**Example 3: apparent weight**

Problem: That rock has a mass of 8.0 kg. How much does it seem to weigh while hanging from a spring scale under water?

1. Its real weight.

   $$
   W = mg = (8.0\ \text{kg})(9.8\ \text{m/s}^2) = 78.4\ \text{N}
   $$
2. The buoyant force from Example 2 pushes up, so the scale reads less.

   $$
   W_{\text{apparent}} = 78.4\ \text{N} - 29.4\ \text{N} = 49.0\ \text{N}
   $$

Answer: 49 N

**Example 4: how much of a block floats under water**

Problem: A wooden block has a density of 600 kg/m³. What fraction of it is under water when it floats?

1. Floating means buoyant force equals weight.

   $$
   \rho_{\text{water}} V_{\text{sub}}\, g = \rho_{\text{wood}} V\, g
   $$
2. Solve for the fraction under water.

   $$
   \frac{V_{\text{sub}}}{V} = \frac{\rho_{\text{wood}}}{\rho_{\text{water}}} = \frac{600}{1000} = 0.60
   $$

Answer: 0.60, so 60% of the block is under water

## Common mistakes and how to fix them

- **Using the object's density in the buoyancy formula.** Fix: \(F_b\) uses the **fluid's** density and the volume of fluid pushed aside.
- **Thinking heavy things always sink.** A steel ship floats because its hollow shape gives it a low average density. Fix: compare densities, not weights.
- **Mixing g/cm³ and kg/m³.** Fix: multiply g/cm³ by 1000 to get kg/m³.
- **Using the whole volume for a floating object.** Fix: only the part under the surface displaces water.

**Practice problems**

1. A sample has a mass of 250 g and a volume of 100. cm³. What is its density?
   A. 0.400 g/cm³
   B. 2.50 g/cm³
   C. 25.0 g/cm³
   D. 350 g/cm³

   Answer: 2.50 g/cm³. \(\rho = m/V = 250 \div 100. = 2.50\) g/cm³.

2. Ice has a density of 0.917 g/cm³. Does it float in fresh water?
   A. Yes, it floats
   B. No, it sinks

   Answer: Yes, it floats. 0.917 g/cm³ is less than water's 1.00 g/cm³, so ice floats with most of its volume under the surface.

3. What fraction of an iceberg (0.917 g/cm³) floating in seawater (1.025 g/cm³) is under the surface?

   Answer: 0.895 (about 89%). Fraction under = \(\rho_{\text{ice}} / \rho_{\text{sea}} = 0.917 \div 1.025 = 0.895\). Only about a tenth shows above the water.

4. A 5.00 × 10⁻⁴ m³ object is fully under water. What is the buoyant force on it?

   Answer: 4.90 N. \(F_b = 1000 \times (5.00 \times 10^{-4}) \times 9.8 = 4.90\) N.

5. What is the mass of 2.0 L of water?
   A. 0.50 kg
   B. 2.0 kg
   C. 20. kg
   D. 2000 kg

   Answer: 2.0 kg. 2.0 L = 2000 cm³, and \(m = \rho V = 1.00 \times 2000 = 2000\) g = 2.0 kg.

## Frequently asked questions

### Why does a steel ship float?

A ship's hull is mostly air inside, so the ship's average density, steel plus air, is less than water's. It sinks until the water it pushes aside weighs as much as the whole ship. If the hull fills with water, its average density rises above water's and it sinks.

### Why is it easier to float in the ocean than in a pool?

Seawater is denser than fresh water because of the dissolved salt, about 1.025 g/cm³ instead of 1.00. A denser fluid gives more buoyant force for the same displaced volume, so you float a little higher.

### Does buoyancy happen in air too?

Yes. Air is a fluid, so it pushes up on everything with a force equal to the weight of air displaced. It is tiny for most objects, but a helium balloon rises because its total density is less than the air's.

## Sources

- [OpenStax College Physics 2e, 11.2 Density](https://openstax.org/books/college-physics-2e/pages/11-2-density), accessed 2026-10-01
- [OpenStax College Physics 2e, 11.7 Archimedes' Principle](https://openstax.org/books/college-physics-2e/pages/11-7-archimedes-principle), accessed 2026-10-01

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## Try asking Ducky

- "Check my buoyancy problem. Did I use the density of the water or the object?"
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