Contents

# Density|Definition & Meaning

## Definition

The **quantity of matter** packed inside a **given space** (length, plane or volume) is measured by using **density.** Its mathematically defined as the **mass per unit volume** and measured in SI units of **kilograms per cubic meter.**

The **mathematical definition** of density is very straight forward but the concept is very critical and important when we are dealing with **quantity** of **matter. **Density plays an **important roles** in many mathematical and scientific **applications.** It is the property of the material. This means that **every material** has a **fixed density** depending upon its internal structure. For example the density of **iron** and **aluminum** is **7800** and **2700** kilogram per cubic meters respectively.

Given the density information and the dimensions (or volume) of the material, we can easily find the **mass** by using the **definition of density.** This is explained in detail later in this article. This article covers a comprehensive introduction to the concept of **density** and its **mathematical concepts.** It also introduces some examples.Â

## Explanation of Density

Lets **consider** an **example** as shown in the figure below.

**Figure 1: Unit Cube Example of Density**

The above figure shows **two boxes.** Each box has fixed dimensions of **1 meter x 1 meter x 1 meter**. In other words its a **unit cube.** The **spheres** inside the box **represent** one **kilogram** of matter each.

Now from mathematical definition, in the **left box,** there are **four spheres** packed in a **unit cube** which means that there are four kilograms packed in a unit cube. We can say that the density of this cube is **four kilogram per cubic meter**.

In the **right box,** there are **eight spheres** packed in a **unit cube** which means that there are eight kilograms packed in a unit cube. We can say that the density of this cube is **eight kilogram per cubic meter.**

## Mathematical Form of Density

The above example was very intuitive and limited to unit cube for developing some **basic understanding.** This section introduces the concept of density in a more **mathematically formal way.**

Lets say that you are given with a piece of **volume ‘V’** of some material that has **density ‘d’**. If **‘m’ kilograms** of the matter are packed inside the given volume ‘V’, then the **relationship** between these quantity is formulated as follows:

d = $\mathsf{\dfrac{ m }{ V }}$

In SI units, the density ‘d’ is measured in **‘kilogram per cubic meter’**, ‘m’ is measured in **‘kilogram’** and ‘V’ is measured in **‘cubic meter’**.

The above formula can also be described as, “**The mass of material per unit volume is called density**“. This formula can be used in solving the numerical problems involving density.

Lets consider the similar example of figure 1 with a different **arbitrary volume** of the cube as shown in the figure 2.

**Figure 2: Example of Density with Arbitrary Cube Size**

In the **left box,** the density can be using the above formula. Here, the **volume is 2 meter x 5 meter x 1 meter** which is equal to **10 cubic meters**. As stated earlier, there are **four kilograms** of matter packed inside this **10 cubic meters** volume. Mathematically:

d = $\mathsf{\dfrac{ m }{ V }}$ = $\mathsf{\dfrac{ 4 }{ 10 }}$ = 0.4 kg/m^{3}

In the **right box,** the volume is **3 meter x 0.5 meter x 10 meter** which is equal to **15 cubic meters**. As stated earlier, there are **eight kilograms** of matter packed inside this **15 cubic meters** volume. Mathematically:

d = $\mathsf{\dfrac{ m }{ V }}$ = $\mathsf{\dfrac{ 8 }{ 15 }}$ = 0.533 kg/m^{3}

## Examples of Density

The concept of density can be extended to many **real life situations.** Consider the case of **population density** for example. We have often heard that a particular area is **densely populated** while the other is **sparsely populated.** This is dependent on the **value of the density** (relative density to be specific).

Following figure illustrates this point. **Two areas** are shown in the figure. The **left** one is **sparsely populated** while the one on the **right** is **densely populated.** This decision is made on the basis of the density. The density of right one is higher than the left one since it contains **more people** in the **same space** compared to the left park.

**Figure 3: Example of Population Density**

## Density as a Measure of Heaviness

The density of a body can be used as a **measure** of **heaviness.** Lets say that someone asks you to pick up a **two blocks** of same size. One of the block is made of some **heavy metal** say **iron** while the other is made of some **light metal** say **aluminum.**

**Figure 4: Example of Density as a Measure of Heaviness**

Now we know from our experience that the **iron block will be heavy** compared to the aluminum one. However, the mathematical explanation may be found in the **density.** The density of iron is **7800 kilogram per cubic meter** while that of aluminum is **2700 kilogram per cubic meter.**

Since the **density of iron is higher,** it will have more matter in the same size volume. Hence, the iron block is heavier.

## Numerical Problems of Density

### Example 1

An **iron block** has a **width of 2 meters**, **height of 1 meters** and **depth of 0.5 meters**. If the mass of the **7800 kilograms**, find the value of its **density**.

### Solution

**Given that:**

m = 7800 kilogram

V = 2 meters x 1 meter x 0.5 meters = 1 cubic meter

**Recall:**

d = $\mathsf{\dfrac{ m }{ V }}$

**Substituting values:**

d = $\mathsf{\dfrac{ 7800 \textsf{ kilogram} }{ 1 \textsf{ cubic meter} }}$

d = 7800 kilogram per cubic meter

### Example 2

An **aluminum block** has a **width of 4 meters, height of 0.5 meters and depth of 0.25 meters**. If the density of the aluminum is **2700 kilograms per cubic meters**, find the value of its **mass.**

### Solution

**Given that:**

d = 2700 kilogram per cubic meter

V = 4 meters x 0.5 meters x 0.25 meters = 0.5 cubic meter

**Recall:**

d = $\mathsf{\dfrac{ m }{ V }}$

**Substituting values:**

2700 kilogram per cubic meter = $\mathsf{\dfrac{ m }{ 0.5 \textsf{ cubic meter} }}$

**Rearranging:**

m = (2700 kilogram per cubic meter) x (0.5 cubic meter)

m = 1350 kilogram

*All images/mathematical drawings were created with GeoGebra.*