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# Mass, weight and density

## Mass

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 Mass / $\text{kg}$ Hydrogen atom Apple Human Earth Universe $10^{-27}$ $0.1$ $70$ $10^{24}$ $10^{53}$
A simple balance used to measure mass.

## Weight

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 weight of the astronaut is lower on the Moon than on Earth.

## Density as mass over volume

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 Density ($\text{kg/m}^{3}$) Helium Air Water (liquid) Gold $0.18$ $1.29$ $1,000$ $19,300$
Gold has a very high density.

## Converting units of density

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 Value $\text{g/cm}^{3}$ $\text{kg/m}^{3}$ $\text{lbs/ft}^{3}$ $\text{kg/}L$ $1$ $1000$ $62.4$ $1$
Osmium is the densest stable element. It has a density of $22.6\text{ g/cm}^{3}$ or $22,600\text{ kg/m}^{3}$

## Measuring density

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.

The volume of a small irregular piece of plastic is measured. This allows the density to be calculated.