In common parlance, the word "weight" usually refers to mass. However, mass and weight are different quantities. The unit of mass is the kilogram, whereas the unit of weight is the newton. A weight of one kilogram is attracted to Earth by [[$ 9,8 $]] newton. Scales measure the weight of a piece and not its mass, although they display the result in mass units. Since the attraction is about the same all over the Earth's surface, a scale can be calibrated to display weights directly in kilograms.

The weight of a body only matters when the body is located in a gravitational field. Your weight on the Moon is one-sixth of your weight on Earth, because the Moon can only attract you with only one-sixth of the force that keeps you on the surface of the Earth. Whether you were on the Moon or Earth, however, your mass remains the same. You are made up of exactly the same ingredients in both places.

The mass of an object depends not only on the size of the object, but also on the material from which the object is made. The density of a material is denoted by the Greek letter [[$ \rho $]] (pronounced: "Roo").

The unit of density in the SI system is kg/m[[$ ^3 $]]​. The density of water is [[$ 1000 \;\text{kg/m}^3 $]]​ and the density of air at sea level is [[$ 1,29 \;\text {kg/m}^3 $]]​.

Population density

Population density can be used to describe how many people live in a certain area. Its unit is usually peoples per square kilometer or [[$ \text{p/km}^2 $]]​.

Example 1

From what substance is a sphere whose mass is [[$ 21,7 \;\text {kg} $]]​ and radius is [[$ 87 \:\text {mm} $]]​ made?

Solution:

If we calculate the density of the sphere, we can use a density table to find out the substance from which the sphere is made. The density table shows the densities of materials in [[$ \:\text {kg/dm}^3 $]]​. To avoid subsequent conversion of the units, it is advisable to place the radius of the sphere into the calculation formula in decimetres.

First, calculate the volume of the sphere:[[$ V = \displaystyle\frac {4} {3} \pi r^3 = \displaystyle\frac {4} {3} \pi (0,87 \;\text {dm})^3 ≈ 2 758 \;\text {dm}^3 $]]​

After this, the density of the sphere can be calculated:
[[$ \rho = \displaystyle\frac {m} {V} = \displaystyle\frac {21,7 \;\text {kg}} {2,758 \;\text {dm}^3} \approx 7,868 \displaystyle {\text {kg}} {\text {dm}^3} $]]​

The density of iron is [[$ 7,87 \;\text {kg/dm}^3 $]]​, which is very close to the density we received.

Answer: The sphere is made of iron.

Example 2

The roof of a detached house with an area of [[$ 150 \;\text{m}^2 $]]​, receives a [[$ 15 \;\text {cm} $]]​ thick snow cover. [[$ 1 \;\text{cm} $]]​ of snow is the same as [[$ 1 \;\text{mm} $]]​ of water when melted. What is the mass of snow on the roof?

Solution:

Snow can be thought of as forming a right angled polyhedron on the roof, the volume of which is obtained as follows:[[$ V = 150 \;\text{m}^2 \cdot 0.015\;\text{m} = 2,25 \;\text{m}^3 $]]​

The density of water is [[$ 1000 \;\text {kg/m}^3 $]]​. From the density formula, the mass can be solved with the formula:

[[$ m= \rho \cdot V = 1000 \displaystyle\frac {\text{kg}} {\text{m}^3} \cdot 2,25 \;\text {m}^3 = 2250 \;\text{kg} $]].​

Answer: The mass of the snow is 2300 kg.

Example 3

How many inhabitants does France have, when its population density is [[$ 107,2 \;\text {as/km}^2 $]]​ and its area is [[$ 543 965 \;\text{km}^2 $]]​?

Solution:

From the population density equation, we get the solution formula for the population: population = population density · area.

[[$ = 58313048 \displaystyle\frac {\;\text {as}} {\;\text {km}^2} \cdot 543965 \;\text{km}^2 = 58313048 \;\text{as} $]]​

Answer: [[$ 58,31 $]]​ million inhabitants.