# Mass, weight and density

The mass $$(m)$$ of an object is a measure of the amount of substance contained within the object. It is a **scalar quantity.**

A bowling ball has a larger mass than a ping pong ball.

The sun has a larger mass than the Earth.

The **SI unit** of mass is the **kilogram $$(\text{kg})$$**.

The kilogram is defined so that a litre of water is approximately one kilogram.

The table shows the masses of some different items.

Object | Hydrogen atom | Apple | Human | Earth | Universe |
---|---|---|---|---|---|

Mass / $$\text{kg}$$ | $$10^{-27}$$ | $$0.1$$ | $$70$$ | $$10^{24}$$ | $$10^{53}$$ |

Weight $$(W)$$ is the **gravitational force** acting on a body of mass $$\Tblue{m}$$. \begin{gather*} \Tgreen{\text{weight}}=\Tblue{\text{mass}} \times \Tred{\text{gravitational field strength}} \\ \Tgreen{W}= \Tblue{m} \times \Tred{g} \end{gather*}

As weight is a **force**, its SI unit is the **newton** $$(N)$$.

**Mass** is an intrinsic property of an object and does not depend on where the object is.

**Weight** is the **force** on an object and depends on the gravitational field around the object.

The mass of a $$10 \text{ kg}$$ stone is the same on Earth and on Mars. But the weight is approximately $$98 \text{ N}$$ on Earth and $$37 \text{ N}$$ on Mars.

The density ($$\rho$$) of an object is its mass $$\Tred{(m)}$$ divided by its volume $$\Tblue{(V)}$$.$$$\rho = \frac{\Tred{m}}{\Tblue{V}}$$$

The symbol $$\rho$$ comes from the Greek alphabet and is called 'rho'. It should not be confused with $$p$$ from the Latin alphabet.

A $$1 \ucm^{3}$$ piece of lead is heavier than a $$1 \ucm^{3}$$ piece of plastic because lead is **denser** than plastic.

The SI unit of density is $$\text{kg/m}^{3}$$.

A person has a mass of $$55 \text{ kg}$$ and a volume of $$0.05 \text{ m}^{3}$$. Their density is $$$\rho = \frac{\Tred{55 \text{ kg}}} {\Tblue{0.05 \text{m}^{3}}} = 1100 \text{kg/m}^{3}$$$

The table below shows the density of some well-known materials at room temperature.Material | Helium | Air | Water (liquid) | Gold |
---|---|---|---|---|

Density ($$\text{kg/m}^{3}$$) | $$0.18$$ | $$1.29$$ | $$1,000$$ | $$19,300$$ |

The SI unit of density is $$\text{kg/m}^{3}$$. Density can also be measured in any units of mass divided by any units of volume.

$$\text{g/cm}^{3}$$ or $$\text{kg/}L$$ are other commonly used units for density.

To convert between different units of density you have to convert the mass units and convert the volume units.

\begin{gather*}1\text{ kg} = \Tred{1000 \text{ g}}\\ 1 \text{ m}^{3} = 100^{3}\text{ cm}^{3} = \Tblue{1,\!000,\!000 \text{ cm}^{3}} \\ \rightarrow 1 \text{ kg/m}^{3} = \frac{\Tred{1000 \text{ g}}}{\Tblue{1,\!000,\!000 \text{ cm}^{3}} } = 0.001 \text{ g/cm}^{3} \end{gather*}

The table below shows the density of **liquid** water using some different units.

Units | $$\text{g/cm}^{3}$$ | $$\text{kg/m}^{3}$$ | $$\text{lbs/ft}^{3}$$ | $$\text{kg/}L$$ |
---|---|---|---|---|

Value | $$1$$ | $$1000$$ | $$62.4$$ | $$1$$ |

To find the ** density ** of an object you have to measure its ** mass ** and its ** volume **.

The mass of an object can be found using a balance or a set of scales.

The volume of an object can be found in two ways:

- If the object is a regular shape like a cuboid then its volume can be calculated with a formula such as $$\text{volume} = \text{height} \times \text{width} \times \text{length}$$
- If an object is not a regular shape then its volume can be measured by
**placing it in a liquid**. This method is called**displacement**.

Pour some water into a measuring container and record its **volume**. Place the object into the container with the water and record the **new volume**. **The difference** between these two values is the ** volume of the object**.